a ам>> from sympy import SeqFormula >>> from sympy.abc import n, m >>> SeqFormula(m*n**2, (n, 0, 5)).free_symbols {m} cs$h|]}|jаИjбD]}|ТqqSr&)┌free_symbols┌ differencer5)┌.0┌i┌jr0r&r'┌ bs z'SeqBase.free_symbols..й┌argsr0r&r0r'r6TszSeqBase.free_symbolscCs0||jks||jkr&td||jfГВ|а|бS)z#Returns the coefficient at point ptzIndex %s out of bounds %s)r r3┌ IndexErrorr)┌_eval_coeffйr,┌ptr&r&r'┌coeffesz SeqBase.coeffcCstd|jГВdS)NzhThe _eval_coeff method should be added to%s to return coefficient so it is availablewhen coeff calls it.)r!┌funcr@r&r&r'r?ls¤zSeqBase._eval_coeffcCs<|jtjur|j}n|j}|jtjur,d}nd}|||S)aпReturns the i'th point of a sequence. Explanation =========== If start point is negative infinity, point is returned from the end. Assumes the first point to be indexed zero. Examples ========= >>> from sympy import oo >>> from sympy.series.sequences import SeqPer bounded >>> SeqPer((1, 2, 3), (-10, 10))._ith_point(0) -10 >>> SeqPer((1, 2, 3), (-10, 10))._ith_point(5) -5 End is at infinity >>> SeqPer((1, 2, 3), (0, oo))._ith_point(5) 5 Starts at negative infinity >>> SeqPer((1, 2, 3), (-oo, 0))._ith_point(5) -5 щ щ)r r┌NegativeInfinityr3)r,r9┌initial┌stepr&r&r'┌ _ith_pointrs zSeqBase._ith_pointcCsdS)aI Should only be used internally. Explanation =========== self._add(other) returns a new, term-wise added sequence if self knows how to add with other, otherwise it returns ``None``. ``other`` should only be a sequence object. Used within :class:`SeqAdd` class. Nr&йr,r-r&r&r'┌_addЮszSeqBase._addcCsdS)aS Should only be used internally. Explanation =========== self._mul(other) returns a new, term-wise multiplied sequence if self knows how to multiply with other, otherwise it returns ``None``. ``other`` should only be a sequence object. Used within :class:`SeqMul` class. Nr&rJr&r&r'┌_mulоszSeqBase._mulcCs t||ГS)aЭ Should be used when ``other`` is not a sequence. Should be defined to define custom behaviour. Examples ======== >>> from sympy import SeqFormula >>> from sympy.abc import n >>> SeqFormula(n**2).coeff_mul(2) SeqFormula(2*n**2, (n, 0, oo)) Notes ===== '*' defines multiplication of sequences with sequences only. rrJr&r&r'┌ coeff_mul╛szSeqBase.coeff_mulcCs$t|tГstdt|ГГВt||ГS)a4Returns the term-wise addition of 'self' and 'other'. ``other`` should be a sequence. Examples ======== >>> from sympy import SeqFormula >>> from sympy.abc import n >>> SeqFormula(n**2) + SeqFormula(n**3) SeqFormula(n**3 + n**2, (n, 0, oo)) zcannot add sequence and %sй┌ isinstancer┌ TypeError┌type┌SeqAddrJr&r&r'┌__add__╥s zSeqBase.