A201279 a(n) = 6n^2 + 10n + 5.

5, 21, 49, 89, 141, 205, 281, 369, 469, 581, 705, 841, 989, 1149, 1321, 1505, 1701, 1909, 2129, 2361, 2605, 2861, 3129, 3409, 3701, 4005, 4321, 4649, 4989, 5341, 5705, 6081, 6469, 6869, 7281, 7705, 8141, 8589, 9049, 9521, 10005, 10501, 11009, 11529, 12061

Offset

0,1

Comments

Numbers n where 6n-5 is a square of a number type 6n-1.

Also sequence found by reading the line from 5, in the direction 5, 21,..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. - Omar E. Pol, Jul 18 2012

The spiral mentioned above naturally appears on a "graphene" like lattice (planar net 6^3). The opposite diagonal is A080859. - Yuriy Sibirmovsky, Oct 04 2016

First differences of A048395. - Leo Tavares, Nov 24 2021 [Corrected by Omar E. Pol, Dec 26 2021]

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Yuriy Sibirmovsky, Diagonals of a 'graphene' number spiral.

Leo Tavares, Illustration: Diamond Frame Stars

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

Formula

G.f.: (1+x)*(5+x)/(1-x)^3. - Colin Barker, Jan 09 2012

a(n) = 1 + A033579(n+1). - Omar E. Pol, Jul 18 2012

a(n) = (n+1)*A001844(n+1)-n*A001844(n). [Bruno Berselli, Jan 15 2013]

From Leo Tavares, Nov 24 2021: (Start)

a(n) = A003154(n+2) - A022144(n+1). See Diamond Frame Stars illustration.

a(n) = A016754(n) + A046092(n+1). (End)

Mathematica

LinearRecurrence[{3, -3, 1}, {5, 21, 49}, 50] (* Vincenzo Librandi, Dec 01 2011 *)
Table[6 n^2 + 10 n + 5, {n, 0, 44}] (* or *)
CoefficientList[Series[(1 + x) (5 + x)/(1 - x)^3, {x, 0, 44}], x] (* Michael De Vlieger, Oct 04 2016 *)

Prog

(PARI) a(n)=6*n^2+10*n+5 \\ Charles R Greathouse IV, Nov 29 2011
(Magma) [6*n^2 + 10*n + 5: n in [0..60]]; // Vincenzo Librandi, Dec 01 2011

Crossrefs

Cf. A136392, A080859.

Cf. A003154, A022144, A016754, A046092, A048395.

Sequence in context: A296200 A041825 A022268 * A146721 A099979 A366723

Adjacent sequences: A201276 A201277 A201278 * A201280 A201281 A201282

Keyword

nonn,easy

Author

Eleonora Echeverri-Toro, Nov 29 2011

Status

approved