A084432 Expansion of 2/(1-x) + Sum_{k>=0} t^2(3-t)/(1+t)/(1-t)^2, where t=x^2^k.

2, 5, 4, 10, 6, 11, 8, 19, 10, 17, 12, 24, 14, 23, 16, 36, 18, 29, 20, 38, 22, 35, 24, 49, 26, 41, 28, 52, 30, 47, 32, 69, 34, 53, 36, 66, 38, 59, 40, 79, 42, 65, 44, 80, 46, 71, 48, 98, 50, 77, 52, 94, 54, 83, 56, 109, 58, 89, 60, 108, 62, 95, 64, 134, 66, 101, 68, 122, 70, 107, 72, 139, 74, 113

Offset

1,1

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Formula

a(1)=2, a(2*n) = a(n)+2*n+1, a(2*n+1) = 2*n+2.

Dirichlet g.f.: 2^s/(2^s-1) * (zeta(s)+zeta(s-1)). - Ralf Stephan, Jun 17 2007

From Seiichi Manyama, May 27 2024: (Start)

G.f. A(x) satisfies A(x) = 1/(1 - x)^2 - 1 + A(x^2).

G.f.: A(x) = Sum_{k>=0} (1/(1 - x^(2^k))^2 - 1). (End)

Conjecture: a(n) is the number of integer solutions (x,y,z) to the Diophantine equation 2^x*(y+z) = n, where 0 <= x,y,z <= n. - Joseph M. Shunia, Aug 27 2024

Sum_{k=1..n} a(k) ~ 2*n^2/3 + 2*n. - Vaclav Kotesovec, Aug 28 2024

Prog

(PARI) for(n=1, 100, l=ceil(log(n)/log(2)); t=polcoeff(sum(k=0, l, 1/(1-x^2^k)^2) + O(x^(n+1)), n); print1(t", "))

Crossrefs

Cf. A069895, A373184, A373185.

Sequence in context: A339809 A360896 A375888 * A071297 A356875 A114393

Adjacent sequences: A084429 A084430 A084431 * A084433 A084434 A084435

Keyword

nonn,easy

Author

Ralf Stephan, Jun 27 2003

Status

approved