A362546 Number of odd chordless cycles of length >=5 in the n-Goldberg graph.

78, 296, 991, 3828, 15807, 63792, 259255, 1077860, 4515523, 18953864, 79937235, 338577316, 1437145747, 6108973856, 26001352815, 110768535036, 472147600567, 2013300270136, 8587441864815, 36635516602300, 156313089749627, 667000223816592, 2846311911402811

Offset

3,1

Links

Andrew Howroyd, Table of n, a(n) for n = 3..500

Eric Weisstein's World of Mathematics, Goldberg Graph.

Eric Weisstein's World of Mathematics, Odd Chordless Cycle.

Index entries for linear recurrences with constant coefficients, signature (12,-54,118,-153,112,327,-1122,358,1816,-1021,-1472,692,680,-132,-144,-16).

Formula

G.f.: x^3*(78 - 640*x + 1651*x^2 - 1284*x^3 + 391*x^4 + 434*x^5 - 11410*x^6 + 11758*x^7 + 23383*x^8 - 17740*x^9 - 23609*x^10 + 5880*x^11 + 8624*x^12 - 1048*x^13 - 1876*x^14 - 336*x^15 - 16*x^16)/((1 - x)^3*(1 + x)*(1 - 2*x)*(1 - x - x^2)*(1 - 2*x - 2*x^2)^2*(1 - 3*x - 19*x^3 - 17*x^4 - 2*x^5)). - Andrew Howroyd, May 26 2025

a(n) = 12*a(n-1)-54*a(n-2)+118*a(n-3)-153*a(n-4)+112*a(n-5)+327*a(n-6)-1122*a(n-7)+358*a(n-8)+1816*a(n-9)-1021*a(n-10)-1472*a(n-11)+692*a(n-12)+680*a(n-13)-132*a(n-14)-144*a(n-15)-16*a(n-16) for n > 17. - Eric W. Weisstein, Sep 04 2025

Mathematica

Join[{78}, Table[(4 n (6 n - 7) - 3 (2 (-1)^n + 2^n) - 3 LucasL[n] + 2^(n/2 + 1) n LucasL[n, Sqrt[2]] + 3 RootSum[-2 - 17 # - 19 #^2 - 3 #^4 + #^5 &, #^n &])/6, {n, 4, 20}] // Round] (* Eric W. Weisstein, Sep 04 2025 *)
Join[{78}, LinearRecurrence[{12, -54, 118, -153, 112, 327, -1122, 358, 1816, -1021, -1472, 692, 680, -132, -144, -16}, {296, 991, 3828, 15807, 63792, 259255, 1077860, 4515523, 18953864, 79937235, 338577316, 1437145747, 6108973856, 26001352815, 110768535036, 472147600567}, 20]] (* Eric W. Weisstein, Sep 04 2025 *)
CoefficientList[Series[-(-78 + 640 x - 1651 x^2 + 1284 x^3 - 391 x^4 - 434 x^5 + 11410 x^6 - 11758 x^7 - 23383 x^8 + 17740 x^9 + 23609 x^10 - 5880 x^11 - 8624 x^12 + 1048 x^13 + 1876 x^14 + 336 x^15 + 16 x^16)/((-1 + x)^3 (1 + x) (-1 + 2 x) (-1 + x + x^2) (-1 + 2 x + 2 x^2)^2 (-1 + 3 x + 19 x^3 + 17 x^4 + 2 x^5)), {x, 0, 20}], x] (* Eric W. Weisstein, Sep 04 2025 *)

Crossrefs

Cf. A362542.

Sequence in context: A257484 A068130 A118938 * A206004 A317412 A231393

Adjacent sequences: A362543 A362544 A362545 * A362547 A362548 A362549

Keyword

nonn,easy

Author

Eric W. Weisstein, Apr 24 2023

Extensions

a(10) from Eric W. Weisstein, May 23 2023

a(11) from Eric W. Weisstein, Feb 06 2024

a(12) onwards from Andrew Howroyd, May 26 2025

Status

approved