A305539 a(n) is a generalized pentagonal number such that 2*a(n) is also a generalized pentagonal number.
0, 1, 35, 1190, 40426, 1373295, 46651605, 1584781276, 53835911780, 1828836219245, 62126595542551, 2110475412227490, 71694037420192110, 2435486796874304251, 82734857056306152425, 2810549653117534878200, 95475953348939879706376, 3243371864210838375138585
Offset: 1
Examples
For n = 56 and k = -80 there is k*(3*k + 2) - 6*n^2 - 4*n = 0, hence A001318(56) = 1190 is in this sequence. And indeed also 2380 is a generalized pentagonal number, A001318(79).
Links
- G. C. Greubel, Table of n, a(n) for n = 1..650
- Andreas Enge, William Hart and Fredrik Johansson, Short addition sequences for theta functions, Journal of Integer Sequences, Vol. 21 (2018), Article 18.2.4. Also available as arXiv:1608.06810 [math.NT], 2016-2018.
- Simon Plouffe Conjectures of the OEIS, as of June 20, 2018.
- Index entries for linear recurrences with constant coefficients, signature (35,-35,1).
Programs
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Magma
I:=[0,1,35]; [n le 3 select I[n] else 35*Self(n-1) -35*Self(n-2) +Self(n-3): n in [1..41]]; // G. C. Greubel, Jun 05 2023
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Maple
a := proc(searchlimit) local L, g, n, s; L := NULL; g := n -> ((6*n^2+6*n+1)-(2*n+1)*(-1)^n)/16; for n from 0 to searchlimit do s := isolve(k*(3*k+2)-6*n^2-4*n = irem(n,2)*(4*n+2)); if s <> NULL then L:=L,g(n); fi od: L end: a(12000); -
Mathematica
LinearRecurrence[{35,-35,1}, {0,1,35}, 18] (* Jean-François Alcover, Jul 14 2019, after A029546 *) (Fibonacci[2*Range[40]-1,2]^2 -1)/24 (* G. C. Greubel, Jun 05 2023 *) -
SageMath
[(lucas_number1(2*n-1,2,-1)^2 -1)/24 for n in range(1,41)] # G. C. Greubel, Jun 05 2023
Formula
If for given n there is an integer k such that k*(3*k + 2) - 6*n^2 - 4*n = (n mod 2)*(4*n + 2) then A001318(n) is in this sequence.
G.f.: x^2/(1 - 35*x + 35*x^2 -x^3). - Simon Plouffe, Jun 20 2018
a(n) = (Pell(2*n-1)^2 - 1)/24, n > 0. - G. C. Greubel, Jun 05 2023
Comments