A353657 -id:A353657 - cp's OEIS frontend

A353655 Number of terms in the Fibonacci-Lucas representation of n.

1, 1, 1, 2, 1, 2, 3, 1, 2, 3, 2, 2, 1, 2, 3, 2, 2, 3, 3, 2, 1, 2, 3, 2, 2, 3, 3, 2, 3, 3, 3, 2, 3, 1, 2, 3, 2, 2, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 4, 3, 4, 2, 3, 3, 1, 2, 3, 2, 2, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 4, 3, 4, 2, 3, 3, 3, 4, 3, 4, 5, 3, 4, 5, 2, 3, 3

Offset: 1

Clark Kimberling, May 02 2022

nonn

The Fibonacci-Lucas representation of n, denoted by FL(n), is defined for n>=1 as the sum t(1) + t(2) + ... + t(k), where t(1) is the greatest Fibonacci number (A000045(n), with n>=2) that is <= n, and t(2) is the greatest Lucas number (A000032(n), with n >= 1) that is <= n - t(1), and so on; that is, the greedy algorithm is applied to find successive greatest Fibonacci and Lucas numbers, in alternating order, with sum n. (See Example.)

			  n      FL(n)
  1   =  1
  2   =  2
  3   =  3
  4   =  3 + 1
  5   =  5
  6   =  5 + 1
  33  =  21 + 11 + 1
  47  =  34 + 11 + 2
  83  =  55 + 18 + 8 + 1 + 1

Cf. A000032, A000045, A007895, A116543, A353656, A353657.

z = 120; fib = Map[Fibonacci, Range[2, 51]];
luc = Map[LucasL, Range[1, 50]];
t = Map[(n = #; fl = {}; f = 0; l = 0;
     While[IntegerQ[l], n = n - f - l;
      f = fib[[NestWhile[# + 1 &, 1, fib[[#]] <= n &] - 1]];
      l = luc[[NestWhile[# + 1 &, 1, luc[[#]] <= n - f &] - 1]];
      AppendTo[fl, {f, l}]];
     {Total[#], #} &[Select[Flatten[fl], IntegerQ]]) &, Range[z]];
u = Take[Map[Last, t], z];
u1 = Map[Length, u]  (* A353655 *)
t = Map[(n = #; lf = {}; f = 0; l = 0;
     While[IntegerQ[f], n = n - l - f;
      l = luc[[NestWhile[# + 1 &, 1, luc[[#]] <= n &] - 1]];
      f = fib[[NestWhile[# + 1 &, 1, fib[[#]] <= n - l &] - 1]];
      AppendTo[lf, {l, f}]];
     {Total[#], #} &[Select[Flatten[lf], IntegerQ]]) &, Range[z]];
v = Take[Map[Last, t], z];
v1 = Map[Length, v]   (* A353656 *)
u1 - v1  (* A353657 *)
(* Peter J. C. Moses *)

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

Original entry on oeis.org

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, 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, 3, 4, 3, 3, 4, 4, 3, 1, 2, 2, 2, 3, 2, 3, 4, 2, 3, 4

Offset: 1

Clark Kimberling, May 04 2022

nonn

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.)

			   n     LF(n)
   1  =  1
   2  =  1 + 1
   3  =  3
   4  =  4
   5  =  4 + 1
   6  =  4 + 2
  17  =  11 + 5 + 1
  66  =  47 + 13 + 4 + 2

Cf. A000032, A000045, A007895, A116543, A353655, A353657, A353658, A353659.

z = 120; fib = Map[Fibonacci, Range[2, 51]];
luc = Map[LucasL, Range[1, 50]];
t = Map[(n = #; fl = {}; f = 0; l = 0;
     While[IntegerQ[l], n = n - f - l;
      f = fib[[NestWhile[# + 1 &, 1, fib[[#]] <= n &] - 1]];
      l = luc[[NestWhile[# + 1 &, 1, luc[[#]] <= n - f &] - 1]];
      AppendTo[fl, {f, l}]];
     {Total[#], #} &[Select[Flatten[fl], IntegerQ]]) &, Range[z]];
u = Take[Map[Last, t], z];
u1 = Map[Length, u]  (* A353655 *)
t = Map[(n = #; lf = {}; f = 0; l = 0;
     While[IntegerQ[f], n = n - l - f;
      l = luc[[NestWhile[# + 1 &, 1, luc[[#]] <= n &] - 1]];
      f = fib[[NestWhile[# + 1 &, 1, fib[[#]] <= n - l &] - 1]];
      AppendTo[lf, {l, f}]];
     {Total[#], #} &[Select[Flatten[lf], IntegerQ]]) &, Range[z]];
v = Take[Map[Last, t], z];
v1 = Map[Length, v]   (* A353656 *)
u1 - v1  (* A353657 *)
(* Peter J. C. Moses, May 04 2022 *)

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A353655 Number of terms in the Fibonacci-Lucas representation of n.

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