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 A090889 Double partial sums of (n * its dyadic valuation). 1
 0, 0, 2, 4, 14, 24, 40, 56, 96, 136, 186, 236, 310, 384, 472, 560, 712, 864, 1034, 1204, 1414, 1624, 1856, 2088, 2392, 2696, 3026, 3356, 3742, 4128, 4544, 4960, 5536, 6112, 6722, 7332, 8014, 8696, 9416, 10136, 10976, 11816, 12698, 13580 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS FORMULA 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). G.f.: 1/(1-x)^2 * sum(k>=0, 2^k*t^2/(1-t^2)^2, t=x^2^k). PROG (PARI) a(n)=sum(k=1, n, bitand(k, n-k)+k*(n-k)) (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))) (PARI) a(n)=sum(l=0, n, sum(k=0, l, k*valuation(k, 2))) CROSSREFS A006851(n)+A000292(n-2). Cf. A007814. Equals A006581(n) + A000292(n-2). Cf. A007814. Sequence in context: A074716 A283290 A283379 * A096012 A261562 A102930 Adjacent sequences:  A090886 A090887 A090888 * A090890 A090891 A090892 KEYWORD nonn,easy AUTHOR Ralf Stephan, Feb 13 2004 STATUS approved

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Last modified May 16 22:04 EDT 2021. Contains 343955 sequences. (Running on oeis4.)