[seqfan] Index of linear recurrences

Richard Mathar mathar at strw.leidenuniv.nl
Sun Jan 11 23:36:02 CET 2009

```To uncover relations between sequences with linear recurrences (of constant
coefficients) of finite order, one could add an index which groups sequences
with the same recurrence into a single line. A start of this
http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html

After scanning roughly 1 percent (!) of the database for generating functions
with a rational format (and elimination of common factors where possible),
a first version  would look as follows. The integers in parenthesis list
the coefficients d(1), d(2), d(3) etc in the recurrence
a(n)=d(1)*a(n-1)+d(2)*a(n-2)+d(3)*a(n-3)+... :

recurrence, linear, order 1, (1): A000004, A000012, A007395, A010701, A010709, A010716, A010722, A010727, A010731, A010734, A010692, A010850
recurrence, linear, order 1, (-1): A033999
recurrence, linear, order 1, (2): A000079, A007283, A081808, A020707, A020714, A091629, A005009, A005010, A005015, A005029, A110286, A110287, A110288, A084215
recurrence, linear, order 1, (3): A000244, A008776, A005030, A005032, A005051, A005052, A120354, A116530
recurrence, linear, order 1, (4): A000302, A004171, A002023, A002042, A002063, A002066, A002089
recurrence, linear, order 1, (5): A000351, A020729, A005055
recurrence, linear, order 1, (6): A000400
recurrence, linear, order 1, (7): A000420, A109808
recurrence, linear, order 1, (16): A090411
recurrence, linear, order 1, (-2): A122803
recurrence, linear, order 2, (1,1): A000032, A000045, A020695, A020701, A020712, A000285, A104449, A022095, A090991, A078642, A001060, A013655, A022112, A022113, A022114, A0022367, A022115, A022368, A022116, A022086
recurrence, linear, order 2, (1,-1): A057079, A117373, A119910, A117378
recurrence, linear, order 2, (1,6): A087451
recurrence, linear, order 2, (1,12): A087452
recurrence, linear, order 2, (2,1): A048654, A000129, A001333, A048655, A078057, A048693, A048694, A048695, A048696, A048697, A002203, A078343, A105082
recurrence, linear, order 2, (2,-1): A086570, A092535
recurrence, linear, order 2, (2,-2): A099087
recurrence, linear, order 2, (2,-8): A090591
recurrence, linear, order 2, (3,1): A052924, A006190, A003688, A108300, A006497, A097924
recurrence, linear, order 2, (3,-2): A068156, A052996
recurrence, linear, order 2, (-3,-2): A104934
recurrence, linear, order 2, (4,1): A001077, A033887, A001076, A015448, A048875, A048876, A048877, A048878, A048879, A014448, A014445
recurrence, linear, order 2, (5,1): A100237, A052918, A015449, A087130
recurrence, linear, order 2, (6,1): A154451, A005667, A005668, A085447, A078469
recurrence, linear, order 2, (7,1): A015453, A054413, A086902
recurrence, linear, order 2, (8,1): A015454, A041024, A088317, A041025, A086594
recurrence, linear, order 2, (3,-1): A001519, A048575, A001906, A002878, A054486, A054492, A055267, A055273, A055849, A055850, A056123, A005248, A025169, A106729, A100545, A097512
recurrence, linear, order 2, (4,-1): A001075, A001835, A079935, A001353, A001834, A054491, A054485, A055845, A003500, A077234, A082841, A005320, A057819, A077236, A106707, A077235
recurrence, linear, order 2, (5,-1): A002310, A002320, A004253, A004254, A030221, A055271, A099867, A054477, A003501, A099868
recurrence, linear, order 2, (5,-3): A095934
recurrence, linear, order 2, (6,-1): A038725, A001541, A038723, A001653, A001109, A002315, A054488, A038761, A054489, A054490, A001542, A003499, A075870, A077413, A077444, A106329, A038762, A075841, A106328, A077445, A100525, A005319, A077240, A101386, A075848, A077239, A081554
recurrence, linear, order 2, (7,-1): A033889, A049685, A004187, A033890, A056914, A056854, A033891, A033888
recurrence, linear, order 2, (8,-1): A001091, A105426, A070997, A001090, A057080, A077245, A086903, A077246, A077243, A077244
recurrence, linear, order 2, (9,-1): A070998, A018913, A057081, A056918, A065100