__add__rScCs||SйNr&rJr&r&r'┌__radd__уszSeqBase.__radd__cCs&t|tГstdt|ГГВt||ГS)a7Returns the term-wise subtraction of ``self`` and ``other``. ``other`` should be a sequence. Examples ======== >>> from sympy import SeqFormula >>> from sympy.abc import n >>> SeqFormula(n**2) - (SeqFormula(n)) SeqFormula(n**2 - n, (n, 0, oo)) zcannot subtract sequence and %srNrJr&r&r'┌__sub__чs zSeqBase.__sub__rVcCs ||SrTr&rJr&r&r'┌__rsub__°szSeqBase.__rsub__cCs |аdбS)z╙Negates the sequence. Examples ======== >>> from sympy import SeqFormula >>> from sympy.abc import n >>> -SeqFormula(n**2) SeqFormula(-n**2, (n, 0, oo)) rD)rMr0r&r&r'┌__neg__№szSeqBase.__neg__cCs$t|tГstdt|ГГВt||ГS)a{Returns the term-wise multiplication of 'self' and 'other'. ``other`` should be a sequence. For ``other`` not being a sequence see :func:`coeff_mul` method. Examples ======== >>> from sympy import SeqFormula >>> from sympy.abc import n >>> SeqFormula(n**2) * (SeqFormula(n)) SeqFormula(n**3, (n, 0, oo)) zcannot multiply sequence and %s)rOrrPrQ┌SeqMulrJr&r&r'┌__mul__ s zSeqBase.__mul__rZcCs||SrTr&rJr&r&r'┌__rmul__szSeqBase.__rmul__ccs*t|jГD]}|а|б}|а|бVq dSrT)┌ranger4rIrB)r,r9rAr&r&r'┌__iter__s zSeqBase.__iter__cstt|tГrИа|б}Иа|бSt|tГrp|j|j}}|durBd}|durPИj}ЗfddДt|||j phdГDГSdS)Nrcsg|]}ИаИа|ббСqSr&)rBrI)r8r9r0r&r'┌ .єz'SeqBase.__getitem__..rE) rO┌intrIrB┌slicer r3r4r\rH)r,┌indexr r3r&r0r'┌__getitem__$s zSeqBase.__getitem__NcCs8ddlm}ddД|d|ЕDГ}t|Г}|dur<|d}nt||dГ}g}td|dГD]ш} d| } g}t| ГD]}|а|||| Ебqt||Г} | абdkr\t| а||| | ЕГбГ}|| kr▐t |dddЕГ}РqFg}t| || ГD]}|а|||| ЕбqЁ||Г} | |||| dЕГkr\t |dddЕГ}РqFq\|duРrT|St|Г} | dkРrngdfS|| d|| dd|| d|| }}t| dГD]n}|||||7}t| |dГD]*}||||||||d8}Рq╘|||||d8}Рqм|tt |Гt |ГГfSdS) aг Finds the shortest linear recurrence that satisfies the first n terms of sequence of order `\leq` ``n/2`` if possible. If ``d`` is specified, find shortest linear recurrence of order `\leq` min(d, n/2) if possible. Returns list of coefficients ``[b(1), b(2), ...]`` corresponding to the recurrence relation ``x(n) = b(1)*x(n-1) + b(2)*x(n-2) + ...