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0, 6, 20, 42, 72, 110, 156, 210, 272, 342, 420, 506, 600, 702, 812, 930, 1056, 1190, 1332, 1482, 1640, 1806, 1980, 2162, 2352, 2550, 2756, 2970, 3192, 3422, 3660, 3906, 4160, 4422, 4692, 4970, 5256, 5550
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OFFSET
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0,2
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COMMENTS
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a(n) = number of edges in (n+1)X(n+1) square grid with all horizontal, vertical and diagonal segments filled in - Asher Auel (asher.auel(AT)reed.edu) Jan 12, 2000.
Write 0,1,2,... in clockwise spiral; sequence gives numbers on one of 4 diagonals.
Twice second hexagonal numbers (Cf. A014105). - Omar E. Pol, May 21 2008
a(n) = A007494(n) + A173511(n) = A007742(n) + n. [From Reinhard Zumkeller, Feb 20 2010]
The identity (4*n+1)^2-(4*n^2+2*n)*(2)^2 = 1 can be written as A016813(n)^2-a(n)*2^2 = 1. - Vincenzo Librandi, Jul 20 2010 - Nov 25 2012
Starting with "6" = binomial transform of [6, 14, 8, 0, 0, 0,...] [From Gary W. Adamson, Aug 27 2010]
a(n+1) = A045896(2*n+1). [Reinhard Zumkeller, Dec 12 2011]
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REFERENCES
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R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd ed., 1994, p. 99.
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LINKS
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T. D. Noe, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Queen's Tour Graph
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FORMULA
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4n^2+2n.
a(n)=A014105(n)*2. - Omar E. Pol, May 21 2008
a(n) = floor((2*n + 1/2)^2). [From Reinhard Zumkeller, Feb 20 2010]
a(n)=8*n+a(n-1)-2 with a(0)=0 [From Vincenzo Librandi, Jul 20 2010]
a(0)=0, a(1)=6, a(2)=20, a(n)=3*a(n-1)-3*a(n-2)+a(n-3) [From Harvey P. Dale, Aug 11 2011]
G.f.: 2*x*(3+x)/(1-x)^3. [Colin Barker, Jan 14 2012]
Sum_{n>=1} 1/a(n) = 1-log(2). Sum_{n>=1} 1/a(n)^2 = 2*log(2)+Pi^2/6-3. - R. J. Mathar, Jan 15 2013
a(n) = A118729(8n+5). - Philippe Deléham, Mar 26 2013
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EXAMPLE
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16 17 18 19 ...
15 4 5 6 ...
14 3 0 7 ...
13 2 1 8 ...
For n=1, a(1)=8*1+0-2=6; n=2, a(2)=8*2+6-2=20; n=3, a(3)=8*3+20-2=42 [From Vincenzo Librandi, Jul 20 2010]
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MAPLE
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a:=n->sum(n+1, j=1..n): seq(a(n*2), n=0..37); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 03 2007
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MATHEMATICA
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s=0; lst={s}; Do[s+=n++ +6; AppendTo[lst, s], {n, 0, 7!, 8}]; lst [From Vladimir Joseph Stephan Orlovsky, Nov 16 2008]
LinearRecurrence[{3, -3, 1}, {0, 6, 20}, 40] (* or *) Table[2n(2n+1), {n, 0, 40}] (* From Harvey P. Dale, Aug 11 2011 *)
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PROG
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(PARI) a(n)=2*n*(2*n+1) \\ Charles R Greathouse IV, Nov 20 2012
(MAGMA) [ 4*n^2+2*n: n in [0..50]]; // Vincenzo Librandi, Nov 25 2012
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CROSSREFS
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Cf. A007742, A033954, A046092, A054000, A014105, A007395, A016813.
Same as A033951 except start at 0.
Sequences from spirals: A001107, A002939, A007742, A033951, A033952, A033953, A033954, A033989, A033990, A033991, A002943, A033996, A033988.
Sequence in context: A097811 A143711 A077539 * A068377 A009946 A094274
Adjacent sequences: A002940 A002941 A002942 * A002944 A002945 A002946
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Formula fixed by Reinhard Zumkeller, Apr 09 2010
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STATUS
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approved
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