%I #11 Oct 29 2022 11:50:26

%S 0,0,1,0,2,4,3,0,4,8,13,16,14,12,7,0,8,16,25,32,42,52,59,64,60,56,53,

%T 48,38,28,15,0,16,32,49,64,82,100,115,128,148,168,189,208,222,236,247,

%U 256,248,240,233,224,218,212,203,192,172,152,133,112,86,60,31,0,32,64

%N Differences between partial sums of Gray code (A048641) and triangular numbers (A000217).

%C a(2^n-1) = 0 for all n.

%H Michael De Vlieger, <a href="/A048644/b048644.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. 40.

%F a(n) = sum(XORnos(j, floor(j/2)), j=0..n)-((n^2+n)/2).

%t {0}~Join~MapIndexed[#1 - PolygonalNumber[First[#2]] &, Accumulate[Array[BitXor[#, Floor[#/2]] &, 65]]] (* _Michael De Vlieger_, Oct 29 2022 *)

%o (PARI) a(n) = sum(k=0, n, bitxor(k, k>>1)) - n*(n+1)/2; \\ _Michel Marcus_, Oct 02 2015

%Y Cf. A000217, A048461, A048643.

%K easy,nonn

%O 0,5

%A _Antti Karttunen_, Jul 14 1999

%E Corrected by _Don Reble_, May 01 2006