A090889 - OEIS

%I #24 Jul 22 2024 03:26:05

%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).

%C 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

%H Michael De Vlieger, <a href="/A090889/b090889.txt">Table of n, a(n) for n = 0..10000</a>

%H Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, <a href="https://arxiv.org/abs/2210.10968">Identities and periodic oscillations of divide-and-conquer recurrences splitting at half</a>, arXiv:2210.10968 [cs.DS], 2022, p. 39.

%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 where t=x^2^k.

%F a(n) = A006581(n) + A000292(n-1).

%t {0}~Join~Accumulate@ Accumulate@ Array[# IntegerExponent[#, 2] &, 43] (* _Michael De Vlieger_, Oct 30 2022 *)

%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)))

%o (Python)

%o 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

%Y Cf. A006581, A000292, A007814.

%K nonn,easy

%O 0,3

%A _Ralf Stephan_, Feb 13 2004