A099885 - OEIS

%I #12 May 04 2023 02:21:36

%S 1,2,6,12,20,40,120,240,272,544,1632,3264,5440,10880,32640,65280,

%T 65792,131584,394752,789504,1315840,2631680,7895040,15790080,17895424,

%U 35790848,107372544,214745088,357908480,715816960,2147450880,4294901760

%N 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 XOR BINOMIAL transform of this sequence is A099886.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Rule102.html">Rule 102</a>

%F a(n) = 2^floor((n+1)/2)*A001317(floor(n/2)), where A001317 forms the XOR BINOMIAL transform of the powers of 2.

%F It appears that a(2*n) = A117998(n). - _Peter Bala_, Feb 01 2017

%e XOR difference triangle of the powers of 2 (A099884) begins:

%e .

%e             (central terms)

%e                    |

%e                    |

%e                    1;

%e                    2,   3;

%e               4,   6,   5;

%e               8,  12,  10,  15;

%e         16,  24,  20,  30,  17;

%e         32,  48,  40,  60,  34,  51;

%e    64,  96,  80, 120,  68, 102,  85;

%e   128, 192, 160, 240, 136, 204, 170, 255;

%e   ...

%o (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 (Python)

%o 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 Cf. A099884, A001317, A099886, A117998.

%K nonn,easy

%O 0,2

%A _Paul D. Hanna_, Oct 28 2004