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

%I A124148 #14 Jun 16 2016 16:59:01
%S A124148 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,
%T A124148 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,
%U A124148 2,3,5,1,1,2,3,5,8,1,1,2,3,5,8,13,1,1
%N A124148 Fibonacci triangle read by rows; the triangles below read by rows. Analog of A124171.
%C A124148 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.
%F A124148 a(n) = F(A124171(n)) = A000045(A124171(n)).
%F A124148 For k>0, max(row(T(k))) = F(k) where T(n) = A000217(k), F(k) = A000045(k).
%F A124148 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).
%e A124148 1
%e A124148 1
%e A124148 1, 1
%e A124148 1
%e A124148 1, 1
%e A124148 1, 1, 2
%e A124148 1
%e A124148 1, 1
%e A124148 1, 1, 2
%e A124148 1, 1, 2, 3
%e A124148 1
%e A124148 1, 1
%e A124148 1, 1, 2
%e A124148 1, 1, 2, 3
%e A124148 1, 1, 2, 3, 5
%t A124148 Flatten[((Fibonacci@ Range@ # &) /@ Range@# &) /@ Range[10]] (* _Giovanni Resta_, Jun 16 2016 *)
%Y A124148 Cf. A000045, A000292, A124171.
%K A124148 easy,nonn,tabf
%O A124148 1,10
%A A124148 _Jonathan Vos Post_, Dec 13 2006
%E A124148 Data corrected by _Giovanni Resta_, Jun 16 2016