A124148 Fibonacci triangle read by rows; the triangles below read by rows. Analog of A124171.

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 3, 5, 1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 3, 5, 1, 1, 2, 3, 5, 8, 1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 3, 5, 1, 1, 2, 3, 5, 8, 1, 1, 2, 3, 5, 8, 13, 1, 1

Offset: 1

Author

Jonathan Vos Post, Dec 13 2006

Keywords

• easy
• nonn
• tabf

Comments

The function is slow-growing at first. The smallest n such that a(n) > n occurs when a(816) = 987. But eventually, the superpolynomial Fibonacci dominates the merely cubic tetrahedral numbers and the mean value of a(n)/n exceeds any fixed bound. There is a slower-starting such analog that starts with F(0) = 0 and F(1) = 1, the triangles beginning: 0 0 0, 1 0 0, 1 0, 1, 1 0 0, 1 0, 1, 1 0, 1, 1, 2 0 0, 1 0, 1, 1 0, 1, 1, 2 0, 1, 1, 2, 3; reading by rows gives offset 0,36 and many zeros.

Examples

			1
1
1, 1
1
1, 1
1, 1, 2
1
1, 1
1, 1, 2
1, 1, 2, 3
1
1, 1
1, 1, 2
1, 1, 2, 3
1, 1, 2, 3, 5
		

Crossrefs

Cf. A000045, A000292, A124171.

Programs

Mathematica

Flatten[((Fibonacci@ Range@ # &) /@ Range@# &) /@ Range[10]] (* Giovanni Resta, Jun 16 2016 *)

Formula

a(n) = F(A124171(n)) = A000045(A124171(n)).

For k>0, max(row(T(k))) = F(k) where T(n) = A000217(k), F(k) = A000045(k).

Records for a(n) after a(1) = 1 are given by a(A000292(n)) = C(n+2,3) = n(n+1)(n+2)/6 = F(n+1) = A000045(n+1).

Extensions

Data corrected by Giovanni Resta, Jun 16 2016