A144117 Number of Fibonacci parts in the last section of the set of partitions of n.

1, 2, 3, 5, 8, 13, 17, 28, 37, 55, 72, 104, 135, 187, 243, 327, 419, 557, 705, 922, 1163, 1494, 1871, 2383, 2960, 3730, 4611, 5754, 7073, 8766, 10710, 13180, 16036, 19600, 23736, 28859, 34788, 42075, 50529, 60811, 72747, 87184, 103907, 124019, 147330

Offset: 1

Author

Omar E. Pol, Sep 11 2008

Keywords

• nonn

Comments

First differences of A144115.

Also number of Fibonacci parts in the n-th section of the set of partitions of any positive integer >= n. - Omar E. Pol, Jul 30 2015

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000045, A135010, A138121, A138137, A144115, A144116, A144118, A144120.

Programs

Maple

b:= proc(n) option remember; false end: l:= [0, 1]: for k to 100 do b(l[1]):= true; l:= [l[2], l[1]+l[2]] od: aa:= proc(n, i) option remember; local g, h; if n=0 then [1, 0] elif i=0 or n<0 then [0, 0] else g:= aa(n, i-1); h:= aa(n-i, i); [g[1]+h[1], g[2]+h[2] +`if`(b(i), h[1], 0)] fi end: a:= n-> aa(n, n)[2] -aa(n-1, n-1)[2]: seq(a(n), n=1..60); # Alois P. Heinz, Jul 28 2009

Mathematica

Clear[b]; b[] = False; l = {0, 1}; For[k = 1, k <= 100, k++, b[l[[1]]] = True; l = {l[[2]], l[[1]] + l[[2]]}]; aa[n, i_] := aa[n, i] = Module[{g, h}, If[n == 0, {1, 0}, If[i == 0 || n < 0, {0, 0}, g = aa[n, i - 1]; h = aa[n - i, i]; {g[[1]] + h[[1]], g[[2]] + h[[2]] + If[b[i], h[[1]], 0]}]]] ; a[n_] := aa[n, n][[2]] - aa[n - 1, n - 1][[2]]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Jul 30 2015, after Alois P. Heinz *)

Formula

a(n) = A144115(n) - A144115(n-1).

Extensions

More terms from Alois P. Heinz, Jul 28 2009