`` Returns ``[]`` if no recurrence is found. If gfvar is specified, also returns ordinary generating function as a function of gfvar. Examples ======== >>> from sympy import sequence, sqrt, oo, lucas >>> from sympy.abc import n, x, y >>> sequence(n**2).find_linear_recurrence(10, 2) [] >>> sequence(n**2).find_linear_recurrence(10) [3, -3, 1] >>> sequence(2**n).find_linear_recurrence(10) [2] >>> sequence(23*n**4+91*n**2).find_linear_recurrence(10) [5, -10, 10, -5, 1] >>> sequence(sqrt(5)*(((1 + sqrt(5))/2)**n - (-(1 + sqrt(5))/2)**(-n))/5).find_linear_recurrence(10) [1, 1] >>> sequence(x+y*(-2)**(-n), (n, 0, oo)).find_linear_recurrence(30) [1/2, 1/2] >>> sequence(3*5**n + 12).find_linear_recurrence(20,gfvar=x) ([6, -5], 3*(5 - 21*x)/((x - 1)*(5*x - 1))) >>> sequence(lucas(n)).find_linear_recurrence(15,gfvar=x) ([1, 1], (x - 2)/(x**2 + x - 1)) r)┌MatrixcSsg|]}tt|ГГСqSr&)rr)r8┌tr&r&r'r^Tr_z2SeqBase.find_linear_recurrence..NщrErD)Zsympy.matricesrd┌len┌minr\┌appendZdetrZLUsolverr)r,┌n┌dZgfvarrd┌xZlx┌rZcoeffs┌l┌l2Zmlist┌k┌m┌yr9r:r&r&r'┌find_linear_recurrence1sJ" 2(zSeqBase.find_linear_recurrence)NN)#┌__name__┌ __module__┌__qualname__┌__doc__Zis_commutativeZ_op_priority┌staticmethodr(r.┌propertyr1r)r r3r4r5r6rrBr?rIrKrLrMrSrrUrVrWrXrZr[r]rcrsr&r&r&r'rsP , rc@s8eZdZdZeddДГZeddДГZddДZdd ДZd S)┌ EmptySequencea╠Represents an empty sequence. The empty sequence is also available as a singleton as ``S.EmptySequence``. Examples ======== >>> from sympy import EmptySequence, SeqPer >>> from sympy.abc import x >>> EmptySequence EmptySequence >>> SeqPer((1, 2), (x, 0, 10)) + EmptySequence SeqPer((1, 2), (x, 0, 10)) >>> SeqPer((1, 2)) * EmptySequence EmptySequence >>> EmptySequence.coeff_mul(-1) EmptySequence cCstjSrT)r┌EmptySetr0r&r&r'r)СszEmptySequence.intervalcCstjSrT)rZZeror0r&r&r'r4ХszEmptySequence.lengthcCs|S)·"See docstring of SeqBase.coeff_mulr&)r,rBr&r&r'rMЩszEmptySequence.coeff_mulcCstgГSrT)┌iterr0r&r&r'r]ЭszEmptySequence.__iter__N) rtrurvrwryr)r4rMr]r&r&r&r'rz|s rz)┌ metaclassc@sXeZdZdZeddДГZeddДГZeddДГZedd ДГZed dДГZ edd ДГZ dS)┌SeqExpra┘Sequence expression class. Various sequences should inherit from this class. Examples ======== >>> from sympy.series.sequences import SeqExpr >>> from sympy.abc import x >>> from sympy import Tuple >>> s = SeqExpr(Tuple(1, 2, 3), Tuple(x, 0, 10)) >>> s.gen (1, 2, 3) >>> s.interval Interval(0, 10) >>> s.length 11 See Also ======== sympy.series.sequences.SeqPer sympy.series.sequences.SeqFormula cCs |jdSйNrr<r0r&r&r'r1╗szSeqExpr.gencCst|jdd|jddГS)NrErf)rr=r0r&r&r'r)┐szSeqExpr.intervalcCs|jjSrTйr)r*r0r&r&r'r ├sz SeqExpr.startcCs|jjSrTйr)r+r0r&r&r'r3╟szSeqExpr.stopcCs|j|jdSйNrEйr3r r0r&r&r'r4╦szSeqExpr.lengthcCs|jddfS)NrErr<r0r&r&r'r5╧szSeqExpr.variablesN)rtrurvrwryr1r)r r3r4r5r&r&r&r'rбs rc@sReZdZdZdddДZeddДГZeddДГZd d ДZddДZ d dДZ ddДZdS)┌SeqPeraт