A091519 G.f.: Sum_{k>=0} (2^k*t*(1+t)/(1-t)^3, t=x^2^k).

1, 6, 9, 28, 25, 54, 49, 120, 81, 150, 121, 252, 169, 294, 225, 496, 289, 486, 361, 700, 441, 726, 529, 1080, 625, 1014, 729, 1372, 841, 1350, 961, 2016, 1089, 1734, 1225, 2268, 1369, 2166, 1521, 3000, 1681, 2646, 1849, 3388, 2025, 3174, 2209, 4464, 2401, 3750

Offset: 1

Author

Ralf Stephan, Jan 18 2004

Keywords

• nonn
• mult
• easy

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000265, A038712, A065442.

Programs

Maple

nmax:=47: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 1 to ceil(nmax/(p+2)) do a((2*n-1)*2^p) := 2^p*(2^(p+1) - 1)*(2*n-1)^2 od: od: seq(a(n), n=1..nmax); # Johannes W. Meijer, Jan 28 2013

Mathematica

a[n_] := n^2*(2 - 1/2^IntegerExponent[n, 2]); Array[a, 50] (* Amiram Eldar, Nov 29 2022 *)

PARI

a(n)=2*n*n-n*n/2^valuation(n,2)

PARI

a(n)=if(n<1,0,if(n%2==0,2*a(n/2)+n^2,n^2))

Formula

a(n) = 2*n^2 - n*A000265(n) = n*A000265(n)*A038712(n).

Recurrence: a(0) = 0, a(2*n) = 2*a(n) + (2*n)^2, a(2*n+1) = (2*n+1)^2.

a((2*n-1)*2^p) = 2^p*(2^(p+1) - 1)*(2*n-1)^2, p >= 0. - Johannes W. Meijer, Jan 28 2013

Sum_{k=1..n} a(k) ~ (4/9) * n^3. - Amiram Eldar, Nov 29 2022

From Amiram Eldar, Jan 05 2023: (Start)

Multiplicative with a(2^e) = 2^e*(2^(e+1)-1), and a(p^e) = p^(2*e) for p >= 3.

Dirichlet g.f.: zeta(s-2)*2^s/(2^s-2).

Sum_{n>=1} 1/a(n) = (c-1)*Pi^2/4, where c = A065442 is Erdős-Borwein constant. (End)