A214980 -id:A214980 - cp's OEIS frontend

A214979 A179180 - A214977.

0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 2, 1, 1, 1, 0, 0, 1, 2, 3, 2, 2, 1, 0, 1, 0, 0, 0, 1, 2, 3, 4, 6, 4, 3, 3, 2, 2, 1, 2, 2, 1, 1, 1, 0, 0, 1, 2, 3, 4, 6, 6, 7, 9, 7, 6, 5, 5, 5, 4, 4, 4, 2, 1, 2, 2, 3, 2, 2, 1, 0, 1, 0, 0, 0, 1, 2, 3, 4, 6, 6, 7, 9, 9, 10, 11, 13, 15, 13, 12, 11, 9, 8, 8, 8, 8

Offset: 1

Clark Kimberling, Oct 22 2012

nonn

Let U(n) and V(n) be the number of terms in the Lucas representations and Zeckendorf (Fibonacci) representations, respectively, of all the numbers 1,2,...,n. Then a(n) = V(n) - U(n). Conjecture: a(n) >= 0 for all n, and a(n) = 0 for infinitely many n.

			(See A214977 and A179180.)

Clark Kimberling, Table of n, a(n) for n = 1..10000
Clark Kimberling, Lucas Representations of Positive Integers, J. Int. Seq., Vol. 23 (2020), Article 20.9.5.

Cf. A214977, A179180, A214980, A214981, A000032, A000045.

z = 200;
s = Reverse[Sort[Table[LucasL[n - 1], {n, 1, 70}]]];
t1 = Map[Length[Select[Reap[FoldList[(Sow[Quotient[#1, #2]]; Mod[#1, #2]) &, #, s]][[2,1]], # > 0 &]] &, Range[z]];
u[n_] := Sum[t1[[k]], {k, 1, n}]
u1 = Table[u[n], {n, 1, z}] (* A214977 *)
s = Reverse[Table[Fibonacci[n + 1], {n, 1, 70}]];
t2 = Map[Length[Select[Reap[FoldList[(Sow[Quotient[#1, #2]]; Mod[#1, #2]) &, #, s]][[2,1]], # > 0 &]] &, Range[z]];
v[n_] := Sum[t2[[k]], {k, 1, n}]
v1 = Table[v[n], {n, 1, z}]  (* A179180 *)
w=v1-u1 (* A214979 *)
Flatten[Position[w, 0]]  (* A214980 *)
(* Peter J. C. Moses, Oct 18 2012 *)

A214977 Number of terms in Lucas representations of 1,2,...,n.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 11, 13, 15, 16, 18, 20, 22, 24, 27, 30, 31, 33, 35, 37, 39, 42, 45, 47, 50, 53, 56, 57, 59, 61, 63, 65, 68, 71, 73, 76, 79, 82, 84, 87, 90, 93, 96, 100, 104, 105, 107, 109, 111, 113, 116, 119, 121, 124, 127, 130, 132, 135, 138, 141, 144

Offset: 1

Clark Kimberling, Oct 22 2012

nonn

See the conjecture at A214979.

			n..Lucas(n)..# terms...A214977(n)
1..1.........1.........1
2..2.........1.........2
3..3.........1.........3
4..4.........1.........4
5..4+1.......2.........6
6..4+2.......2.........8
7..7.........1.........9
8..7+1.......2.........11
9..7+2.......2.........13

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A214978, A214979, A214980, A214981, A000032.

z = 200; s = Reverse[Sort[Table[LucasL[n - 1], {n, 1, 70}]]]; t1 = Map[Length[Select[Reap[FoldList[(Sow[Quotient[#1, #2]]; Mod[#1, #2]) &, #, s]][[2,1]], # > 0 &]] &, Range[z]]; u[n_] := Sum[t1[[k]], {k, 1, n}]; u1 = Table[u[n], {n, 1, z}]
(* Peter J. C. Moses, Oct 18 2012 *)

A214981 Number of terms in the greedy Lucas-and-Fibonacci representations of 1,2,...,n; partial sums of A214973.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 14, 16, 17, 19, 21, 23, 25, 26, 28, 30, 31, 33, 35, 37, 39, 41, 44, 46, 47, 49, 51, 53, 55, 56, 58, 60, 62, 64, 66, 69, 71, 73, 76, 79, 81, 84, 85, 87, 89, 91, 93, 95, 98, 100, 101, 103, 105, 107, 109, 111, 114, 116, 118, 121, 124

Offset: 1

Clark Kimberling, Oct 22 2012

nonn

For comparison with Zeckendorf (Fibonacci) representations, it is conjectured that the limit of A179180(n)/A214981(n) exists and is between 1.2 and 1.4.

			The basis is B = (1,2,3,4,5,7,8,11,13,18,21,29,34,47,55,...), composed of Fibonacci numbers and Lucas numbers. Representations of positive integers using the greedy algorithm on B:
   n  repres.   # terms  a(n)
   1   1        1        1
   2   2        1        2
   3   3        1        3
   4   4        1        4
   5   5        1        5
   6   5+1      2        7
   7   7        1        8
   8   8        1        9
   9   8+1      2       11
  10   8+2      2       13
  27   21+5+1   3       44

Clark Kimberling, Table of n, a(n) for n = 1..10000
Clark Kimberling, Lucas Representations of Positive Integers, J. Int. Seq., Vol. 23 (2020), Article 20.9.5.

Cf. A000032, A000045, A179180, A214973, A214977, A214979, A214980, A214981.

Mathematica
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(See the program at A214973.)
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Edited by Clark Kimberling, Jun 13 2020

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A214979 A179180 - A214977.

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A214977 Number of terms in Lucas representations of 1,2,...,n.

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A214981 Number of terms in the greedy Lucas-and-Fibonacci representations of 1,2,...,n; partial sums of A214973.

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