A118654 Square array T(n,k) read by antidiagonals: T(n,k) = 2^n*Fibonacci(k) - Fibonacci(k-2).

1, 1, 0, 1, 1, 1, 1, 3, 2, 1, 1, 7, 4, 3, 2, 1, 15, 8, 7, 5, 3, 1, 31, 16, 15, 11, 8, 5, 1, 63, 32, 31, 23, 18, 13, 8, 1, 127, 64, 63, 47, 38, 29, 21, 13, 1, 255, 128, 127, 95, 78, 61, 47, 34, 21, 1, 511, 256, 255, 191, 158, 125, 99, 76, 55, 34

Offset: 0

Author

Ross La Haye, May 17 2006

Keywords

• nonn
• tabl

Comments

Inverse binomial transform (by columns) of A090888.

Examples

			T(2,3) = 7 because 2^2(Fibonacci(3)) - Fibonacci(3-2) = 4*2 - 1 = 7.
{1};
{1,  0};
{1,  1,  1};
{1,  3,  2,  1};
{1,  7,  4,  3,  2};
{1, 15,  8,  7,  5,  3};
{1, 31, 16, 15, 11,  8,  5};
{1, 63, 32, 31, 23, 18, 13,  8};
		

Crossrefs

Rows: T(0,k) = A000045(k-1), for k > 0; T(1,k) = A000045(k+1); T(2,k) = A000032(k+1); T(3,k) = A022097(k); T(4,k) = A022105(k); T(5,k) = A022401(k).

Columns: T(n,1) = A000225(n); T(n,2) = A000079(n); T(n,3) = A000225(n+1); T(n,4) = A055010(n+1); T(n,5) = A051633(n); a(T,6) = A036563(n+3).

Formula

T(n,k) = 2^n*Fibonacci(k) - Fibonacci(k-2).

T(n,k) = (2^n-2)*Fibonacci(k) + Fibonacci(k+1).

T(n,0) = 1; T(n,1) = 2^n - 1; T(n,k) = T(n,k-1) + T(n,k-2), for k > 1.

T(0,k) = Fibonacci(k-1); T(1,k) = Fibonacci(k+1); T(n,k) = 3T(n-1,k) - 2T(n-2,k), for n > 1.

T(n,k) = 2T(n-1,k) + Fibonacci(k-2), for n > 0.

T(n,k) = A109754(2^n-2, k+1) = A101220(2^n-2, 0, k+1), for n > 0.

O.g.f. (by rows) = (1+(-2+2^n)x)/(1-x-x^2).

Sum_{k=0..n} T(n-k,k) = A119587(n+1). - Ross La Haye, May 31 2006