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A033586
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a(n) = 4*n*(2*n + 1).
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23
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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
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OFFSET
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0,2
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COMMENTS
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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
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
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REFERENCES
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E. Bonsdorff, K. Fabel and O. Riihimaa, Schach und Zahl (Chess and numbers), Walter Rau Verlag, Dusseldorf, 1966.
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LINKS
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FORMULA
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Binomial transform of [12, 28, 16, 0, 0, 0, ...] = (12, 40, 84, 144, 220, ...). - Gary W. Adamson, Oct 24 2007
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
a(n) = Sum_{i=3n..5n} i.
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)
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MAPLE
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MATHEMATICA
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LinearRecurrence[{3, -3, 1}, {0, 12, 40}, 60] (* Harvey P. Dale, May 19 2011 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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