A116543 -id:A116543 - cp's OEIS frontend

A214974 Numbers k for which A116543(k) = A007895(k); i.e., the Lucas and Zeckendorf representations of k have the same length.

1, 2, 3, 6, 9, 10, 14, 15, 17, 22, 27, 28, 36, 38, 41, 43, 44, 46, 52, 58, 59, 61, 62, 66, 69, 74, 75, 81, 84, 94, 95, 96, 98, 107, 112, 114, 117, 119, 120, 122, 128, 131, 136, 139, 148, 152, 153, 154, 155, 159, 161, 164, 173, 175, 176, 181, 182, 184, 185

Offset: 1

Clark Kimberling, Oct 20 2012

nonn

			k...Lucas.....Zeckendorf....counter
1...1.........1.............a(1)= 1
2...2.........2.............a(2)= 2
3...3.........3.............a(3)= 3
4...4.........3+1
5...4+1.......5
6...4+2.......5+1...........a(4)= 6
7...7.........5+2
8...7+1.......8
9...7+2.......8+1...........a(5)= 9

Clark Kimberling, Table of n, a(n) for n = 1..8000

Cf. A116543, A007895, A214975, A214976, A214973.

u = Reverse[Sort[Table[LucasL[n - 1], {n, 1, 50}]]];
u1 = Map[Length[Select[Reap[FoldList[(Sow[Quotient[#1, #2]]; Mod[#1, #2]) &, #, u]][[2,1]], # > 0 &]] &, Range[1000]];
v = Reverse[Table[Fibonacci[n + 1], {n, 1, 50}]];
v1 = Map[Length[Select[Reap[FoldList[(Sow[Quotient[#1, #2]]; Mod[#1, #2]) &, #, v]][[2,1]], # > 0 &]] &, Range[1000]];
s[n_] := If[u1[[n]] == v1[[n]], 1, 0];
s1 = Table[s[n], {n, 1, 200}];
f1 = Flatten[Position[s1, 1]] (* A214974 *)
s[n_] := If[u1[[n]] < v1[[n]], 1, 0];
s2 = Table[s[n], {n, 1, 200}];
f2 = Flatten[Position[s2, 1]] (* A214975 *)
s[n_] := If[u1[[n]] > v1[[n]], 1, 0];
s3 = Table[s[n], {n, 1, 200}];
f3 = Flatten[Position[s3, 1]] (* A214976 *)
(* Peter J. C. Moses *)

A214975 Numbers k for which A116543(k) < A007895(k); i.e., the Lucas representation of k is shorter than the Zeckendorf representation.

Original entry on oeis.org

4, 7, 11, 12, 18, 19, 20, 25, 29, 30, 31, 32, 33, 40, 47, 48, 49, 50, 51, 53, 54, 65, 67, 72, 76, 77, 78, 79, 80, 82, 83, 85, 86, 87, 88, 101, 105, 106, 108, 109, 116, 123, 124, 125, 126, 127, 129, 130, 132, 133, 134, 135, 137, 138, 140, 141, 142, 143, 156

Offset: 1

Clark Kimberling, Oct 20 2012

nonn

			k...Lucas.....Zeckendorf....counter
1...1.........1
2...2.........2
3...3.........3
4...4.........3+1...........a(1) = 4
5...4+1.......5
6...4+2.......5+1
7...7.........5+2...........a(2) = 7
8...7+1.......8
9...7+2.......8+1

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

Cf. A116543, A007895, A214974, A214976.

Mathematica
```
(See A214974.)
```

A214976 Numbers k for which A116543(k) > A007895(k); i.e., the Lucas representation of k is longer than the Zeckendorf representation.

Original entry on oeis.org

5, 8, 13, 16, 21, 23, 24, 26, 34, 35, 37, 39, 42, 45, 55, 56, 57, 60, 63, 64, 68, 70, 71, 73, 89, 90, 91, 92, 93, 97, 99, 100, 102, 103, 104, 110, 111, 113, 115, 118, 121, 144, 145, 146, 147, 149, 150, 151, 157, 158, 160, 162, 165, 166, 167, 168, 169, 178

Offset: 1

Clark Kimberling, Oct 20 2012

nonn

			k...Lucas.....Zeckendorf....counter
1...1.........1
2...2.........2
3...3.........3
4...4.........3+1
5...4+1.......5.............a(1) = 5
6...4+2.......5+1
7...7.........5+2
8...7+1.......8.............a(2) = 8
9...7+2.......8+1

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

Cf. A116543, A007895, A214974, A214975.

Mathematica
```
(See A214974.)
```

A130310 Minimal (or "greedy") Lucas representation of n, in which L(0) = 2 and L(2) = 3 are not allowed in the same representation (hence the correct representation of the integer 5 is 1010 rather than 101). A binary system of integers with Lucas numbers (A000032) as a base.

