A099900 XOR difference triangle, read by rows, of A099901 (in leftmost column) such that the main diagonal equals A099901 shift left and divided by 2.

1, 2, 3, 6, 4, 7, 14, 8, 12, 11, 22, 24, 16, 28, 23, 46, 56, 32, 48, 44, 59, 118, 88, 96, 64, 112, 92, 103, 206, 184, 224, 128, 192, 176, 236, 139, 278, 472, 352, 384, 256, 448, 368, 412, 279, 558, 824, 736, 896, 512, 768, 704, 944, 556, 827, 1654, 1112, 1888, 1408

Offset: 0

Author

Paul D. Hanna, Oct 29 2004

Keywords

• eigen
• nonn
• tabl

Comments

Central terms of rows equal powers of 2: T(n,[n/2]) = 2^n for n>=0. The leftmost column is A099901. The diagonal forms A099902 and equals the XOR BINOMIAL transform of A099901.

Examples

			Rows begin:
[_1],
[_2,3],
[6,_4,7],
[14,_8,12,11],
[22,24,_16,28,23],
[46,56,_32,48,44,59],
[118,88,96,_64,112,92,103],
[206,184,224,_128,192,176,236,139],
[278,472,352,384,_256,448,368,412,279],
[558,824,736,896,_512,768,704,944,556,827],
[1654,1112,1888,1408,1536,_1024,1792,1472,1648,1116,1895],...
notice that the column terms equal twice the diagonal (with offset), and that the central terms in the rows form the powers of 2.
		

Crossrefs

Cf. A099884, A099901, A099902.

Programs

PARI

T(n,k)=if(n
				

Formula

T(n, [n/2]) = 2^n. T(n+1, 0) = 2*T(n, n) (n>=0); T(0, 0)=1; T(n, k) = T(n, k-1) XOR T(n-1, k-1) for n>k>0. T(n, k) = SumXOR_{i=0..k} (C(k, i)mod 2)*T(n-i, 0), where SumXOR is the analog of summation under the binary XOR operation and C(k, i)mod 2 = A047999(k, i).