Represents a periodic sequence. The elements are repeated after a given period. Examples ======== >>> from sympy import SeqPer, oo >>> from sympy.abc import k >>> s = SeqPer((1, 2, 3), (0, 5)) >>> s.periodical (1, 2, 3) >>> s.period 3 For value at a particular point >>> s.coeff(3) 1 supports slicing >>> s[:] [1, 2, 3, 1, 2, 3] iterable >>> list(s) [1, 2, 3, 1, 2, 3] sequence starts from negative infinity >>> SeqPer((1, 2, 3), (-oo, 0))[0:6] [1, 2, 3, 1, 2, 3] Periodic formulas >>> SeqPer((k, k**2, k**3), (k, 0, oo))[0:6] [0, 1, 8, 3, 16, 125] See Also ======== sympy.series.sequences.SeqFormula NcCs$t|Г}ddД}d\}}}|dur8||Гdtj}}}t|tГrvt|ГdkrZ|\}}}nt|Гdkrv||Г}|\}}t|ttfГrФ|dusФ|durдt dt |ГГВ|tjur└|tjur└t dГВt|||fГ}t|tГrъttt |ГГГ}nt d |ГВt|d |dГtjuРrtjStа|||бS)NcSs(|j}t|jГdkr|абStdГSdS)NrErp)r6rg┌popr)┌ periodical┌freer&r&r'┌_find_xszSeqPer.__new__.._find_xйNNNrщrf·Invalid limits given: %sz/Both the start and end valuecannot be unboundedz6invalid period %s should be something like e.g (1, 2) rE)rrr$rrrgrOrrr#┌strrF┌tuplerrr{rzr┌__new__)┌clsrЗ┌limitsrЙrlr r3r&r&r'rПs0 zSeqPer.__new__cCs t|jГSrT)rgr1r0r&r&r'┌period-sz SeqPer.periodcCs|jSrTйr1r0r&r&r'rЗ1szSeqPer.periodicalcCsF|jtjur|j||j}n||j|j}|j|а|jd|бSrА)r rrFr3rТrЗ┌subsr5)r,rA┌idxr&r&r'r?5szSeqPer._eval_coeffc CsРt|tГrМ|j|j}}|j|j}}t||Г}g}t|ГD]*}|||} |||} |а| | бq<|а|б\}}t||jd||fГSdSйzSee docstring of SeqBase._addrNй rOrЕrЗrТrr\rir.r5й r,r-Zper1Zlper1Zper2Zlper2Z per_lengthZnew_perrlZele1Zele2r r3r&r&r'rK<s zSeqPer._addc CsРt|tГrМ|j|j}}|j|j}}t||Г}g}t|ГD]*}|||} |||} |а| | бq<|а|б\}}t||jd||fГSdSйzSee docstring of SeqBase._mulrNrЧrШr&r&r'rLMs zSeqPer._mulcs,tИГЙЗfddД|jDГ}t||jdГS)r|csg|]}|ИСqSr&r&йr8rlйrBr&r'r^ar_z$SeqPer.coeff_mul..rE)rrЗrЕr=)r,rBZperr&rЫr'rM^szSeqPer.coeff_mul)N)rtrurvrwrПryrТrЗr?rKrLrMr&r&r&r'rЕ╘s0 ( rЕc@sNeZdZdZdddДZeddДГZddДZd d ДZddДZ d dДZ ddДZdS)┌ SeqFormulaaf Represents sequence based on a formula. Elements are generated using a formula. Examples ======== >>> from sympy import SeqFormula, oo, Symbol >>> n = Symbol('n') >>> s = SeqFormula(n**2, (n, 0, 5)) >>> s.formula n**2 For value at a particular point >>> s.coeff(3) 9 supports slicing >>> s[:] [0, 1, 4, 9, 16, 25] iterable >>> list(s) [0, 1, 4, 9, 16, 25] sequence starts from negative infinity >>> SeqFormula(n**2, (-oo, 0))[0:6] [0, 1, 4, 9, 16, 25] See Also ======== sympy.series.sequences.SeqPer NcCs·t|Г}ddД}d\}}}|dur8||Гdtj}}}t|tГrvt|ГdkrZ|\}}}nt|Гdkrv||Г}|\}}t|ttfГrФ|dusФ|durдt dt |ГГВ|tjur└|tjur└t dГВt|||fГ}t|d |dГtj urьtjStа|||бS) NcSs6|j}t|Гdkr|абS|s&tdГStd|ГВdS)NrErpzж specify dummy variables for %s. If the formula contains more than one free symbol, a dummy variable should be supplied explicitly e.g., SeqFormula(m*n**2, (n, 0, 5)))r6rgrЖrr#)┌formularИr&r&r'rЙСs¤ z#SeqFormula.