A099897 XOR difference triangle, read by rows, of A099898 (in leftmost column) such that the main diagonal equals A099898 shift left and divided by 4.

1, 4, 5, 20, 16, 21, 84, 64, 80, 69, 276, 320, 256, 336, 277, 1108, 1344, 1024, 1280, 1104, 1349, 5396, 4416, 5120, 4096, 5376, 4432, 5141, 20564, 17728, 21504, 16384, 20480, 17664, 21584, 16453, 65812, 86336, 70656, 81920, 65536, 86016, 70912

Offset

0,2

Comments

Central terms of rows equal powers of 4: T(n,[n/2]) = 4^n for n>=0. The leftmost column is A099898. The diagonal forms A099899 and equals the XOR BINOMIAL transform of A099898. See A099884 for the definitions of XOR difference triangle and the XOR BINOMIAL transform.

Links

Table of n, a(n) for n=0..42.

Formula

T(n, [n/2]) = 4^n. T(n+1, 0) = 4*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],
[_4,5],
[20,_16,21],
[84,_64,80,69],
[276,320,_256,336,277],
[1108,1344,_1024,1280,1104,1349],
[5396,4416,5120,_4096,5376,4432,5141],
[20564,17728,21504,_16384,20480,17664,21584,16453],
[65812,86336,70656,81920,_65536,86016,70912,82256,65813],...
notice that the column terms equal 4 times the diagonal (with offset), and that the central terms in the rows form the powers of 4.

Prog

(PARI) T(n, k)=if(n<k || k<0, 0, if(k==0, if(n==0, 1, 4*T(n-1, n-1)), bitxor(T(n, k-1), T(n-1, k-1))); )

Crossrefs

Cf. A099884, A099898, A099899, A099900.

Sequence in context: A042835 A193964 A337443 * A050251 A125995 A080610

Adjacent sequences: A099894 A099895 A099896 * A099898 A099899 A099900

Keyword

nonn,tabl

Author

Paul D. Hanna, Oct 30 2004

Status

approved