A363378 Third Lie-Betti number of a cycle graph on n vertices.

12, 25, 41, 68, 105, 152, 210, 280, 363, 460, 572, 700, 845, 1008, 1190, 1392, 1615, 1860, 2128, 2420, 2737, 3080, 3450, 3848, 4275, 4732, 5220, 5740, 6293, 6880, 7502, 8160, 8855, 9588, 10360, 11172, 12025, 12920, 13858, 14840, 15867, 16940, 18060

Offset

3,1

Comments

Sequence T(n,3) in A360572.

Links

Andrew Howroyd, Table of n, a(n) for n = 3..1000

M. Aldi and S. Bevins, L_oo-algebras and hypergraphs, arXiv:2212.13608 [math.CO], 2022. See page 9.

M. Mainkar, Graphs and two step nilpotent Lie algebras, arXiv:1310.3414 [math.DG], 2013. See page 1.

Eric Weisstein's World of Mathematics, Cycle Graph.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Formula

a(3) = 12, a(4) = 25, a(5) = 41, a(n) = n*(n+11)*(n-2)/6 for n >= 6.

a(n) = A005581(n-4) + A054000(n-1) + A028347(n-2) + A000027(n) for n >= 6.

a(n) = A106058(n+1) - 2 for n >= 6. - Hugo Pfoertner, Jun 02 2023

G.f.: x^3*(12 - 23*x + 13*x^2 + 6*x^3 - 9*x^4 + x^5 + x^6)/(1 - x)^4. - Andrew Howroyd, Nov 13 2025

Mathematica

LinearRecurrence[{4, -6, 4, -1}, {12, 25, 41, 68, 105, 152, 210}, 50] (* Paolo Xausa, Nov 14 2025 *)

Prog

(Python)
def A363378(n):
    values = [12, 25, 41]
    for i in range(6, n+1):
        result = (i*(i+11)*(i-2))/6
        values.append(result)
    return values
(PARI) a(n) = if(n<6, [12, 25, 41][n-2], n*(n+11)*(n-2)/6) \\ Andrew Howroyd, Nov 13 2025

Crossrefs

Cf. A005581, A054000, A028347, A000027, A360572 (cycle graph triangle)

Sequence in context: A041280 A132754 A250666 * A224669 A164577 A195143

Adjacent sequences: A363375 A363376 A363377 * A363379 A363380 A363381

Keyword

nonn,easy

Author

Samuel J. Bevins, Jun 01 2023

Status

approved