__new__.._find_xrКrrЛrfrМz0Both the start and end value cannot be unboundedrE)rrr$rrrgrOrrr#rНrFrr{rzrrП)rРrЭrСrЙrlr r3r&r&r'rПОs& zSeqFormula.__new__cCs|jSrTrУr0r&r&r'rЭ╡szSeqFormula.formulacCs|jd}|jа||бSrА)r5rЭrФ)r,rArkr&r&r'r?╣s zSeqFormula._eval_coeffc Cs`t|tГr\|j|jd}}|j|jd}}||а||б}|а|б\}}t||||fГSdSrЦйrOrЬrЭr5rФr.й r,r-Zform1┌v1Zform2┌v2rЭr r3r&r&r'rK╜s zSeqFormula._addc Cs`t|tГr\|j|jd}}|j|jd}}||а||б}|а|б\}}t||||fГSdSrЩrЮrЯr&r&r'rL╞s zSeqFormula._mulcCs"t|Г}|j|}t||jdГS)r|rE)rrЭrЬr=)r,rBrЭr&r&r'rM╧s zSeqFormula.coeff_mulcOs$tt|jg|вRi|дО|jdГSrГ)rЬrrЭr=)r,r=┌kwargsr&r&r'r╒szSeqFormula.expand)N)rtrurvrwrПryrЭr?rKrLrMrr&r&r&r'rЬes( ' rЬc@sЦeZdZdZdddДZeddДГZedd ДГZed dДГZedd ДГZ eddДГZ eddДГZeddДГZeddДГZ eddДГZddДZddДZdS)┌RecursiveSeqaМ A finite degree recursive sequence. Explanation =========== That is, a sequence a(n) that depends on a fixed, finite number of its previous values. The general form is a(n) = f(a(n - 1), a(n - 2), ..., a(n - d)) for some fixed, positive integer d, where f is some function defined by a SymPy expression. Parameters ========== recurrence : SymPy expression defining recurrence This is *not* an equality, only the expression that the nth term is equal to. For example, if :code:`a(n) = f(a(n - 1), ..., a(n - d))`, then the expression should be :code:`f(a(n - 1), ..., a(n - d))`. yn : applied undefined function Represents the nth term of the sequence as e.g. :code:`y(n)` where :code:`y` is an undefined function and `n` is the sequence index. n : symbolic argument The name of the variable that the recurrence is in, e.g., :code:`n` if the recurrence function is :code:`y(n)`. initial : iterable with length equal to the degree of the recurrence The initial values of the recurrence. start : start value of sequence (inclusive) Examples ======== >>> from sympy import Function, symbols >>> from sympy.series.sequences import RecursiveSeq >>> y = Function("y") >>> n = symbols("n") >>> fib = RecursiveSeq(y(n - 1) + y(n - 2), y(n), n, [0, 1]) >>> fib.coeff(3) # Value at a particular point 2 >>> fib[:6] # supports slicing [0, 1, 1, 2, 3, 5] >>> fib.recurrence # inspect recurrence Eq(y(n), y(n - 2) + y(n - 1)) >>> fib.degree # automatically