OFFSET
1,1
LINKS
Christoph Koutschan, Table of n, a(n) for n = 1..75 (first 33 terms from R. H. Hardin)
Manuel Kauers and Christoph Koutschan, Some D-finite and some possibly D-finite sequences in the OEIS, arXiv:2303.02793 [cs.SC], 2023, pp. 19-23.
FORMULA
Conjectured recurrence of order 3 and degree 9: (n + 2)*(n + 3)^5*(2*n + 5)*(62*n^2 + 217*n + 191)*a(n + 3) - 9*(n + 2)*(3*n + 7)*(3*n + 8)*(5084*n^6 + 68634*n^5 + 383756*n^4 + 1137319*n^3 + 1884032*n^2 + 1653960*n + 601185)*a(n + 2) - 9*(3*n + 4)*(3*n + 5)*(3*n + 7)*(3*n + 8)*(31372*n^5 + 313720*n^4 + 1227805*n^3 + 2354425*n^2 + 2220988*n + 827860)*a(n + 1) + 20250*(n + 1)*(3*n + 1)*(3*n + 2)*(3*n + 4)*(3*n + 5)*(3*n + 7)*(3*n + 8)*(62*n^2 + 341*n + 470)*a(n) = 0. - Christoph Koutschan, Feb 26 2023
Conjecture: a(n) ~ 3^(3*n + 1/2) * 5^(3*n + 5/2) / (128 * Pi^(5/2) * n^(5/2)), based on the recurrence by Christoph Koutschan. - Vaclav Kotesovec, Feb 26 2023
From Peter Bala, Oct 28 2024: (Start)
Conjectures: 1) a(p) == 90 (mod p^2) for all primes p (checked up to p = 73).
2) a(n*p^r) == a(n*p^(r-1)) (mod p^(2*r)) for all primes p and all positive integers n and r. (End)
MATHEMATICA
Walks6D[n_Integer, steps_List] := Walks6D[n, steps] =
Module[{vals = {{{{{1}}}}}, seq = {}, n0, n1},
Do[
vals = Table[
n0 = Ceiling[(nsum - n1 - n2 - n3 - n4)/2];
Join[
Table[0, {n0}],
Table[Total[Function[s,
pos = Reverse[Sort[{n1, n2, n3, n4, n5, nsum - n1 - n2 - n3 - n4 - n5} - s]];
If[Min[pos] < 0, 0, vals[[##]]& @@ Most[pos + 1]]] /@ steps],
{n5, n0, Min[n4, nsum - n1 - n2 - n3 - n4]}]
],
{n1, 0, Min[n, nsum]},
{n2, 0, Min[n1, nsum - n1]},
{n3, 0, Min[n2, nsum - n1 - n2]},
{n4, 0, Min[n3, nsum - n1 - n2 - n3]}];
If[IntegerQ[n1 = nsum/6 + 1], AppendTo[seq, vals[[n1, n1, n1, n1, n1]]]],
{nsum, 2, 6 n, 2}];
Return[seq];
];
Walks6D[20, Permutations[{1, 1, 0, 0, 0, 0}]] (* Christoph Koutschan, Feb 26 2023 *)
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
R. H. Hardin, Feb 06 2010
STATUS
approved