A096657 a(n) = (2^n)*a(n-1) + (2^(n-1))*a(n-2), a(0)=1, a(1)=3.

1, 3, 14, 124, 2096, 69056, 4486656, 578711552, 148724449280, 76295068188672, 78202296743231488, 160236429879963287552, 656488575092059763900416, 5378610735570941915498020864, 88128536246001466497105446043648

Offset: 0

Author

Clark Kimberling, Jul 01 2004

Keywords

• nonn

Comments

This is the sequence of numerators of self-convergents to the number 1.40861... whose self-continued fraction is (1,2,4,8,16,...)=A000079. See A096658 for denominators and A096654 for definitions.

Examples

			a(2)=4*3+2*1=14, a(3)=8*14+4*3=124.
		

Crossrefs

Cf. A000079, A096654, A096658.

Programs

Mathematica

a[0] = 1; a[1] = 3; a[n_] := (2^n)*a[n-1] + (2^(n-1))*a[n-2]; Table[ a[n], {n, 0, 14}] (* Robert G. Wilson v, Jul 03 2004 *)
b[n_, k_] := k^2 - k (1 + n) +  n (1 + n)/2;
a[n_] := Sum[2^b[n, k] QBinomial[n - k + 1, k, 2], {k, 0, n + 1}] ;
Table[a[n], {n, 0, 14}] (* After Vladimir Kruchinin, Peter Luschny, Jan 19 2020 *)

Formula

a(n) is asymptotic to c*2^(n(n+1)/2) where c = 2.1726687508496636560169136... - Benoit Cloitre, Jul 02 2004

c = 1 + Sum_{k>=1} (Product_{j=1..k} 1/(2^(j-1)*(2^j-1))) = 2.172668750849663656016913609859312820656436935109608860295... . - Vaclav Kotesovec, Nov 27 2015

a(n) = Sum_{k=0..n+1} q-binomial(n-k+1,k)*2^(binomial(n-k+1,2)+binomial(k,2)), where q-binomial is triangle A022166, that is, with q=2. - Vladimir Kruchinin, Jan 19 2020

Extensions

More terms from Benoit Cloitre, Jul 02 2004