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A305539
a(n) is a generalized pentagonal number such that 2*a(n) is also a generalized pentagonal number.
2
0, 1, 35, 1190, 40426, 1373295, 46651605, 1584781276, 53835911780, 1828836219245, 62126595542551, 2110475412227490, 71694037420192110, 2435486796874304251, 82734857056306152425, 2810549653117534878200, 95475953348939879706376, 3243371864210838375138585
OFFSET
1,3
COMMENTS
Enge, Hart and Johansson prove that every generalized pentagonal number c >= 5 is the sum of a smaller one and twice a smaller one, that is, there are generalized pentagonal numbers a, b < c such that c = 2a + b (see link, theorem 5). We look here at those c >= 0 which have b = 0. A305538 lists the smallest b > 0 for a given c.
LINKS
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.
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
EXAMPLE
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).
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 *)
PROG
(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
(SageMath) [(lucas_number1(2*n-1, 2, -1)^2 -1)/24 for n in range(1, 41)] # G. C. Greubel, Jun 05 2023
CROSSREFS
Essentially A029546.
Sequence in context: A352183 A162847 A029546 * A163218 A163600 A164068
KEYWORD
nonn,easy
AUTHOR
Peter Luschny, Jun 04 2018
STATUS
approved