determine degree 2 >>> for x in zip(range(10), fib): # supports iteration ... print(x) (0, 0) (1, 1) (2, 1) (3, 2) (4, 3) (5, 5) (6, 8) (7, 13) (8, 21) (9, 34) See Also ======== sympy.series.sequences.SeqFormula Nrcs^t|tГstdа|бГВt|tГr(|js6tdа|бГВ|j|fkrJtdГВ|jЙtd|fdН}d}|а Иб}|D]f} t | jГdkrКtdГВ| jdа||б|} | абr╕| j r╕| dks╞td а| бГВ| |krp| }qp|sюd dДt|ГDГ}t |Г|kРrtdГВt|Г}tИГЙtd dД|DГО}tа|||||Иб}ЗЗfddДt|ГDГ|_||_|S)NzErecurrence sequence must be an applied undefined function, found `{}`z0recurrence variable must be a symbol, found `{}`z)recurrence sequence does not match symbolrp)┌excluderrEz)Recurrence should be in a single variablezDRecurrence should have constant, negative, integer shifts (found {})cSsg|]}tdа|бГСqS)zc_{})r┌format)r8rpr&r&r'r^Ir_z(RecursiveSeq.__new__..z)Number of initial terms must equal degreecss|]}t|ГVqdSrTrrЪr&r&r'┌ Qr_z'RecursiveSeq.__new__..csi|]\}}ИИ|Г|УqSr&r&)r8rp┌initйr rrr&r'┌ Ur_z(RecursiveSeq.__new__..)rOrrPrеrZ is_symbolr=rCr┌findrg┌match┌is_constant┌ is_integerr\r#r rrrП┌ enumerate┌cache┌degree)rР┌ recurrence┌ynrjrGr rpr░Zprev_ysZprev_y┌shift┌seqr&rиr'rП%sF ■ zRecursiveSeq.__new__cCs |jdSйzEquation defining recurrence.rr<r0r&r&r'┌_recurrenceZszRecursiveSeq._recurrencecCst|j|jdГSr╡)rr▓r=r0r&r&r'r▒_szRecursiveSeq.recurrencecCs |jdS)z*Applied function representing the nth termrEr<r0r&r&r'r▓dszRecursiveSeq.yncCs|jjS)z3Undefined function for the nth term of the sequence)r▓rCr0r&r&r'rriszRecursiveSeq.ycCs |jdS)zSequence index symbolrfr<r0r&r&r'rjnszRecursiveSeq.ncCs |jdS)z"The initial values of the sequencerЛr<r0r&r&r'rGsszRecursiveSeq.initialcCs |jdS)r2щr<r0r&r&r'r xszRecursiveSeq.startcCstjS)z&The ending point of the sequence. (oo))rr$r0r&r&r'r3}szRecursiveSeq.stopcCs|jtjfS)z&Interval on which sequence is defined.)r rr$r0r&r&r'r)ВszRecursiveSeq.intervalcCsМ||jt|jГkr$|j|а|бStt|jГ|dГD]<}|j|}|jа|j|iб}|а|jб}||j|а|б<q8|j|а|j|бSrГ)r rgrпrrr\r╢Zxreplacerj)r,rb┌currentZ seq_indexZcurrent_recurrenceZnew_termr&r&r'r?Зs zRecursiveSeq._eval_coeffccs |j}|а|бV|d7}qdSrГ)r r?)r,rbr&r&r'r]ЦszRecursiveSeq.__iter__)Nr)rtrurvrwrПryr╢r▒r▓rrrjrGr r3r)r?r]r&r&r&r'rг╪s,L 5 rгNcCs*t|Г}t|tГrt||ГSt||ГSdS)a Returns appropriate sequence object. Explanation =========== If ``seq`` is a SymPy sequence, returns :class:`SeqPer` object otherwise returns :class:`SeqFormula` object. Examples ======== >>> from sympy import sequence >>> from sympy.abc import n >>> sequence(n**2, (n, 0, 5)) SeqFormula(n**2, (n, 0, 5)) >>> sequence((1, 