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)).

1, 2, 6, 12, 20, 40, 120, 240, 272, 544, 1632, 3264, 5440, 10880, 32640, 65280, 65792, 131584, 394752, 789504, 1315840, 2631680, 7895040, 15790080, 17895424, 35790848, 107372544, 214745088, 357908480, 715816960, 2147450880, 4294901760

Offset: 0

Author

Paul D. Hanna, Oct 28 2004

Keywords

• nonn
• easy

Comments

XOR BINOMIAL transform of this sequence is A099886.

Examples

			XOR difference triangle of the powers of 2 (A099884) begins:
.
            (central terms)
                   |
                   |
                   1;
                   2,   3;
              4,   6,   5;
              8,  12,  10,  15;
        16,  24,  20,  30,  17;
        32,  48,  40,  60,  34,  51;
   64,  96,  80, 120,  68, 102,  85;
  128, 192, 160, 240, 136, 204, 170, 255;
  ...
		

Links

• Eric Weisstein's World of Mathematics, Rule 102

Crossrefs

Cf. A099884, A001317, A099886, A117998.

Programs

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}

Python

def A099885(n): return sum((bool(~(m:=n>>1)&m-k)^1)<>1)+1))<<(n+1>>1) # Chai Wah Wu, May 03 2023

Formula

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

It appears that a(2*n) = A117998(n). - Peter Bala, Feb 01 2017