A236267 a(n) = 8*n^2 + 3*n + 1.

1, 12, 39, 82, 141, 216, 307, 414, 537, 676, 831, 1002, 1189, 1392, 1611, 1846, 2097, 2364, 2647, 2946, 3261, 3592, 3939, 4302, 4681, 5076, 5487, 5914, 6357, 6816, 7291, 7782, 8289, 8812, 9351, 9906, 10477, 11064, 11667, 12286, 12921, 13572, 14239, 14922, 15621, 16336

Offset

0,2

Comments

Positions a(n) of hexagonal numbers such that h(a(n)) = h(a(n)-1) + h(4*n+1), where h = A000384.

First bisection of A057029. The sequence contains infinitely many squares: 1, 676, 779689, 899760016, ... [Bruno Berselli, Jan 24 2014]

Links

Table of n, a(n) for n=0..45.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

From Colin Barker, Jan 21 2014: (Start)

G.f.: -(6*x^2 + 9*x + 1)/(x-1)^3.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)

E.g.f.: exp(x)*(1 + 11*x + 8*x^2). - Elmo R. Oliveira, Oct 19 2024

Example

For n=5, A000384(a(5)) = 93096 = A000384(a(5)-1) + A000384(4*5+1) = 92235 + 861.

Mathematica

Table[8 n^2 + 3 n + 1, {n, 0, 50}] (* Bruno Berselli, Jan 24 2014 *)
LinearRecurrence[{3, -3, 1}, {1, 12, 39}, 50] (* Harvey P. Dale, May 26 2019 *)

Prog

(PARI) Vec(-(6*x^2+9*x+1)/(x-1)^3 + O(x^100)) \\ Colin Barker, Jan 21 2014
(Magma) [8*n^2+3*n+1: n in [0..50]]; // Bruno Berselli, Jan 24 2014

Crossrefs

Cf. A000384, A057029, A064225, A152948, A236257.

Sequence in context: A209872 A186779 A154266 * A119094 A226348 A359023

Adjacent sequences: A236264 A236265 A236266 * A236268 A236269 A236270

Keyword

nonn,easy

Author

Vladimir Shevelev, Jan 21 2014

Extensions

More terms from Colin Barker, Jan 21 2014

a(44)-a(45) from Elmo R. Oliveira, Oct 19 2024

Status

approved