A118654 - OEIS

%I #20 Dec 05 2016 06:12:58

%S 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,

%T 31,23,18,13,8,1,127,64,63,47,38,29,21,13,1,255,128,127,95,78,61,47,

%U 34,21,1,511,256,255,191,158,125,99,76,55,34

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

%C Inverse binomial transform (by columns) of A090888.

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

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

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

%F 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.

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

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

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

%F Sum_{k=0..n} T(n-k,k) = A119587(n+1). - _Ross La Haye_, May 31 2006

%e T(2,3) = 7 because 2^2(Fibonacci(3)) - Fibonacci(3-2) = 4*2 - 1 = 7.

%e {1};

%e {1,  0};

%e {1,  1,  1};

%e {1,  3,  2,  1};

%e {1,  7,  4,  3,  2};

%e {1, 15,  8,  7,  5,  3};

%e {1, 31, 16, 15, 11,  8,  5};

%e {1, 63, 32, 31, 23, 18, 13,  8};

%Y 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).

%Y 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).

%K nonn,tabl

%O 0,8

%A _Ross La Haye_, May 17 2006