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A090889
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Double partial sums of (n * its dyadic valuation).
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2
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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
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OFFSET
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0,3
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COMMENTS
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Hwang-Janson-Tsai paper, p. 39: "Note that the recurrence provided on OEIS for A090889 is incorrect (and the generating function misses a factor of 2)." - Michael De Vlieger, Oct 30 2022
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LINKS
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FORMULA
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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 where t=x^2^k.
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MATHEMATICA
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{0}~Join~Accumulate@ Accumulate@ Array[# IntegerExponent[#, 2] &, 43] (* Michael De Vlieger, Oct 30 2022 *)
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PROG
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(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)))
(Python)
def A090889(n): return (sum(k&n-k for k in range(1, n+1>>1))<<1)+(0 if n&1 else n>>1)+n*(n-1)*(n+1)//6 # Chai Wah Wu, May 08 2023
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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