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

Author

Reinhard Zumkeller, Dec 20 2012

Keywords

• nonn
• easy

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).

Crossrefs

For n > 0: odd terms in A001859.

Cf. A001082.

Cf. A016754, A000384.

Programs

Haskell

a185212 = (+ 1) . (* 4) . a000567

Mathematica

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

PARI

a(n)=12*n^2-8*n+1 \\ Charles R Greathouse IV, Jun 17 2017

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)