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
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).
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
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
KEYWORD
nonn,easy
AUTHOR
Peter Luschny, Jun 04 2018
STATUS
approved