1,1

Seiichi Manyama, Table of n, a(n) for n = 1..24

Germain Kreweras, Complexité et circuits Eulériens dans les sommes tensorielles de graphes, J. Combin. Theory, B 24 (1978), 202-212. See p. 210, Parag. 4.

From Vaclav Kotesovec, Feb 26 2021: (Start)

a(n) ~ (21 + 12*sqrt(3) + 2*sqrt(2*(97 + 56*sqrt(3))))^n * n/4.

G.f.: 4*x*(1 + 310*x - 33278*x^2 + 785814*x^3 + 4923451*x^4 - 476492324*x^5 + 8394222196*x^6 - 74272031652*x^7 + 371582629705*x^8 - 981246223862*x^9 + 441533151262*x^10 + 6161037199338*x^11 - 23802532730757*x^12 + 46995963516168*x^13 - 58240430817576*x^14 + 46995963516168*x^15 - 23802532730757*x^16 + 6161037199338*x^17 + 441533151262*x^18 - 981246223862*x^19 + 371582629705*x^20 - 74272031652*x^21 + 8394222196*x^22 - 476492324*x^23 + 4923451*x^24 + 785814*x^25 - 33278*x^26 + 310*x^27 + x^28)/ ((1 - x)^2*(1 - 14*x + x^2)^2*(1 - 6*x + x^2)^2*(1 - 4*x + x^2)^2* (1 - 84*x + 230*x^2 - 84*x^3 + x^4)^2*(1 - 24*x + 50*x^2 - 24*x^3 + x^4)^2). (End)

Table[2^(6*n-4)*n*Product[Sin[j*Pi/4]^2 + Sin[k*Pi/n]^2, {j, 1, 3}, {k, 1, n-1}], {n, 1, 20}]//Round (* Vaclav Kotesovec, Feb 26 2021 *)

(Python)

# Using graphillion

from graphillion import GraphSet

def make_CnXCk(n, k):

grids = []

for i in range(1, k + 1):

for j in range(1, n):

grids.append((i + (j - 1) * k, i + j * k))

grids.append((i + (n - 1) * k, i))

for i in range(1, k * n, k):

for j in range(1, k):

grids.append((i + j - 1, i + j))

grids.append((i + k - 1, i))

return grids

def A212799(n):

if n == 1: return 4

if n == 2: return 2304

universe = make_CnXCk(4, n)

GraphSet.set_universe(universe)

spanning_trees = GraphSet.trees(is_spanning=True)

return spanning_trees.len()

print([A212799(n) for n in range(1, 8)]) # Seiichi Manyama, Nov 22 2020

Cf. A003696, A212796, A212798.

Sequence in context: A180993 A192889 A062407 * A343695 A097476 A047676

Adjacent sequences: A212796 A212797 A212798 * A212800 A212801 A212802

N. J. A. Sloane, May 27 2012

a(10)-a(15) from Seiichi Manyama, Nov 22 2020

approved