A054000 -id:A054000 - cp's OEIS frontend

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

Offset: 3

Samuel J. Bevins, Jun 01 2023

nonn

Sequence T(n,3) in A360572.

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.

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

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

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

A384125 Array read by antidiagonals: T(n,m) is the number of edges in the n X m rook graph K_n X K_m.

Original entry on oeis.org

0, 1, 1, 3, 4, 3, 6, 9, 9, 6, 10, 16, 18, 16, 10, 15, 25, 30, 30, 25, 15, 21, 36, 45, 48, 45, 36, 21, 28, 49, 63, 70, 70, 63, 49, 28, 36, 64, 84, 96, 100, 96, 84, 64, 36, 45, 81, 108, 126, 135, 135, 126, 108, 81, 45, 55, 100, 135, 160, 175, 180, 175, 160, 135, 100, 55

Offset: 1

Andrew Howroyd, May 20 2025

			Array begins:
=======================================
n\m |  1  2   3   4   5   6   7   8 ...
----+----------------------------------
  1 |  0  1   3   6  10  15  21  28 ...
  2 |  1  4   9  16  25  36  49  64 ...
  3 |  3  9  18  30  45  63  84 108 ...
  4 |  6 16  30  48  70  96 126 160 ...
  5 | 10 25  45  70 100 135 175 220 ...
  6 | 15 36  63  96 135 180 231 288 ...
  7 | 21 49  84 126 175 231 294 364 ...
  8 | 28 64 108 160 220 288 364 448 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 antidiagonals)
Eric Weisstein's World of Mathematics, Edge Count.
Eric Weisstein's World of Mathematics, Rook Graph.

Main diagonal is A045991.
Columns 1..6 are A000217(n-1), A000290, A045943, A054000, A269457(n-1), A067707.
Cf. A003991 (number of vertices), A360855 (triangles), A384120 (all cliques).

Table[#*Binomial[m, 2] + m*Binomial[#, 2] &[n - m + 1], {n, 11}, {m, n}] // Flatten (* Michael De Vlieger, May 22 2025 *)

T(n,m) = n*binomial(m,2) + m*binomial(n,2)

T(n,m) = n*binomial(m,2) + m*binomial(n,2).
T(n,m) = binomial(n*m,2) - 2*binomial(n,2)*binomial(m,2).
T(n,m) = T(m,n).

cp's OEIS Frontend

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

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