A137712 Triangle read by rows: T(n,k) = T(n-1, k-1) - T(n-k, k-1); with left border = the Fibonacci sequence.

1, 1, 1, 2, 0, 1, 3, 1, 0, 1, 5, 1, 0, 0, 1, 8, 2, 1, 0, 0, 1, 13, 3, 1, 0, 0, 0, 1, 21, 5, 2, 1, 0, 0, 0, 1, 34, 8, 3, 2, 0, 0, 0, 0, 1, 55, 13, 5, 2, 2, 0, 0, 0, 0, 1, 89, 21, 8, 4, 2, 1, 0, 0, 0, 0, 1, 144, 34, 13, 6, 4, 2, 1, 0, 0, 0, 0, 1, 233, 55, 21, 10, 5, 4, 1, 1, 0, 0, 0, 0, 1

Offset: 1

Author

Gary W. Adamson, Feb 08 2008

Keywords

• nonn
• tabl

Comments

Row sums = A137713: (1, 2, 3, 5, 7, 12, 18, 30, 48, 78, 126, ...).

A137710 is the analogous triangle with left border = (1, 2, 4, 8, 16, 32, ...).

Examples

			First few rows of the triangle:
    1;
    1,  1;
    2,  0,  1;
    3,  1,  0,  1;
    5,  1,  0,  0, 1;
    8,  2,  1,  0, 0, 1;
   13,  3,  1,  0, 0, 0, 1;
   21,  5,  2,  1, 0, 0, 0, 1;
   34,  8,  3,  2, 0, 0, 0, 0, 1;
   55, 13,  5,  2, 2, 0, 0, 0, 0, 1;
   89, 21,  8,  4, 2, 1, 0, 0, 0, 0, 1;
  144, 34, 13,  6, 4, 2, 1, 0, 0, 0, 0, 1;
  233, 55, 21, 10, 5, 4, 1, 1, 0, 0, 0, 0, 1;
  377, 89, 34, 16, 8, 5, 4, 1, 1, 0, 0, 0, 0, 1;
  ...
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10011(rows 1 to 141, flattened)

Crossrefs

Cf. A137713, A137710.

Programs

Maple

for n from 1 to 20 do
  T[n,1]:= combinat:-fibonacci(n);
  for k from 2 to n do
    if n >= 2*k-1 then T[n,k]:= T[n-1,k-1] - T[n-k,k-1]
    else T[n,k]:= T[n-1,k-1]
    fi
  od:
od:
seq(seq(T[n,k],k=1..n),n=1..20); # Robert Israel, Aug 20 2018

Formula

T(n,k) = T(n-1, k-1) - T(n-k, k-1), given left border = (1, 1, 2, 3, 5, 8, 13, ...).

Here T(n,k) = T(n-1,k-1) if n-k < k-1. - Robert Israel, Aug 20 2018

Extensions

Corrected by Robert Israel, Aug 20 2018