2, 3), (n, 0, 5)) SeqPer((1, 2, 3), (n, 0, 5)) See Also ======== sympy.series.sequences.SeqPer sympy.series.sequences.SeqFormula N)rrrrЕrЬ)r┤rСr&r&r'┌sequenceЭs r╣c@sXeZdZdZeddДГZeddДГZeddДГZedd ДГZed dДГZ edd ДГZ dS)┌ SeqExprOpaс Base class for operations on sequences. Examples ======== >>> from sympy.series.sequences import SeqExprOp, sequence >>> from sympy.abc import n >>> s1 = sequence(n**2, (n, 0, 10)) >>> s2 = sequence((1, 2, 3), (n, 5, 10)) >>> s = SeqExprOp(s1, s2) >>> s.gen (n**2, (1, 2, 3)) >>> s.interval Interval(5, 10) >>> s.length 6 See Also ======== sympy.series.sequences.SeqAdd sympy.series.sequences.SeqMul cCstddД|jDГГS)zjGenerator for the sequence. returns a tuple of generators of all the argument sequences. css|]}|jVqdSrTrУйr8┌ar&r&r'rжуr_z SeqExprOp.gen..)rОr=r0r&r&r'r1▌sz SeqExprOp.gencCstddД|jDГОS)zeSequence is defined on the intersection of all the intervals of respective sequences css|]}|jVqdSrTйr)r╗r&r&r'rжъr_z%SeqExprOp.interval..)rr=r0r&r&r'r)хszSeqExprOp.intervalcCs|jjSrTrБr0r&r&r'r ьszSeqExprOp.startcCs|jjSrTrВr0r&r&r'r3ЁszSeqExprOp.stopcCsttddД|jDГГГS)z%Cumulative of all the bound variablescSsg|] }|jСqSr&)r5r╗r&r&r'r^ўr_z'SeqExprOp.variables..)rОrr=r0r&r&r'r5ЇszSeqExprOp.variablescCs|j|jdSrГrДr0r&r&r'r4∙szSeqExprOp.lengthN)rtrurvrwryr1r)r r3r5r4r&r&r&r'r║─s r║c@s,eZdZdZddДZeddДГZddДZdS) rRaЪRepresents term-wise addition of sequences. Rules: * The interval on which sequence is defined is the intersection of respective intervals of sequences. * Anything + :class:`EmptySequence` remains unchanged. * Other rules are defined in ``_add`` methods of sequence classes. Examples ======== >>> from sympy import EmptySequence, oo, SeqAdd, SeqPer, SeqFormula >>> from sympy.abc import n >>> SeqAdd(SeqPer((1, 2), (n, 0, oo)), EmptySequence) SeqPer((1, 2), (n, 0, oo)) >>> SeqAdd(SeqPer((1, 2), (n, 0, 5)), SeqPer((1, 2), (n, 6, 10))) EmptySequence >>> SeqAdd(SeqPer((1, 2), (n, 0, oo)), SeqFormula(n**2, (n, 0, oo))) SeqAdd(SeqFormula(n**2, (n, 0, oo)), SeqPer((1, 2), (n, 0, oo))) >>> SeqAdd(SeqFormula(n**3), SeqFormula(n**2)) SeqFormula(n**3 + n**2, (n, 0, oo)) See Also ======== sympy.series.sequences.SeqMul csР|аdtjб}t|Г}ЗfddДЙИ|Г}ddД|DГ}|sBtjStddД|DГОtjur`tjS|rntа |бStt |tjГГ}t j|g|вRОS)N┌evaluatecsPt|tГr,t|tГr&ttИ|jГgГS|gSt|ГrDttИ|ГgГStdГВdSйNz2Input must be Sequences or iterables of Sequences)rOrrR┌sum┌mapr=rrPй┌argй┌_flattenr&r'r┼"s z SeqAdd.__new__.._flattencSsg|]}|tjur|СqSr&)rrzr╗r&r&r'r^.r_z"SeqAdd.