A330638 a(n) = P(n)*a(n-1) + a(n-2), with a(0) = 0, a(1) = 1 where P(n) is the n-th Pell number.

0, 1, 2, 11, 134, 3897, 272924, 46128053, 18820518548, 18538256897833, 44083993723565422, 253086226505245985535, 3507775143446703083080522, 117373664327956358368203332177, 9481679355248745685146904663002936, 1849164516374760291573731440270350925577

Offset

0,3

Comments

The ratio of the ratios of the consecutive numbers of this sequence tends to the silver ratio (A014176 = 1 + sqrt(2)). This property can be proven.

Links

Table of n, a(n) for n=0..15.

Formula

a(n) = A000129(n)*a(n-1) + a(n-2) for n > 1.

a(n)*a(n-2)/a(n-1)^2 ~ 1 + sqrt(2).

a(n) ~ c * (1 + sqrt(2))^(n*(n+1)/2) / 2^(3*n/2), where c = 1.2795822677496757181586660872019163393334688780614201258263927413618707127... - Vaclav Kotesovec, Dec 29 2019

Mathematica

Nest[Append[#1, Fibonacci[#2, 2] #1[[-1]] + #1[[-2]] ] & @@ {#, Length@ #} &, {0, 1}, 14] (* Michael De Vlieger, Dec 22 2019 *)

Prog

(PARI) seq(n)={my(u=Vec(1/(1 - 2*x - x^2) + O(x^n)), v=vector(#u+1)); v[2]=1; for(n=2, #u, v[n+1] = u[n]*v[n] + v[n-1]); v} \\ Andrew Howroyd, Dec 22 2019

Crossrefs

Cf. A000129 (Pell numbers), A014176 (silver ratio), A135829.

Sequence in context: A282855 A139387 A041725 * A296571 A077544 A087480

Adjacent sequences: A330635 A330636 A330637 * A330639 A330640 A330641

Keyword

nonn

Author

Archit Handa, Dec 22 2019

Status

approved