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

0,2

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.

Links

Table of n, a(n) for n=0..58.

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).

Example

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.

Prog

(PARI) T(n, k)=if(n<k || k<0, 0, if(k==0, if(n==0, 1, 2*T(n-1, n-1)), bitxor(T(n, k-1), T(n-1, k-1))); )

Crossrefs

Cf. A099884, A099901, A099902.

Sequence in context: A092283 A266191 A273338 * A193903 A138728 A291604

Adjacent sequences: A099897 A099898 A099899 * A099901 A099902 A099903

Keyword

eigen,nonn,tabl

Author

Paul D. Hanna, Oct 29 2004

Status

approved