Fibonacci numbers F(n) are the sequence 0,1,1,2,3,5,8,13,...
recursively defined as
F(0) = 0 F(1) = 1 F(n) = F(n-1) + F(n-2)The Cassini's identity for Fibonacci numbers is F(n+1) F(n-1) - F(n) F(n) = (-1)n and could be taken as an alternate recursive definition of the Fibonacci numbers, namely
F(n+1) = [ F(n)^2 + (-)^n ] / F(n-1)e.g.,
2 = [ 1^1 + (-)^2 ] / 1 3 = [ 2^2 + (-)^3 ] / 1 5 = [ 3^2 + (-)^4 ] / 2 ...Cassini identity is generalized by Catalan's identity,
F(n+r) = [ F(n)^2 - (-)^(n-r) F(r)^2 ] / F(n-r)We can look at Catalan's formula as a recursive definition of the terms of a sequence, either with alternating signs,
C'(0) = 1 C'(1) = A C'(n+1) = [ C'(n)^2 + (-)^n D ] / C'(n-1)or with fixed sign
C"(0) = 1 C"(1) = A C"(n+1) = [ C"(n)^2 + D ] / C"(n-1)It turns out that only certain choices of (A,D) makes one of these into an integer sequence. The table below lists the first few terms of some of these sequences. Several of these sequences are already known sequences. Their OEIS numbers are in the table. The sequences with D=0 are trivial solutions, consisting of the powers f A: C(k) = Ak. Therefore we consider only sequences with A, and D not zero.
Question.
Is the number of non-zero integer pairs (A,D) that make C' or C" into an
integer sequence finite or infinite ?
Is it possible to describe the values of the non-zero integer pairs (A,D)
that make C' or C" into an integer sequence ?
By looking at the sequences in the table below it seems that the values of D that make C' and C" into integer sequences are spaced by A.
Let's begin with a lemma about the sequences, defined as
A(0) = 1 A(1) = a A(k) = ( A(k-1)^2 + (-)^k ) / A(k-2)has the properties
(a') a A(k+1) = A(k+2) - A(k) (b') A(k+3) A(k) - A(k+2) A(k+1) = (-)^k aLet's compute
A(2) = a^2 + 1 A(3) = a^3 + 2 a A(4) = a^4 + 3 a^2 + 1 A(5) = a^5 + 4 a^3 + 3 a ...
( A(k+1)^2 + (-)^k - A(k)^2 )/A(k) = ( A(k+1)^2 - A(k+1) A(k-1) ) / A(k) = A(k+1) ( A(k+1) - A(k-1) ) / A(k) = A(k+1) ( a A(k) ) / A(k)
(b') then follows by induction, using (a'). For k=0 it is A(3) A(0) - A(2) A(1) = (a3 + 2 a) - (a2 + 1) a = a. For k=1,
A(4) A(1) - A(3) A(2) = (a^4 + 3 a^2 + 1) a - (a^3 + 2 a) (a^2 + 1) = a^5 + 3 a^3 + a - a^5 - 3 a^3 - 2 a = - a
A(k+3) A(k) - A(k+2) A(k+1) = ( A(k+2)^2 + (-)^(k+3) )/A(k+1) * A(k) - A(k+2) A(k+1) = [(A(k+2)^2 + (-)^(k+3)) A(k) - A(k+2) (A(k+2) A(k) - (-)^(k+2))]/A(k+1) = (-)^k (-A(k) + A(k+2))/A(k+1) = (-)^k a
Now the Catalan's sequences with alternating signs defined by A=n, D=1 + a n - n2 are integer. Let's write the first few terms:
C'(0) = 1, C'(1) = n, C'(2) = A(0) + A(1) n C'(3) = A(1) + A(2) n ... C'(k) = A(k-1) n + A(k-2) (for k > 1)We try to verify that this expression antually satisfies the defining recursive relation of C'(k).
( A(k-1) n + A(k-2) )^2 - (-)^k ( 1 + a n - n^2 ) = A(k-1)^2 n^2 + 2 A(k-1) A(k-2) n + A(k-2)^2 + (-)^(k+1) + (-)^(k+1) a n + (-)^k n^2 = [A(k-1)^2 + (-)^k] n^2 + [2 A(k-1) A(k-2) + a (-)^(k+1)] n + [ A(k-1)^2 + (-)^(k+1) ] = A(k) A(k-2) n^2 + A(k-1) A(k-2) n + A(k-1) A(k-3) + [ A(k-1)A(k-2) + a (-)^(k+1) ] n = ( A(k) n + A(k-1) ) * ( A(k-2) n + A(k-3) ) by prop. (b)
This result (almost) answers the question: there are infinitely many Catalan's sequences with alternating sign and have D = 1 + a n - n2. We still have not proved that other values of D do not give integer sequences.
