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A010006
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Coordination sequence for C_3 lattice: a(n) = 16*n^2 + 2 (n>0), a(0)=1.
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14
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1, 18, 66, 146, 258, 402, 578, 786, 1026, 1298, 1602, 1938, 2306, 2706, 3138, 3602, 4098, 4626, 5186, 5778, 6402, 7058, 7746, 8466, 9218, 10002, 10818, 11666, 12546, 13458, 14402, 15378, 16386, 17426, 18498, 19602, 20738, 21906, 23106, 24338, 25602, 26898
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OFFSET
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0,2
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COMMENTS
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If Y_i (i=1,2,3) are 2-blocks of a (2n+1)-set X then a(n-1) is the number of 5-subsets of X intersecting each Y_i (i=1,2,3). - Milan Janjic, Oct 28 2007
Also sequence found by reading the segment (1, 18) together with the line from 18, in the direction 18, 66, ..., in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Nov 02 2012
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LINKS
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FORMULA
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a(0)=1, a(n) = 16*n^2 + 2, n >= 1.
G.f.: (1 + 15*x + 15*x^2 + x^3)/(1-x)^3.
G.f. for coordination sequence of C_n lattice: (1/(1-z)^n)*Sum_{i=0..n} binomial(2*n, 2*i)*z^i.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=1, a(1)=18, a(2)=66, a(3)=146. - Harvey P. Dale, Oct 15 2012
G.f. for sequence with interpolated zeros: cosh(6*arctanh(x)) = (1/2)*( ((1 - x)/(1 + x))^3 + ((1 + x)/(1 - x))^3) = 1 + 18*x^2 + 66*x^4 + 146*x^6 + .... More generally, cosh(2*n*arctanh(sqrt(x))) is the o.g.f. for the coordination sequence of the C_n lattice. Note that exp(t*arctanh(x)) is the e.g.f. for the Mittag_Leffler polynomials. See A137513. - Peter Bala, Apr 09 2017
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MATHEMATICA
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PROG
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane, mbaake(AT)sunelc3.tphys.physik.uni-tuebingen.de (Michael Baake)
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STATUS
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approved
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