A185212 a(n) = 12*n^2 - 8*n + 1.

1, 5, 33, 85, 161, 261, 385, 533, 705, 901, 1121, 1365, 1633, 1925, 2241, 2581, 2945, 3333, 3745, 4181, 4641, 5125, 5633, 6165, 6721, 7301, 7905, 8533, 9185, 9861, 10561, 11285, 12033, 12805, 13601, 14421, 15265, 16133, 17025, 17941, 18881, 19845, 20833

Offset

0,2

Comments

Sequence found by reading the line from 1, in the direction 1, 5, and the same line from 5, in the direction 5, 33, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. - Omar E. Pol, May 08 2018

Links

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Leo Tavares, Illustration: Square Block Rays

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

Formula

a(n) = 4*A000567(n) + 1.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) with a(0)=1, a(1)=5, a(2)=33. - Harvey P. Dale, Jul 07 2015

G.f.: (-1 - 2*x - 21*x^2)/(-1+x)^3. - Harvey P. Dale, Jul 07 2015

E.g.f.: (12*x^2 + 4*x + 1)*exp(x). - G. C. Greubel, Jun 25 2017

a(n) = A016754(n-1) + 4*A000384(n). - Leo Tavares, May 21 2022

From Amiram Eldar, May 28 2022: (Start)

Sum_{n>=0} 1/a(n) = sqrt(3)*Pi/8 - 3*log(3)/8 + 1.

Sum_{n>=0} (-1)^n/a(n) = Pi/8 - sqrt(3)*arccoth(sqrt(3))/2 + 1. (End)

Mathematica

Table[12n^2-8n+1, {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {1, 5, 33}, 50] (* Harvey P. Dale, Jul 07 2015 *)

Prog

(Haskell)
a185212 = (+ 1) . (* 4) . a000567
(PARI) a(n)=12*n^2-8*n+1 \\ Charles R Greathouse IV, Jun 17 2017

Crossrefs

For n > 0: odd terms in A001859.

Cf. A001082.

Cf. A016754, A000384.

Sequence in context: A299518 A338342 A135111 * A250225 A250273 A050830

Adjacent sequences: A185209 A185210 A185211 * A185213 A185214 A185215

Keyword

nonn,easy

Author

Reinhard Zumkeller, Dec 20 2012

Status

approved