A017125 Table read by antidiagonals of Fibonacci-type sequences.

0, 1, 1, 1, 1, 2, 2, 2, 1, 3, 3, 3, 3, 1, 4, 5, 5, 4, 4, 1, 5, 8, 8, 7, 5, 5, 1, 6, 13, 13, 11, 9, 6, 6, 1, 7, 21, 21, 18, 14, 11, 7, 7, 1, 8, 34, 34, 29, 23, 17, 13, 8, 8, 1, 9, 55, 55, 47, 37, 28, 20, 15, 9, 9, 1, 10, 89, 89, 76, 60, 45, 33, 23, 17, 10, 10, 1, 11, 144, 144, 123, 97, 73

Offset: 0

Author

Henry Bottomley, Jul 31 2000

Keywords

• easy
• nonn
• tabl

Crossrefs

Rows are (essentially) A000045, A000045, A000032, A000285, A022095, A022096, A022097, etc. Columns are (essentially) A001477, A000012, A000027, A005408, A016789, A016885, etc. One of the diagonals is A007502.

Antidiagonal sums are in A019274.

Formula

T(n, k) = T(n, k-1)+T(n, k-2) [with T(n, 0) = n and T(n, 1) = 1] = 2*T(n-1, k)-T(n-2, k) = Fib(k)+n*Fib(k-1) = (s^k*(1+2n/s)-t^k*(1+2n/t))/(2^k*sqrt(5)) where s = (1+sqrt(5))/2 and t = (1-sqrt(5))/2 = 1-s.

G.f. for n-th row: (n+x-nx)/(1-x-x^2).