OFFSET
1,2
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
FORMULA
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)
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 *)
PROG
(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))
CROSSREFS
KEYWORD
nonn,mult,easy
AUTHOR
Ralf Stephan, Jan 18 2004
STATUS
approved