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

1,4

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, ...).

Links

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

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

Example

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;
  ...

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

Crossrefs

Cf. A137713, A137710.

Sequence in context: A079221 A168019 A026794 * A194711 A168532 A181940

Adjacent sequences: A137709 A137710 A137711 * A137713 A137714 A137715

Keyword

nonn,tabl

Author

Gary W. Adamson, Feb 08 2008

Extensions

Corrected by Robert Israel, Aug 20 2018

Status

approved