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 A010006 Coordination sequence for C_3 lattice: a(n)=16*n^2+2 (n>0), a(0)=1. 13
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..10000 R. Bacher, P. de la Harpe and B. Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142. Milan Janjic, Two Enumerative Functions Joan Serra-Sagrista, Enumeration of lattice points in l_1 norm, Inf. Proc. Lett. 76 (1-2) (2000) 39-44. Index entries for linear recurrences with constant coefficients, signature (3, -3, 1). FORMULA 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: Sum(binomial(2*n, 2*i)*z^i, i=0..n)/(1-z)^n. E.g.f.: (x*(x+1)*16+2)*e^x-1. - Gopinath A. R., Feb 14 2012 a(0)=1, a(1)=18, a(2)=66, a(3)=146, a(n) = 3*a(n-1)-3*a(n-2)+a(n-3). - Harvey P. Dale, Oct 15 2012 From Peter Bala, Apr 09 2017: (Start) 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. (End) MATHEMATICA t = Table[n^2 + (n + 1)^2, {n, 1, 100}]; Join[{1}, Table[t[[n]] + t[[n + 1]], {n, 1, Length[t] - 1, 2}]] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2012 *) Join[{1}, Table[16n^2+2, {n, 50}]] (* or *) Join[{1}, LinearRecurrence[ {3, -3, 1}, {18, 66, 146}, 50]] (* Harvey P. Dale, Oct 15 2012 *) PROG (PARI) A010006(n)=16*n^2+2-!n   \\ M. F. Hasler, Feb 14 2012 (MAGMA) [1], [16*n^2+2: n in [1..50]]; // Vincenzo Librandi, Feb 20 2012 CROSSREFS Cf. A206399. For the coordination sequences of other C_n lattices see A022144 (C_2), A010006 (C_3), A019560 - A019564 (C_4 through C_8), A035746 - A035787 (C_9 through C_50). Cf. A137513. Sequence in context: A259634 A165029 A264652 * A237616 A044156 A044537 Adjacent sequences:  A010003 A010004 A010005 * A010007 A010008 A010009 KEYWORD nonn,easy AUTHOR N. J. A. Sloane, mbaake(AT)sunelc3.tphys.physik.uni-tuebingen.de (Michael Baake) STATUS approved

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