A similar result should hold for Catalan's sequences with fixed sign, using D = -1 + a n - n2. We consider the sequence defined by B(k+1) = [ B(k)2 - 1 ]/B(k-1)
B(0) = 1 B(1) = b B(2) = b^2 - 1 B(3) = b^3 - 2 b B(4) = b^4 - 3 b^2 + 1 B(5) = b^5 - 4 b^3 + 3 bThen the two lemma are
(a") b B(k+1) = B(k) B(k+2) (b") B(k+3) B(k) - B(k+2) B(k+1) = -b
As before, these lemma can be proven by induction. Then a solution for the recurrence relation with fixed sign is
C"(0) = 1 C"(1) = n C"(2) = B(1) n - B(0) C"(3) = B(2) n - B(1) ... C"(k) = B(k-1) n - B(k-2)and, ss before this can be proven by induction, using (a") (b").
A D s A016777 4 -9 +: 1,4,7,10,13,16,19,22,25,28 A002878 4 -5 +: 1,4,11,29,76,199,521,1364,3571,9349 A001353 4 -1 +: 1,4,15,56,209,780,2911,10864,40545,151316 A004253 4 3 +: 1,4,19,91,436,2089,10009,47956,229771,1100899 A038723 4 7 +: 1,4,23,134,781,4552,26531,154634,901273,5253004 4 11 +: 1,4,27,185,1268,8691,59569,408292,2798475,19181033 A000285 4 -11 -: 1,4,5,9,14,23,37,60,97,157 A048654 4 -7 -: 1,4,9,22,53,128,309,746,1801,4348 A003688 4 -3 -: 1,4,13,43,142,469,1549,5116,16897,55807 A001076 4 1 -: 1,4,17,72,305,1292,5473,23184,98209,416020 A100237 4 5 -: 1,4,21,109,566,2939,15261,79244,411481,2136649 4 9 -: 1,4,25,154,949,5848,36037,222070,1368457,8432812 A054486 5 -11 +: 1,5,14,37,97,254,665,1741,4558,11933 A001834 5 -6 +: 1,5,19,71,265,989,3691,13775,51409,191861 A004254 5 -1 +: 1,5,24,115,551,2640,12649,60605,290376,1391275 A001653 5 4 +: 1,5,29,169,985,5741,33461,195025,1136689,6625109 A033889 5 9 +: 1,5,34,233,1597,10946,75025,514229,3524578,24157817 A105426 5 14 +: 1,5,39,307,2417,19029,149815,1179491,9286113,73109413 A048655 5 -14 -: 1,5,11,27,65,157,379,915,2209,5333 A108300 5 -9 -: 1,5,16,53,175,578,1909,6305,20824,68777 A015448 5 -4 -: 1,5,21,89,377,1597,6765,28657,121393,514229 A052918 5 1 -: 1,5,26,135,701,3640,18901,98145,509626,2646275 5 6 -: 1,5,31,191,1177,7253,44695,275423,1697233,10458821 5 11 -: 1,5,36,257,1835,13102,93549,667945,4769164,34052093 A054491 6 -13 +: 1,6,23,86,321,1198,4471,16686,62273,232406 A030221 6 -7 +: 1,6,29,139,666,3191,15289,73254,350981,1681651 A001109 6 -1 +: 1,6,35,204,1189,6930,40391,235416,1372105,7997214 A049685 6 5 +: 1,6,41,281,1926,13201,90481,620166,4250681,29134601 6 11 +: 1,6,47,370,2913,22934,180559,1421538,11191745,88112422 6 17 +: 1,6,53,471,4186,37203,330641,2938566,26116453,232109511 A189735 6 -17 -: 1,6,19,63,208,687,2269,7494,24751,81747 A048875 6 -11 -: 1,6,25,106,449,1902,8057,34130,144577,612438 A015449 6 -5 -: 1,6,31,161,836,4341,22541,117046,607771,3155901 A005668 6 1 -: 1,6,37,228,1405,8658,53353,328776,2026009,12484830 6 7 -: 1,6,43,307,2192,15651,111749,797894,5697007,40676943 6 13 -: 1,6,49,398,3233,26262,213329,1732894,14076481,114344742 A055271 7 -15 +: 1,7,34,163,781,3742,17929,85903,411586,1972027 A002315 7 -8 +: 1,7,41,239,1393,8119,47321,275807,1607521,9369319 A004187 7 -1 +: 1,7,48,329,2255,15456,105937,726103,4976784,34111385 A070997 7 6 +: 1,7,55,433,3409,26839,211303,1663585,13097377,103115431 7 13 +: 1,7,62,551,4897,43522,386801,3437687,30552382,271533751 7 20 +: 1,7,69,683,6761,66927,662509,6558163,64919121,642633047 A048876 7 -20 -: 1,7,29,123,521,2207,9349,39603,167761,710647 7 -13 -: 1,7,36,187,971,5042,26181,135947,705916,3665527 A015451 7 -6 -: 1,7,43,265,1633,10063,62011,382129,2354785,14510839 A054413 7 1 -: 1,7,50,357,2549,18200,129949,927843,6624850,47301793 