A201506 -id:A201506 - cp's OEIS frontend

A048634 a(n) = a(n-1)*a(n-3) + a(n-2), with a(0)=a(1)=0 and a(2)=1.

0, 0, 1, 0, 1, 1, 1, 2, 3, 5, 13, 44, 233, 3073, 135445, 31561758, 96989417779, 13136731722638413, 414618347540933702027833, 40213592128486236142855326045681320, 528275171395527518169753769210241662354568290572993

Offset: 0

N. J. A. Sloane, David(AT)interface.co.uk

A048634 := proc(n) option remember; if n<=1 then 0 elif n=2 then 1 else A048634(n-1)*A048634(n-3)+A048634(n-2); fi; end;

RecurrenceTable[{a[n] == a[n-1]*a[n-3] + a[n-2], a[0] == 0, a[1] == 0, a[2] == 1}, a, {n, 0, 20}] (* Vaclav Kotesovec, Aug 16 2021 *)

a(n) ~ c^(A092526^n), where c = A344388 = 1.0574735961... (very close to A201506). - Vaclav Kotesovec, Aug 16 2021

Name clarified by Michel Marcus, Aug 16 2021

A344388 Decimal expansion of a constant related to the asymptotics of A048634.

Original entry on oeis.org

1, 0, 5, 7, 4, 7, 3, 5, 9, 6, 1, 0, 2, 9, 3, 0, 7, 1, 4, 5, 8, 8, 3, 6, 1, 3, 6, 9, 0, 1, 1, 1, 7, 2, 1, 2, 3, 2, 5, 9, 5, 6, 8, 3, 4, 0, 4, 0, 1, 4, 9, 4, 6, 9, 5, 1, 9, 6, 0, 0, 8, 8, 9, 3, 4, 0, 8, 4, 1, 4, 1, 8, 9, 2, 9, 2, 5, 7, 3, 5, 3, 8, 4, 8, 5, 1, 2, 6, 6, 6, 7, 6, 0, 5, 3, 3, 5, 1, 0, 5, 5, 3, 4, 0, 1, 9

Offset: 1

Vaclav Kotesovec, Aug 16 2021

This constant is a very close to A201506.

Conjecture: It is equal to the limit of column "h^2" in the Table 1 in reference by Wright and Trefethen, p. 336.

			1.05747359610293071458836136901117212325956834040149469519600889340841418929257...

T. G. Wright and L. N. Trefethen, Computed Lyapunov constants for random recurrences with smooth coefficients, J. Comp. Appl. Math. 132 (2) (2001), 331-340, Table 1.

Cf. A048634, A092526, A201506.

A092526 = 1/3 + 2/(3*(116 + 12*Sqrt[93])^(1/3)) + (1/6)*(116 + 12*Sqrt[93])^(1/3); terms = 500; b = ConstantArray[0, terms]; b[[7]] = N[Log[2], 1000]; b[[8]] = N[Log[3], 1000]; b[[9]] = N[Log[5], 1000]; Quiet[Do[b[[n]] = b[[n-1]] + b[[n-3]] - Sum[Exp[k*(b[[n-2]] - b[[n-1]] - b[[n-3]])]/k*(-1)^k, {k, 1, 1000}], {n, 10, terms}]; Exp[Table[N[b[[n]]/A092526^n, 110], {n, Length[b] - 20, Length[b]}]]]

Equals exp(limit_{n->infinity} log(A048634(n)) / A092526^n ).

cp's OEIS Frontend

A048634 a(n) = a(n-1)*a(n-3) + a(n-2), with a(0)=a(1)=0 and a(2)=1.

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A344388 Decimal expansion of a constant related to the asymptotics of A048634.

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