OFFSET
1,2
COMMENTS
From Peter Bala, Nov 21 2024: (Start)
Numbers of the form n*(7*n + 3)/2 for n in Z. Cf. A057570.
The sequence terms occur as the exponents in the expansion of Product_{n >= 1} (1 - x^(7*n)) * (1 + x^(7*n-2)) * (1 + x^(7*n-5)) = 1 + x^2 + x^5 + x^11 + x^17 + x^27 + x^36 + .... Cf. A363800. (End)
LINKS
Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).
FORMULA
G.f.: x^2*(2 + 3*x + 2*x^2) / ((1 - x)^3*(1 + x)^2).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>5.
a(n) = (14*(n - 1)*n + (2*n - 1)*(-1)^n + 1)/16. Therefore:
a(n) = n*(7*n - 6)/8 for n even,
a(n) = (n - 1)*(7*n - 1)/8 for n odd.
E.g.f.: (x*(7*x -1)*cosh(x) + (7*x^2 + x + 1)*sinh(x))/8. - Stefano Spezia, Nov 26 2024
MATHEMATICA
Table[(14 (n - 1) n + (2 n - 1) (-1)^n + 1)/16, {n, 1, 60}] (* Bruno Berselli, Jul 08 2016 *)
PROG
(PARI) select(n->ispolygonal(7*n+1, 3), vector(3000, n, n-1))
(PARI) concat(0, Vec(x^2*(2+3*x+2*x^2)/((1-x)^3*(1+x)^2) + O(x^100)))
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Colin Barker, Jul 08 2016
EXTENSIONS
Edited by Bruno Berselli, Jul 08 2016
STATUS
approved