Original entry on oeis.org

0, 10, 1, 100, 1000, 1010, 1001, 10000, 10010, 10001, 10100, 100000, 100010, 100001, 100100, 101000, 101010, 101001, 1000000, 1000010, 1000001, 1000100, 1001000, 1001010, 1001001, 1010000, 1010010, 1010001, 1010100, 10000000, 10000010, 10000001, 10000100, 10001000

Offset: 0

Casey Mongoven, May 21 2007

			a(9) = 10001 because 7 + 2 = 9.
a(10) is 10100 because 7 + 3 = 10.

Richard A. Dunlap, The Golden Ratio and Fibonacci Numbers, Singapore, World Scientific, 1997, pp. 73-77.
Edouard Zeckendorf, Représentation des nombres naturels par une somme des nombres de Fibonacci ou de nombres de Lucas, Bull. Soc. Roy. Sci. Liège, Vol. 41 (1972), pp. 179-182.

Amiram Eldar, Table of n, a(n) for n = 0..10000
Ron Knott, Using Powers of Phi to represent Integers.

Cf. A000032, A014417, A116543, A130311.

a[n_] := Module[{s = {}, m = n, k = 1}, While[m > 0, If[m == 1, k = 1; AppendTo[s, k]; m = 0, If[m == 2, k = 0; AppendTo[s, k]; m = 0, While[LucasL[k] <= m, k++]; k--; AppendTo[s, k]; m -= LucasL[k]; k = 1]]]; FromDigits @ IntegerDigits[Total[2^s], 2]]; Array[a, 30, 0] (* Amiram Eldar, Feb 17 2022 *)

Definition corrected by Casey Mongoven, May 29 2010

a(0) and more terms from Amiram Eldar, Feb 17 2022

A351714 Lucas-Niven numbers: numbers that are divisible by the number of terms in their minimal (or greedy) representation in terms of the Lucas numbers (A130310).

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 10, 11, 12, 14, 18, 20, 22, 24, 27, 29, 30, 32, 36, 39, 40, 42, 47, 48, 50, 54, 57, 58, 60, 64, 66, 69, 72, 76, 78, 80, 81, 84, 90, 92, 94, 96, 100, 104, 108, 120, 123, 124, 126, 129, 130, 132, 134, 135, 138, 140, 144, 152, 153, 156, 159, 160

Offset: 1

Amiram Eldar, Feb 17 2022

Numbers k such that A116543(k) | k.

			6 is a term since its minimal Lucas representation, A130310(6) = 1001, has A116543(6) = 2 1's and 6 is divisible by 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000032, A116543, A130310.
Subsequences: A351715, A351716.
Similar sequences: A005349, A049445, A064150, A064438, A064481, A118363, A328208, A328212, A331085, A333426, A342726, A334308, A331728, A342426, A344341, A351719.

lucasNivenQ[n_] := Module[{s = {}, m = n, k = 1}, While[m > 0, If[m == 1, k = 1; AppendTo[s, k]; m = 0, If[m == 2, k = 0; AppendTo[s, k]; m = 0, While[LucasL[k] <= m, k++]; k--; AppendTo[s, k]; m -= LucasL[k]; k = 1]]]; Divisible[n, Plus @@ IntegerDigits[Total[2^s], 2]]]; Select[Range[160], lucasNivenQ]

A265744 a(n) is the number of Pell numbers (A000129) needed to sum to n using the greedy algorithm (A317204).

Original entry on oeis.org

0, 1, 1, 2, 2, 1, 2, 2, 3, 3, 2, 3, 1, 2, 2, 3, 3, 2, 3, 3, 4, 4, 3, 4, 2, 3, 3, 4, 4, 1, 2, 2, 3, 3, 2, 3, 3, 4, 4, 3, 4, 2, 3, 3, 4, 4, 3, 4, 4, 5, 5, 4, 5, 3, 4, 4, 5, 5, 2, 3, 3, 4, 4, 3, 4, 4, 5, 5, 4, 5, 1, 2, 2, 3, 3, 2, 3, 3, 4, 4, 3, 4, 2, 3, 3, 4, 4, 3, 4, 4, 5, 5, 4, 5, 3, 4, 4, 5, 5, 2, 3, 3, 4, 4, 3, 4, 4, 5, 5, 4, 5, 3

Offset: 0

Antti Karttunen, Dec 17 2015

a(0) = 0, because no numbers are needed to form an empty sum, which is zero.

It would be nice to know for sure whether this sequence also gives the least number of Pell numbers that add to n, i.e., that there cannot be even better nongreedy solutions.

A. F. Horadam, Zeckendorf representations of positive and negative integers by Pell numbers, Applications of Fibonacci Numbers, Springer, Dordrecht, 1993, pp. 305-316.

