A033586 a(n) = 4*n*(2*n + 1).

0, 12, 40, 84, 144, 220, 312, 420, 544, 684, 840, 1012, 1200, 1404, 1624, 1860, 2112, 2380, 2664, 2964, 3280, 3612, 3960, 4324, 4704, 5100, 5512, 5940, 6384, 6844, 7320, 7812, 8320, 8844, 9384, 9940, 10512, 11100, 11704, 12324, 12960, 13612, 14280

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

Number of possible king moves on an (n+1) X (n+1) chessboard. E.g., for a 3 X 3 board: king has 4*5 moves, 4*3 moves and 1*8 moves, so a(2)=40. - Ulrich Schimke (ulrschimke(AT)aol.com)

Sequence found by reading the line from 0, in the direction 0, 12, ..., in the square spiral whose vertices are the triangular numbers A000217. Opposite numbers to the members of A085250 in the same spiral. - Omar E. Pol, Sep 03 2011

Sum of the numbers from 3n to 5n. - Wesley Ivan Hurt, Dec 22 2015

From Emeric Deutsch, Nov 09 2016: (Start)

a(n) is the second Zagreb index of the friendship graph F[n]. The second Zagreb index of a simple connected graph is the sum of the degree products d(i)d(j) over all edges ij of the graph. The friendship graph (or Dutch windmill graph) F[n] can be constructed by joining n copies of the cycle graph C[3] with a common vertex.

For instance, a(2)=40. Indeed, the friendship graph F[2] has 2 edges with end-point degrees 2,2 and 4 edges with end-point degrees 2,4. Then the second Zagreb index is 2*4 + 4*8 = 40. (End)

a(n) is the number of vertices in conjoined n X n dodecagons which are arranged into a square array, a.k.a. 3-4-3-12 tiling. - Donghwi Park, Dec 20 2020

References

• E. Bonsdorff, K. Fabel and O. Riihimaa, Schach und Zahl (Chess and numbers), Walter Rau Verlag, Dusseldorf, 1966.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Tim Krabbe, Open Chess Diary, see item 221
• Wikipedia, Friendship graph
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A035005 (Queen), A035006 (Rook), A035008 (Knight), A002492 (Bishop) and A049450 (Pawn).

Cf. A000217, A005408, A008586, A014105, A085250.

Programs

Magma

[4*n*(2*n + 1) : n in [0..50]]; // Wesley Ivan Hurt, Dec 22 2015

Maple

A033586:=n->4*n*(2*n+1); seq(A033586(n), n=0..60); # Wesley Ivan Hurt, Feb 25 2014

Mathematica

Table[4n*(2n + 1), {n, 0, 60}] (* Stefan Steinerberger, Apr 08 2006 *)
LinearRecurrence[{3,-3,1},{0,12,40},60] (* Harvey P. Dale, May 19 2011 *)

PARI

a(n)=4*n*(2*n+1) \\ Charles R Greathouse IV, Jul 16 2011

Formula

Binomial transform of [12, 28, 16, 0, 0, 0, ...] = (12, 40, 84, 144, 220, ...). - Gary W. Adamson, Oct 24 2007

a(n) = 4 * A014105(n). - Johannes W. Meijer, Feb 04 2010

a(n) = 16*n + a(n-1) - 4 (with a(0)=0). - Vincenzo Librandi, Aug 05 2010

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3. - Harvey P. Dale, May 10 2011

G.f.: 4*x*(3+x)/(1-3*x+3*x^2-x^3). - Colin Barker, Jan 06 2012

From Wesley Ivan Hurt, Feb 25 2014, Dec 22 2015: (Start)

a(n) = A008586(n) * A005408(n).

a(n) = Sum_{i=3n..5n} i.

a(-n) = A085250(n). (End)

E.g.f.: (8*x^2 + 12*x)*exp(x). - G. C. Greubel, Jul 16 2017

From Vaclav Kotesovec, Dec 21 2020: (Start)

Sum_{n>=1} 1/a(n) = (1 - log(2))/2.

Sum_{n>=1} (-1)^n/a(n) = 1/2 - Pi/8 - log(2)/4. (End)

Extensions

More terms from Erich Friedman

Crossref added, minor errors corrected and edited by Johannes W. Meijer, Feb 04 2010