recurrence, linear, order 2, (10,-1): A001078, A001079, A072256, A004189, A054320, A077251, A087799, A077249, A122652, A122653, A077409, A077250
recurrence, linear, order 2, (11,-1): A078922, A004190, A097783, A057076, A075835
recurrence, linear, order 2, (12,-1): A023038, A077417, A004191, A077416, A087800, A065101
recurrence, linear, order 2, (13,-1): A085260, A078362, A078363
recurrence, linear, order 2, (14,-1): A011943, A011944, A122769, A001570, A122571, A007655, A094347, A067902, A094347, A028230, A011945, A067900
recurrence, linear, order 2, (15,-1): A078364, A078365
recurrence, linear, order 2, (16,-1): A001081, A077412, A001080, A090727
recurrence, linear, order 2, (17,-1): A078366, A078367
recurrence, linear, order 2, (18,-1): A023039, A007805, A049660, A049629, A087215, A075796, A075869, A065102, A103134, A060645
recurrence, linear, order 2, (18,-80): A060531
recurrence, linear, order 2, (19,-1): A078368, A078369,
recurrence, linear, order 2, (20,-1): A001084, A001085, A075839, A075843, A083043, A090728, A075844
recurrence, linear, order 2, (21,-1): A092499, A090729
recurrence, linear, order 2, (22,-1): A077422, A077421, A090730
recurrence, linear, order 2, (23,-1): A097778, A090731
recurrence, linear, order 2, (24,-1): A077424, A077423, A090732
recurrence, linear, order 2, (25,-1): A097780, A090733
recurrence, linear, order 2, (26,-1): A097308, A097309, A090247
recurrence, linear, order 2, (2,2): A026150, A002605, A080953, A028859, A080040, A106433, A028860, A083337, A021006, A116556
recurrence, linear, order 2, (3,3): A030195, A125145, A106435, A085480, A108306
recurrence, linear, order 2, (4,4): A057087, A086347, A084128, A094013, A106568
recurrence, linear, order 2, (5,5): A057088, A106565
recurrence, linear, order 2, (6,6): A057089, A010924
recurrence, linear, order 2, (6,-7): A102285
recurrence, linear, order 2, (7,7): A057090
recurrence, linear, order 2, (8,8): A057091
recurrence, linear, order 2, (9,9): A057092
recurrence, linear, order 2, (10,10): A057093
recurrence, linear, order 2, (-1,1): A075193, A061084
recurrence, linear, order 2, (-1,-1): A061347
recurrence, linear, order 2, (-2,1): A077985
recurrence, linear, order 2, (-2,-1): A038608
recurrence, linear, order 2, (-2,-2): A078069
recurrence, linear, order 2, (-2,3): A014983
recurrence, linear, order 2, (-2,-9): A025170
recurrence, linear, order 2, (-3,-1): A099496, A098150, A098149
recurrence, linear, order 2, (-4,1): A099843
recurrence, linear, order 2, (-4,-4): A122803, A085750
recurrence, linear, order 2, (-11,1): A122574
recurrence, linear, order 2, (-14,-1): A122572
recurrence, linear, order 3, (0,0,1): A101825
recurrence, linear, order 3, (1,0,-1): A104770, A104771
recurrence, linear, order 3, (1,1,-1): A136746
recurrence, linear, order 3, (1,2,-2): A063757
recurrence, linear, order 3, (-1,0,1): A104769
recurrence, linear, order 3, (2,0,1): A052980
recurrence, linear, order 3, (2,-1,2): A007910, A100720
recurrence, linear, order 3, (2,2,-1): A001254, A090692, A102714
recurrence, linear, order 3, (2,2,-4): A122746, A014236
recurrence, linear, order 3, (2,3,-2): A046672, A107334
recurrence, linear, order 3, (-2,2,-3): A102785
recurrence, linear, order 3, (-2,6,27): A103644, A103645, A103646
recurrence, linear, order 3, (3,1,-1): A033505
recurrence, linear, order 3, (3,-1,2): A104004
recurrence, linear, order 3, (3,-1,-2): A099166
recurrence, linear, order 3, (3,2,1): A070207
recurrence, linear, order 3, (3,2,-4): A052899
recurrence, linear, order 3, (3,-3,1): A056109
recurrence, linear, order 3, (-3,4,-1): A122600
recurrence, linear, order 3, (3,5,1): A102129
recurrence, linear, order 3, (4,-2,-3): A099167
recurrence, linear, order 3, (5,5,-1): A046729, A105058
recurrence, linear, order 3, (5,-7,2): A061667
recurrence, linear, order 3, (6,0,-8): A111989
recurrence, linear, order 3, (6,-11,6): A118979
recurrence, linear, order 3, (7,-11,5): A094195
recurrence, linear, order 3, (8,-13,6): A094259
recurrence, linear, order 3, (11,-11,1): A095685
