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A099885 Central terms of the rows of the XOR difference triangle of the powers of 2 (A099884) so that a(n) = A099884(n, floor(n/2)).

%I A099885 #13 Feb 16 2025 08:32:55
%S A099885 1,2,6,12,20,40,120,240,272,544,1632,3264,5440,10880,32640,65280,
%T A099885 65792,131584,394752,789504,1315840,2631680,7895040,15790080,17895424,
%U A099885 35790848,107372544,214745088,357908480,715816960,2147450880,4294901760
%N A099885 Central terms of the rows of the XOR difference triangle of the powers of 2 (A099884) so that a(n) = A099884(n, floor(n/2)).
%C A099885 XOR BINOMIAL transform of this sequence is A099886.
%H A099885 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Rule102.html">Rule 102</a>
%F A099885 a(n) = 2^floor((n+1)/2)*A001317(floor(n/2)), where A001317 forms the XOR BINOMIAL transform of the powers of 2.
%F A099885 It appears that a(2*n) = A117998(n). - _Peter Bala_, Feb 01 2017
%e A099885 XOR difference triangle of the powers of 2 (A099884) begins:
%e A099885 .
%e A099885             (central terms)
%e A099885                    |
%e A099885                    |
%e A099885                    1;
%e A099885                    2,   3;
%e A099885               4,   6,   5;
%e A099885               8,  12,  10,  15;
%e A099885         16,  24,  20,  30,  17;
%e A099885         32,  48,  40,  60,  34,  51;
%e A099885    64,  96,  80, 120,  68, 102,  85;
%e A099885   128, 192, 160, 240, 136, 204, 170, 255;
%e A099885   ...
%o A099885 (PARI) {a(n)=local(B);B=0;for(i=0,n\2,B=bitxor(B,binomial(n\2,i)%2*2^(n\2-i)));2^((n+1)\2)*B}
%o A099885 (Python)
%o A099885 def A099885(n): return sum((bool(~(m:=n>>1)&m-k)^1)<<k for k in range((n>>1)+1))<<(n+1>>1) # _Chai Wah Wu_, May 03 2023
%Y A099885 Cf. A099884, A001317, A099886, A117998.
%K A099885 nonn,easy
%O A099885 0,2
%A A099885 _Paul D. Hanna_, Oct 28 2004