

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.


3



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
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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, k1) XOR T(n1, k1) for n>k>0. T(n, k) = SumXOR_{i=0..k} (C(k, i)mod 2)*T(ni, 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(n1, n1)), bitxor(T(n, k1), T(n1, k1))); )


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



