A214979 -id:A214979 - cp's OEIS frontend

A214980 Positions of zeros in A214979.

1, 2, 3, 5, 6, 8, 9, 10, 16, 17, 24, 26, 27, 28, 45, 46, 71, 73, 74, 75, 121, 122, 194, 196, 197, 198, 320, 321, 516, 518, 519, 520, 841, 842, 1359, 1361, 1362, 1363, 2205, 2206, 3566, 3568, 3569, 3570, 5776, 5777, 9344, 9346, 9347, 9348, 15125

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 A214980 is the sequence of zeros of the sequence A214979(n) = V(n) - U(n). It is conjectured at A214979 that A214980 is infinite.

Clark Kimberling, Table of n, a(n) for n = 1..58

Cf. A214977, A214978, A214979, A214981, A000032, A000045.

Mathematica
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(See the program at A214979.)
```

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
```
(See the program at A214973.)
```

Edited by Clark Kimberling, Jun 13 2020

A179180 Partial sums of A007895.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 8, 10, 11, 13, 15, 17, 20, 21, 23, 25, 27, 30, 32, 35, 38, 39, 41, 43, 45, 48, 50, 53, 56, 58, 61, 64, 67, 71, 72, 74, 76, 78, 81, 83, 86, 89, 91, 94, 97, 100, 104, 106, 109, 112, 115, 119, 122, 126, 130, 131, 133, 135, 137, 140, 142, 145

Offset: 0

Walt Rorie-Baety, Jun 30 2010

nonn

Total number of summands in Zeckendorf representations of all the numbers 1,2,...,n (for n>0); see the conjecture at A214979. - Clark Kimberling, Oct 23 2012

			For n = 6, a(n) = 1+1+1+2+1+2 = 8.

Clark Kimberling, Table of n, a(n) for n = 0..10000
Christian Ballot, On Zeckendorf and Base b Digit Sums, Fibonacci Quarterly, Vol. 51, No. 4 (2013), pp. 319-325.
Vaclav Kotesovec, Graph - the asymptotic ratio

Cf. A001622, A007895, A214979.

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}]
(* Peter J. C. Moses, Oct 18 2012 *)
DigitCount[Select[Range[0, 500], BitAnd[#, 2*#] == 0&], 2, 1] // Accumulate (* Jean-François Alcover, Jan 25 2018 *)

a(n) ~ c * n * log(n), where c = (phi-1)/(sqrt(5)*log(phi)) = 0.574369... and phi is the golden ratio (A001622) (Ballot, 2013). - Amiram Eldar, Dec 09 2021

Corrected term a(17); the working list of the terms were not in order. Walt Rorie-Baety, Jun 30 2010

Extended by Clark Kimberling, Oct 23 2012

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A214980 Positions of zeros in A214979.

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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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A179180 Partial sums of A007895.

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