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

Author

Eleonora Echeverri-Toro, Nov 29 2011

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A136392, A080859.

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

Programs

Magma

[6*n^2 + 10*n + 5: n in [0..60]]; // Vincenzo Librandi, Dec 01 2011

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 *)

PARI

a(n)=6*n^2+10*n+5 \\ Charles R Greathouse IV, Nov 29 2011

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)