
    \h                     J    S SK Jr  S SKJr  S SKJrJr  S SKJr  S SK	J
r
  S rg)    )Rational)	fibonacci)Ssymbols)raiseslinrecc            	      v  ^^^^ [        SS/SS/SS9S:X  d   e[        / SQ/ SQSS9S:X  d   e[        / S	QS
S/SS9S:X  d   e[        / SQ/ SQSS9S:X  d   e[        / SQ/ SQSS9S:X  d   e[        S/S-  / SQ-   S/S-  / SQ-   SS9S:X  d   e[        / SQ/ SQSS9(       d   e[        / SQ/ SQSS9(       d   e[        S [        S S!5       5       5      (       d   e[        S" [        S#S$5       5       5      (       d   e[        R                  [        S%S&5      [        S'S(5      S)[        S*S+5      [        S%S'5      /mSS,S*[        S'S)5      [        S%S)5      [        S,S+5      S(/m[        S-5      u  mmn [        TS S' TS S& S.S9[        S/S05      :X  d   e[        TS S& TS S& SS9[        S1S25      :X  d   e[        TS%S  TS S% S3S9[        S4S55      :X  d   e[        S/S6-  [        S,S%5      [        S&S'5      /-   TS7S9[        S8S95      :X  d   e[        [        S: 5        [        [        UU4S; j5        [        [        UU4S< j5        [        [        UU4S= j5        [        [        UU4S> j5        [        TTU // S?QS&S9TS,-  TT-  -   TU -  -   T-   U -   :X  d   e[        / S@QTTU /SS9SAT-  SBT-  -   SCU -  -   :X  d   e[        / SDQTT/SES9SFT-  SGT-  -   :X  d   e[        S/S-  / SQ-   TTU /SHS9SIT-  SJT-  -   SKU -  -   :X  d   e[        / SS/SS9S:X  d   e[        TTU // SQS,S9S%:X  d   eg )LN      coeffsinitni*  r               )r   r   r   r   
   i  )r   r                  l   X;GD7 )r   r   r   r   r   )r   r   r         r   r   r      )r   r   r   r   F   l   Y!Hc"6)`ToNR0s(Wm"j  r   7   )r   r   r   r   2   )r   r   r   i  l   wN;ELM$xqK a>'  i c              3   X   #    U  H   n[        S S /SS /US9[        U5      :H  v   M"     g7f)r   r   r   Nr	   r   .0r   s     ]/var/www/auris/envauris/lib/python3.13/site-packages/sympy/discrete/tests/test_recurrences.py	<genexpr>test_linrec.<locals>.<genexpr>   s2      M=K aV1a&A6)A,F=Ks   (*_   s   c              3   ^   #    U  H#  n[        S S /S S /US9[        US -   5      :H  v   M%     g7f)r   r   Nr%   r&   s     r(   r)   r*      s7      N=L aV1a&A6)AE:JJ=Ls   +-iS  ig  r   r   r            	   r   zx y zP   l   ua#o[|,VNM,e*iG l	             VvRl   9g9Z
(#l         kr#   l   6Nn&`^W*H
UeX`h"l   EW&G <   i  l          Pjej~Tl   yk%+Gk& c                       [        / SQ/ SQSS9$ )Nr   r   r   r   r        r(   <lambda>test_linrec.<locals>.<lambda>,   s    f,<?VWXr7   c                  &   > [        T S S TS S SS9$ )Nr   r   r#   r   r   abs   r(   r8   r9   -   s    fAbqE"1Gr7   c                  &   > [        T S S TS S SS9$ )Nr   ir   r   r;   s   r(   r8   r9   .   s    vQrU2A&Ir7   c                     > [        TT SS9$ Nr#   )r   r   )r=   xs   r(   r8   r9   /       fQU3r7   c                     > [        T TSS9$ r@   r   )r<   ys   r(   r8   r9   0   rB   r7   )r   r   r   )r   r   r   i i#
 i )r   r   r   r      i|i4.\i  l   `7u)l   <=C^l   x)m )
r	   allranger   Halfr   r   r   	TypeError
ValueError)zr<   r=   rA   rD   s    @@@@r(   test_linrecrL      s   !Qq!f3u<<<|rBdJJJr2h"=LLLyB?3FFF)CRS S S!R,.aSVi5G4PAB B B )FFF)FFF M=B2s^M M M M M  N=B3_N N N N N 
!Q!QHQNHQPQNSA	
Aq(1a.(1a.(1a.!DAgGAq!2AQrUb1Q4	66 6 6 2AQrUb157LMN N N 12QrUb1F$	&& & & !R8Aq>8Aq>"BBdS13PQR R R 9XY
9GH
:IJ
934
934!Qa8	1!acA!" " "!Qb9q6!8fQh&' ' 'QFb9
XaZ     !R)+1a)tDq>!++nQ.>>? ? ?1a&B/1444!Qa8A===r7   N)sympy.core.numbersr   %sympy.functions.combinatorial.numbersr   
sympy.corer   r   sympy.testing.pytestr   sympy.discrete.recurrencesr	   rL   r6   r7   r(   <module>rR      s    ' ; ! ' -4>r7   