Antti Karttunen, Table of n, a(n) for n = 0..13860
L. Carlitz, Richard Scoville, and V. E. Hoggatt, Jr., Pellian Representations, The Fibonacci Quarterly, Vol. 10, No. 5 (1972), pp. 449-488.

Cf. A000129, A007953, A317204.
Cf. also A014420, A053610, A265404, A265743, A265745.
Similar sequences: A007895, A116543, A278043.

pell[1] = 1; pell[2] = 2; pell[n_] := pell[n] = 2*pell[n - 1] + pell[n - 2]; a[n_] := Module[{s = {}, m = n, k}, While[m > 0, k = 1; While[pell[k] <= m, k++]; k--; AppendTo[s, k]; m -= pell[k]; k = 1]; Plus @@ IntegerDigits[Total[3^(s - 1)], 3]]; Array[a, 100, 0] (* Amiram Eldar, Mar 12 2022 *)

a(n) = A007953(A317204(n)). - Amiram Eldar, Mar 12 2022

A131343 Number of 1's in maximal Lucas representation (A130311) of n.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 3, 3, 4, 4, 4, 4, 5, 4, 4, 4, 5, 4, 4, 5, 5, 5, 5, 6, 4, 4, 5, 5, 5, 5, 6, 5, 5, 5, 6, 5, 5, 6, 6, 6, 6, 7, 5, 5, 5, 6, 5, 5, 6, 6, 6, 6, 7, 5, 5, 6, 6, 6, 6, 7, 6, 6, 6, 7, 6, 6, 7, 7, 7, 7, 8, 5, 5, 6, 6, 6, 6, 7, 6, 6, 6, 7

Offset: 0

Casey Mongoven, Jun 29 2007

Amiram Eldar, Table of n, a(n) for n = 0..10000
Ron Knott, Using the Fibonacci numbers to represent whole numbers.

Cf. A000032, A007895, A130311, A116543.

lazy = Select[IntegerDigits[Range[400], 2], SequenceCount[#, {0, 0}] == 0 &]; t = Total[# * Reverse@LucasL[Range[0, Length[#] - 1]]] & /@ lazy; Join[{0}, Plus@@@ lazy[[TakeWhile[Flatten[FirstPosition[t, #] & /@ Range[Max[t]]], NumberQ]]]] (* Amiram Eldar, Feb 17 2022 *)

a(n) = A007953(A130311(n)). - Amiram Eldar, Feb 17 2022

a(0) and more terms from Amiram Eldar, Feb 17 2022

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

Original entry on oeis.org

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 *)

A353657 a(n) = A353655(n)- A353656(n).

Original entry on oeis.org

0, -1, 0, 1, -1, 0, 2, -1, 0, 1, 1, 0, -1, 0, 0, 0, -1, 2, 1, 0, -1, -1, 1, -1, -2, 1, 0, -2, 2, 1, 1, 0, 0, -1, -1, -1, 0, -1, -1, 0, -1, 1, 0, -1, -1, 0, 2, 1, 2, 1, 1, 0, 0, -1, -1, -1, -1, -1, -1, 1, 0, -2, 0, 0, -1, -2, 0, 1, 0, 0, 0, 1, -2, -1, 0, 2, 2

Offset: 1

Clark Kimberling, May 04 2022

sign

Conjectures: a(n) = 0 for infinitely many n, and (a(n)) is unbounded below and above.

			a(7) because A353655(u) = 3 and A353656(7) = 1, since the Fibonacci-Lucas representation of 7 is FL(7) = 5 + 1 + 1, and the Lucas-Fibonacci representation of 7 is LF(7) = 7.

Cf. A000032, A000045, A007895, A116543, A353655, A353656, 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 *)

cp's OEIS Frontend

A214974 Numbers k for which A116543(k) = A007895(k); i.e., the Lucas and Zeckendorf representations of k have the same length.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A214975 Numbers k for which A116543(k) < A007895(k); i.e., the Lucas representation of k is shorter than the Zeckendorf representation.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A214976 Numbers k for which A116543(k) > A007895(k); i.e., the Lucas representation of k is longer than the Zeckendorf representation.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A130310 Minimal (or "greedy") Lucas representation of n, in which L(0) = 2 and L(2) = 3 are not allowed in the same representation (hence the correct representation of the integer 5 is 1010 rather than 101). A binary system of integers with Lucas numbers (A000032) as a base.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

Extensions

A351714 Lucas-Niven numbers: numbers that are divisible by the number of terms in their minimal (or greedy) representation in terms of the Lucas numbers (A130310).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A265744 a(n) is the number of Pell numbers (A000129) needed to sum to n using the greedy algorithm (A317204).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

A131343 Number of 1's in maximal Lucas representation (A130311) of n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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