7 8 -: 1,7,57,463,3761,30551,248169,2015903,16375393,133019047 7 15 -: 1,7,64,583,5311,48382,440749,4015123,36576856,333206827 A054488 8 -17 +: 1,8,47,274,1597,9308,54251,316198,1842937,10741424 A033890 8 -9 +: 1,8,55,377,2584,17711,121393,832040,5702887,39088169 A001090 8 -1 +: 1,8,63,496,3905,30744,242047,1905632,15003009,118118440 A070998 8 7 +: 1,8,71,631,5608,49841,442961,3936808,34988311,310957991 8 15 +: 1,8,79,782,7741,76628,758539,7508762,74329081,735782048 8 23 +: 1,8,87,949,10352,112923,1231801,13436888,146573967,1598876749 8 -23 -: 1,8,41,213,1106,5743,29821,154848,804061,4175153 8 -15 -: 1,8,49,302,1861,11468,70669,435482,2683561,16536848 A015453 8 -7 -: 1,8,57,407,2906,20749,148149,1057792,7552693,53926643 A041025 8 1 -: 1,8,65,528,4289,34840,283009,2298912,18674305,151693352 8 9 -: 1,8,73,665,6058,55187,502741,4579856,41721445,380072861 8 17 -: 1,8,81,818,8261,83428,842541,8508838,85930921,867818048 9 -19 +: 1,9,62,425,2913,19966,136849,937977,6428990,44064953 A057080 9 -10 +: 1,9,71,559,4401,34649,272791,2147679,16908641,133121449 A018913 9 -1 +: 1,9,80,711,6319,56160,499121,4435929,39424240,350382231 A072256 9 8 +: 1,9,89,881,8721,86329,854569,8459361,83739041,828931049 9 17 +: 1,9,98,1069,11661,127202,1387561,15135969,165108098,1801053109 9 -26 -: 1,9,55,339,2089,12873,79327,488835,3012337,18562857 9 -17 -: 1,9,64,457,3263,23298,166349,1187741,8480536,60551493 A015454 9 -8 -: 1,9,73,593,4817,39129,317849,2581921,20973217,170367657 A099371 9 1 -: 1,9,82,747,6805,61992,564733,5144589,46866034,426938895 A109108 9 10 -: 1,9,91,919,9281,93729,946571,9559439,96540961,974969049 9 19 -: 1,9,100,1109,12299,136398,1512677,16775845,186046972,2063292537 A077245 10 -21 +: 1,10,79,622,4897,38554,303535,2389726,18814273,148124458 A057081 10 -11 +: 1,10,89,791,7030,62479,555281,4935050,43860169,389806471 A004189 10 -1 +: 1,10,99,980,9701,96030,950599,9409960,93149001,922080050 A078922 10 9 +: 1,10,109,1189,12970,141481,1543321,16835050,183642229,2003229469 10 -29 -: 1,10,71,507,3620,25847,184549,1317690,9408379,67176343 10 -19 -: 1,10,81,658,5345,43418,352689,2864930,23272129,189041962 A015455 10 -9 -: 1,10,91,829,7552,68797,626725,5709322,52010623,473804929 A041041 10 1 -: 1,10,101,1020,10301,104030,1050601,10610040,107151001,1082120050 11 -23 +: 1,11,98,871,7741,68798,611441,5434171,48296098,429230711 A054320 11 -12 +: 1,11,109,1079,10681,105731,1046629,10360559,102558961,1015229051 11 -32 -: 1,11,89,723,5873,47707,387529,3147939,25571041,207716267 11 -21 -: 1,11,100,911,8299,75602,688717,6274055,57155212,520670963 A015456 11 -10 -: 1,11,111,1121,11321,114331,1154631,11660641,117761041,1189271051 A077251 12 -25 +: 1,12,119,1178,11661,115432,1142659,11311158,111968921,1108378052 12 -35 -: 1,12,109,993,9046,82407,750709,6838788,62299801,567536997 A171510 12 -23 -: 1,12,121,1222,12341,124632,1258661,12711242,128371081,1296422052 13 -40 +: 1,13,129,1277,12641,125133,1238689,12261757,121378881,1201527053 13 -51 -: 1,13,118,1075,9793,89212,812701,7403521,67444390,614403031 13 -38 -: 1,13,131,1323,13361,134933,1362691,13761843,138981121,1403573053 14 -57 +: 1,14,139,1376,13621,134834,1334719,13212356,130788841,1294676054 14 -69 -: 1,14,127,1157,10540,96017,874693,7968254,72588979,661269065 14 -55 -: 1,14,141,1424,14381,145234,1466721,14812444,149591161,1510724054 15 -74 -: 1,15,151,1525,15401,155535,1570751,15863045,160201201,1617875055
Marco Corvi - 2014