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%I #18 Jan 05 2023 03:21:19
%S 1,6,9,28,25,54,49,120,81,150,121,252,169,294,225,496,289,486,361,700,
%T 441,726,529,1080,625,1014,729,1372,841,1350,961,2016,1089,1734,1225,
%U 2268,1369,2166,1521,3000,1681,2646,1849,3388,2025,3174,2209,4464,2401,3750
%N G.f.: Sum_{k>=0} (2^k*t*(1+t)/(1-t)^3, t=x^2^k).
%H Amiram Eldar, <a href="/A091519/b091519.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = 2*n^2 - n*A000265(n) = n*A000265(n)*A038712(n).
%F Recurrence: a(0) = 0, a(2*n) = 2*a(n) + (2*n)^2, a(2*n+1) = (2*n+1)^2.
%F a((2*n-1)*2^p) = 2^p*(2^(p+1) - 1)*(2*n-1)^2, p >= 0. - _Johannes W. Meijer_, Jan 28 2013
%F Sum_{k=1..n} a(k) ~ (4/9) * n^3. - _Amiram Eldar_, Nov 29 2022
%F From _Amiram Eldar_, Jan 05 2023: (Start)
%F Multiplicative with a(2^e) = 2^e*(2^(e+1)-1), and a(p^e) = p^(2*e) for p >= 3.
%F Dirichlet g.f.: zeta(s-2)*2^s/(2^s-2).
%F Sum_{n>=1} 1/a(n) = (c-1)*Pi^2/4, where c = A065442 is Erdős-Borwein constant. (End)
%p 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
%t a[n_] := n^2*(2 - 1/2^IntegerExponent[n, 2]); Array[a, 50] (* _Amiram Eldar_, Nov 29 2022 *)
%o (PARI) a(n)=2*n*n-n*n/2^valuation(n,2)
%o (PARI) a(n)=if(n<1,0,if(n%2==0,2*a(n/2)+n^2,n^2))
%Y Cf. A000265, A038712, A065442.
%K nonn,mult,easy
%O 1,2
%A _Ralf Stephan_, Jan 18 2004
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