__new__..css|]}|jVqdSrTr╜r╗r&r&r'rж4r_z!SeqAdd.__new__..)┌getrr╛┌listrrzrr{rR┌reducerrr(rrПйrРr=rвr╛r&r─r'rПs zSeqAdd.__new__csаd}|r|t|ГD]h\}Йd}t|ГD]F\}Й||kr6q$ИаИб}|dur$ЗЗfddД|DГ}|а|бqlq$|r|}qqqt|ГdkrР|абSt|ddНSdS)aSimplify :class:`SeqAdd` using known rules. Iterates through all pairs and ask the constituent sequences if they can simplify themselves with any other constituent. Notes ===== adapted from ``Union.reduce`` TFNcsg|]}|ИИfvr|СqSr&r&r╗й┌srer&r'r^Wr_z!SeqAdd.reduce..rEйr╛)rоrKrirgrЖrRйr=┌new_argsZid1Zid2Znew_seqr&r╩r'r╚?s$ z SeqAdd.reducecstЗfddД|jDГГS)z9adds up the coefficients of all the sequences at point ptc3s|]}|аИбVqdSrTrЫr╗йrAr&r'rжer_z%SeqAdd._eval_coeff..)r└r=r@r&r╧r'r?cszSeqAdd._eval_coeffNйrtrurvrwrПrxr╚r?r&r&r&r'rR■s $ #rRc@s,eZdZdZddДZeddДГZddДZdS) rYa'Represents term-wise multiplication of sequences. Explanation =========== Handles multiplication of sequences only. For multiplication with other objects see :func:`SeqBase.coeff_mul`. Rules: * The interval on which sequence is defined is the intersection of respective intervals of sequences. * Anything \* :class:`EmptySequence` returns :class:`EmptySequence`. * Other rules are defined in ``_mul`` methods of sequence classes. Examples ======== >>> from sympy import EmptySequence, oo, SeqMul, SeqPer, SeqFormula >>> from sympy.abc import n >>> SeqMul(SeqPer((1, 2), (n, 0, oo)), EmptySequence) EmptySequence >>> SeqMul(SeqPer((1, 2), (n, 0, 5)), SeqPer((1, 2), (n, 6, 10))) EmptySequence >>> SeqMul(SeqPer((1, 2), (n, 0, oo)), SeqFormula(n**2)) SeqMul(SeqFormula(n**2, (n, 0, oo)), SeqPer((1, 2), (n, 0, oo))) >>> SeqMul(SeqFormula(n**3), SeqFormula(n**2)) SeqFormula(n**5, (n, 0, oo)) See Also ======== sympy.series.sequences.SeqAdd csВ|аdtjб}t|Г}ЗfddДЙИ|Г}|s4tjStddД|DГОtjurRtjS|r`tа |бStt |tjГГ}t j|g|вRОS)Nr╛csRt|tГr.t|tГr&ttИ|jГgГS|gSnt|ГrFttИ|ГgГStdГВdSr┐)rOrrYr└r┴r=rrPr┬r─r&r'r┼Тs z SeqMul.__new__.._flattencss|]}|jVqdSrTr╜r╗r&r&r'rжвr_z!SeqMul.__new__..)r╞rr╛r╟rrzrr{rYr╚rrr(rrПr╔r&r─r'rПЛs zSeqMul.__new__csаd}|r|t|ГD]h\}Йd}t|ГD]F\}Й||kr6q$ИаИб}|dur$ЗЗfddД|DГ}|а|бqlq$|r|}qqqt|ГdkrР|абSt|ddНSdS)a.Simplify a :class:`SeqMul` using known rules. Explanation =========== Iterates through all pairs and ask the constituent sequences if they can simplify themselves with any other constituent. Notes ===== adapted from ``Union.reduce`` TFNcsg|]}|ИИfvr|СqSr&r&r╗r╩r&r'r^╚r_z!SeqMul.reduce..rEr╠)rоrLrirgrЖrYr═r&r╩r'r╚нs$ z SeqMul.reducecCs"d}|jD]}||а|б9}q |S)zs@c%3sF ':j