A156070 Triangle read by rows based on the Fibonacci sequence A000045: t(n,m) = 1 + Fibonacci[n] - Fibonacci[m] - Fibonacci[n - m].

1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 3, 2, 1, 1, 3, 5, 5, 5, 3, 1, 1, 5, 8, 9, 9, 8, 5, 1, 1, 8, 13, 15, 16, 15, 13, 8, 1, 1, 13, 21, 25, 27, 27, 25, 21, 13, 1, 1, 21, 34, 41, 45, 46, 45, 41, 34, 21, 1

Offset: 0

Author

Roger L. Bagula, Feb 03 2009

Keywords

• nonn
• tabl

Comments

Row sums are {1, 2, 2, 4, 6, 12, 23, 46, 90, 174, 330,...} (see A188538).

More generally, we can define for a sequence with a(n)=0 : add one;

t(n,m)=1+a(n)-a(m)-a(n-m)

or a(0)=1: add two:

t(n,m)=2+a(n)-a(m)-a(n-m).

Examples

			{1},
{1, 1},
{1, 0, 1},
{1, 1, 1, 1},
{1, 1, 2, 1, 1},
{1, 2, 3, 3, 2, 1},
{1, 3, 5, 5, 5, 3, 1},
{1, 5, 8, 9, 9, 8, 5, 1},
{1, 8, 13, 15, 16, 15, 13, 8, 1},
{1, 13, 21, 25, 27, 27, 25, 21, 13, 1},
{1, 21, 34, 41, 45, 46, 45, 41, 34, 21, 1}
		

Crossrefs

Cf. A000045, A188538.

Programs

Mathematica

t[n_, m_] = 1 + Fibonacci[n] - Fibonacci[m] - Fibonacci[n - m];
Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];
Flatten[%]

Formula

t(n,m)=1 + Fibonacci[n] - Fibonacci[m] - Fibonacci[n - m].

Extensions

Edited by N. J. A. Sloane, Apr 03 2011