cp's OEIS Frontend

A353656 Number of terms in the Lucas-Fibonacci representation of n.

%I A353656 #17 Dec 26 2024 21:43:43
%S A353656 1,2,1,1,2,2,1,2,2,2,1,2,2,2,3,2,3,1,2,2,2,3,2,3,4,2,3,4,1,2,2,2,3,2,
%T A353656 3,4,2,3,4,3,3,2,3,4,3,3,1,2,2,2,3,2,3,4,2,3,4,3,3,2,3,4,3,3,4,4,3,2,
%U A353656 3,4,3,3,4,4,3,1,2,2,2,3,2,3,4,2,3,4
%N A353656 Number of terms in the Lucas-Fibonacci representation of n.
%C A353656 The Lucas-Fibonacci representation of n, denoted by LF(n), is defined for n>=1 as the sum t(1) + t(2) + ... + t(k), where t(1) is the greatest Lucas number (A000032(n), with n >= 1) that is <= n, and t(2) is the greatest Fibonacci number (A000045(n), with n >= 2) that is <= n - t(1), and so on; that is, the greedy algorithm is applied to find successive greatest Lucas and Fibonacci numbers, in alternating order, with sum n. (See Example.)
%e A353656    n     LF(n)
%e A353656    1  =  1
%e A353656    2  =  1 + 1
%e A353656    3  =  3
%e A353656    4  =  4
%e A353656    5  =  4 + 1
%e A353656    6  =  4 + 2
%e A353656   17  =  11 + 5 + 1
%e A353656   66  =  47 + 13 + 4 + 2
%t A353656 z = 120; fib = Map[Fibonacci, Range[2, 51]];
%t A353656 luc = Map[LucasL, Range[1, 50]];
%t A353656 t = Map[(n = #; fl = {}; f = 0; l = 0;
%t A353656      While[IntegerQ[l], n = n - f - l;
%t A353656       f = fib[[NestWhile[# + 1 &, 1, fib[[#]] <= n &] - 1]];
%t A353656       l = luc[[NestWhile[# + 1 &, 1, luc[[#]] <= n - f &] - 1]];
%t A353656       AppendTo[fl, {f, l}]];
%t A353656      {Total[#], #} &[Select[Flatten[fl], IntegerQ]]) &, Range[z]];
%t A353656 u = Take[Map[Last, t], z];
%t A353656 u1 = Map[Length, u]  (* A353655 *)
%t A353656 t = Map[(n = #; lf = {}; f = 0; l = 0;
%t A353656      While[IntegerQ[f], n = n - l - f;
%t A353656       l = luc[[NestWhile[# + 1 &, 1, luc[[#]] <= n &] - 1]];
%t A353656       f = fib[[NestWhile[# + 1 &, 1, fib[[#]] <= n - l &] - 1]];
%t A353656       AppendTo[lf, {l, f}]];
%t A353656      {Total[#], #} &[Select[Flatten[lf], IntegerQ]]) &, Range[z]];
%t A353656 v = Take[Map[Last, t], z];
%t A353656 v1 = Map[Length, v]   (* A353656 *)
%t A353656 u1 - v1  (* A353657 *)
%t A353656 (* _Peter J. C. Moses_, May 04 2022 *)
%Y A353656 Cf. A000032, A000045, A007895, A116543, A353655, A353657, A353658, A353659.
%K A353656 nonn
%O A353656 1,2
%A A353656 _Clark Kimberling_, May 04 2022