A362542 Number of chordless cycles of length >=4 in the n-Goldberg graph.

0, 29, 168, 617, 2028, 7755, 31790, 127921, 519144, 2156929, 9033368, 37912247, 159883298, 677172001, 2874325792, 12218015649, 52002840492, 221537338251, 944295735054, 4026601604297, 17174885851616, 73271037438905, 312626187952376, 1334000464514567, 5692623856528338

Offset

1,2

Comments

The number of chordless cycles in the 3-Goldberg graph is one less: 167. - Eric W. Weisstein, Aug 28 2025

Sequence extended to n = 1 using the formula/recurrence. - Eric W. Weisstein, Aug 28 2025

Links

Eric W. Weisstein, Table of n, a(n) for n = 1..500

Eric Weisstein's World of Mathematics, Chordless Cycle.

Eric Weisstein's World of Mathematics, Goldberg Graph.

Index entries for linear recurrences with constant coefficients, signature (9,-27,43,-82,62,171,-57,-231,-161,198,204,-52,-68,-8).

Formula

a(n) = 9*a(n-1)-27*a(n-2)+43*a(n-3)-82*a(n-4)+62*a(n-5)+171*a(n-6)-57*a(n-7)-231*a(n-8)-161*a(n-9)+198*a(n-10)+204*a(n-11)-52*a(n-12)-68*a(n-13)-8*a(n-14). - Eric W. Weisstein, Aug 28 2025

G.f.: -(x^2*(-29+93*x+112*x^2+236*x^3-1316*x^4-2198*x^5+789*x^6+6313*x^7+7528*x^8+4880*x^9+2308*x^10+660*x^11+64*x^12)/((-1+x)^3*(1+x+x^2)*(-1+2*x+2*x^2)^2*(-1+3*x+19*x^3+17*x^4+2*x^5))). - Eric W. Weisstein, Aug 28 2025

Mathematica

LinearRecurrence[{9, -27, 43, -82, 62, 171, -57, -231, -161, 198, 204, -52, -68, -8}, {0, 29, 168, 617, 2028, 7755, 31790, 127921, 519144, 2156929, 9033368, 37912247, 159883298, 677172001}, 20] (* Eric W. Weisstein, Aug 28 2025 *)
CoefficientList[Series[(167 - 886 x + 984 x^2 - 1019 x^3 + 3914 x^4 + 4232 x^5 - 7795 x^6 - 14284 x^7 - 9780 x^8 + 3273 x^9 + 5454 x^10 - 1368 x^11 - 2024 x^12 - 300 x^13 - 8 x^14)/((-1 + x)^3 (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, Aug 28 2025 *)
Table[1 + 2/3 (-20 + (1 - Sqrt[3])^n + (1 + Sqrt[3])^n) n + 9 n^2 + 2 ChebyshevT[n, -1/2] + RootSum[-2 - 17 # - 19 #^2 - 3 #^4 + #^5 & , #^n &], {n, 20}] // Expand (* Eric W. Weisstein, Aug 28 2025 *)
CoefficientList[Series[-(x (-29 + 93 x + 112 x^2 + 236 x^3 - 1316 x^4 - 2198 x^5 + 789 x^6 + 6313 x^7 + 7528 x^8 + 4880 x^9 + 2308 x^10 + 660 x^11 + 64 x^12)/((-1 + x)^3 (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, Aug 28 2025 *)

Crossrefs

Cf. A362546.

Sequence in context: A264252 A264530 A141910 * A111032 A238599 A033219

Adjacent sequences: A362539 A362540 A362541 * A362543 A362544 A362545

Keyword

nonn,easy

Author

Eric W. Weisstein, Apr 24 2023

Extensions

a(10) from Eric W. Weisstein, May 19 2023

a(11) from Eric W. Weisstein, Jun 09 2023

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

Extended to a(1) and g.f. adjusted by Eric W. Weisstein, Aug 28 2025

Status

approved