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 A090889 Double partial sums of (n * its dyadic valuation). 1

%I

%S 0,0,2,4,14,24,40,56,96,136,186,236,310,384,472,560,712,864,1034,1204,

%T 1414,1624,1856,2088,2392,2696,3026,3356,3742,4128,4544,4960,5536,

%U 6112,6722,7332,8014,8696,9416,10136,10976,11816,12698,13580

%N Double partial sums of (n * its dyadic valuation).

%F a(0)=0, a(2n) = 2a(n) + 2a(n-1) + n(n+1)(2n+1)/3, a(2n+1) = 4a(n) + 2/3*(n+1)(n+2)(n+3).

%F G.f.: 1/(1-x)^2 * sum(k>=0, 2^k*t^2/(1-t^2)^2, t=x^2^k).

%o (PARI) a(n)=sum(k=1,n,bitand(k,n-k)+k*(n-k))

%o (PARI) a(n)=if(n<1,0,if(n%2==0,2*a(n/2)+2*a(n/2-1)+n/2*(n/2+1)*(n+1)/3,4*a((n-1)/2)+2/3*((n-1)/2)*((n-1)/2+1)*((n-1)/2+2)))

%o (PARI) a(n)=sum(l=0,n,sum(k=0,l,k*valuation(k,2)))

%Y A006851(n)+A000292(n-2). Cf. A007814.

%Y Equals A006581(n) + A000292(n-2). Cf. A007814.

%K nonn,easy

%O 0,3

%A _Ralf Stephan_, Feb 13 2004

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Last modified April 18 19:07 EDT 2021. Contains 343089 sequences. (Running on oeis4.)