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A124148
Fibonacci triangle read by rows; the triangles below read by rows. Analog of A124171.
0
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
1,10
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.
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).
EXAMPLE
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
MATHEMATICA
Flatten[((Fibonacci@ Range@ # &) /@ Range@# &) /@ Range[10]] (* Giovanni Resta, Jun 16 2016 *)
CROSSREFS
KEYWORD
easy,nonn,tabf
AUTHOR
Jonathan Vos Post, Dec 13 2006
EXTENSIONS
Data corrected by Giovanni Resta, Jun 16 2016
STATUS
approved