login
Triangle read by rows: n-th row is n-th nonzero Fibonacci number repeated n+1 times.
4

%I #13 Nov 07 2024 20:40:16

%S 0,1,1,1,1,1,2,2,2,2,3,3,3,3,3,5,5,5,5,5,5,8,8,8,8,8,8,8,13,13,13,13,

%T 13,13,13,13,21,21,21,21,21,21,21,21,21,34,34,34,34,34,34,34,34,34,34,

%U 55,55,55,55,55,55,55,55,55,55,55,89,89,89,89,89,89,89,89,89,89,89,89,144,144

%N Triangle read by rows: n-th row is n-th nonzero Fibonacci number repeated n+1 times.

%H Reinhard Zumkeller, <a href="/A108037/b108037.txt">Rows n = 0..120 of triangle, flattened</a>

%F G.f.: x*(1+y-x*y)/((1-x-x^2)*(1-x*y-x^2*y^2)). [U coordinates]

%e 0; 1,1; 1,1,1; 2,2,2,2; 3,3,3,3,3; 5,5,5,5,5,5; ...

%t Table[Table[Fibonacci[n],{n+1}],{n,0,12}]//Flatten (* _Harvey P. Dale_, May 07 2017 *)

%o (Haskell)

%o a108037 n k = a108037_tabl !! n !! k

%o a108037_row n = a108037_tabl !! n

%o a108037_tabl = zipWith replicate [1..] a000045_list

%o -- _Reinhard Zumkeller_, Oct 07 2012

%o (Python)

%o from math import isqrt

%o from sympy import fibonacci

%o def A108037(n): return int(fibonacci((m:=isqrt(k:=n+1<<1))-(k<=m*(m+1)))) # _Chai Wah Wu_, Nov 07 2024

%Y Cf. A039913, A108036, A108035.

%Y Cf. A099920 (row sums).

%K nonn,tabl,changed

%O 0,7

%A _N. J. A. Sloane_, Jun 01 2005