recurrence, linear, order 3, (11,-36,36): A001240
recurrence, linear, order 3, (15,-15,1): A007654
recurrence, linear, order 3, (27,-224,528): A020568
recurrence, linear, order 4, (0,1,1,1): A107458
recurrence, linear, order 4, (0,-1,0,-1): A101675, A100434
recurrence, linear, order 4, (0,1,0,1): A082587
recurrence, linear, order 4, (0,3,0,-1): A099255, A099256
recurrence, linear, order 4, (0,3,0,-2): A106624
recurrence, linear, order 4, (0,-5,0,-1): A101463
recurrence, linear, order 4, (1,4,-2,-4): A085903
recurrence, linear, order 4, (1,0,1,-1): A097950, A105077
recurrence, linear, order 4, (-1,-2,-1,-1): A098554
recurrence, linear, order 4, (2,0,-1,1): A059633
recurrence, linear, order 4, (2,0,1,-2): A063823
recurrence, linear, order 3, (2,0,-1,2): A113405
recurrence, linear, order 4, (2,0,-2,1): A002623, A102214, A092634
recurrence, linear, order 4, (2,1,0,-1): A052967
recurrence, linear, order 4, (2,1,-2,-1): A102702
recurrence, linear, order 4, (2,-1,-2,2): A106664
recurrence, linear, order 4, (2,3,-4,-4): A095977
recurrence, linear, order 4, (3,0,2,-2): A052958
recurrence, linear, order 4, (3,0,-4,2): A106666
recurrence, linear, order 4, (3,-2,-1,1): A104161
recurrence, linear, order 4, (4,-2,-2,1): A061703
recurrence, linear, order 4, (4,-6,4,-1): A006000, A136264, A113922
recurrence, linear, order 4, (-4,0,4,1): A097948, A097947, A097949
recurrence, linear, order 4, (-4,5,0,-3): A103135
recurrence, linear, order 4, (6,7,-5,-6): A019484
recurrence, linear, order 4, (6,-13,6,-1): A105660
recurrence, linear, order 4, (8,-21,20,-5): A094865
recurrence, linear, order 4, (8,-22,23,-6): A089883
recurrence, linear, order 5, (1,0,0,1,-1): A008624, A092533
recurrence, linear, order 5, (1,0,1,0,-1): A123552
recurrence, linear, order 5, (1,1,1,1,1): A124312
recurrence, linear, order 5, (2,-1,1,-2,1): A000969, A084683
recurrence, linear, order 5, (2,2,-4,-1,2): A106157
recurrence, linear, order 5, (3,-2,-2,3,-1): A005744, A026035
recurrence, linear, order 5, (3,-2,0,-1,1): A1041087
recurrence, linear, order 5, (3,3,-4,-1,1): A038342
recurrence, linear, order 5, (5,-3,-11,16,-6): A107307
recurrence, linear, order 5, (5,-10,10,-5,1): A081441
recurrence, linear, order 5, (7,-19,25,-16,4): A048503
recurrence, linear, order 6, (0,0,2,0,0,-1): A089598
recurrence, linear, order 6, (0,1,-4,1,2,-1): A005993
recurrence, linear, order 6, (1,1,0,-1,-1,1): A036410
recurrence, linear, order 6, (1,-1,2,-1,1,-1): A091972
recurrence, linear, order 6, (3,-1,-2,-1,1,1): A095681
recurrence, linear, order 6, (6,-15,20,-15,6,-1): A069038, A089830
recurrence, linear, order 6, (9,-30,46,-34,13,-2): A089932
recurrence, linear, order 6, (-4,-2,8,7,-4,-4): A106691
recurrence, linear, order 7, (0,0,1,1,0,0,-1): A008647
recurrence, linear, order 7, (0,1,2,-1,0,1,2): A107854
recurrence, linear, order 7, (0,-1,-2,1,0,-1,-2): A107853
recurrence, linear, order 7, (1,0,0,0,0,1,-1): A097992
recurrence, linear, order 7, (1,0,2,-2,0,-1,1): A092076
recurrence, linear, order 7, (1,-3,-3,3,3,-1,-1): A122576
recurrence, linear, order 7, (2,-2,3,-3,2,-2,1): A014679
recurrence, linear, order 7, (4,-6,5,-2,-1,1,-1): A083839
recurrence, linear, order 7, (7,-21,35,-35,21,-7,1): A006858, A051946, A069039
recurrence, linear, order 8, (0,4,0,-6,0,4,0,1): A115046
recurrence, linear, order 8, (1,2,-1,-2,-1,2,1,-1): A083709
recurrence, linear, order 8, (2,-1,0,0,0,1,-2,1): A008724
recurrence, linear, order 8, (2,-1,2,-4,2,-1,2,-1): A092353
recurrence, linear, order 8, (-2,0,0,2,0,0,1): A104237
recurrence, linear, order 8, (6,-14,14,0,-14,14,-6,1): A053493
recurrence, linear, order 9, (0,2,0,-1,1,0,-2,0,1): A038167
recurrence, linear, order 9, (1,0,1,-1,1,-1,0,-1,1): A091971, A097920, A097923
recurrence, linear, order 9, (1,2,-2,0,0,-2,2,1,-1): A106607
recurrence, linear, order 10, (0,1,0,0,-2,0,0,1,1,-1): A001400
recurrence, linear, order 10, (2,0,-3,0,3,0,0,-2,1): A100779
recurrence, linear, order 10, (4,-3,-8,14,0,-14,8,3,-4,1): A060099

```