A095791 -id:A095791 - cp's OEIS frontend

A095880 Numbers whose lazy Fibonacci representation has an even number of summands.

0, 3, 4, 5, 7, 11, 14, 16, 17, 18, 21, 22, 23, 25, 26, 28, 32, 33, 34, 36, 40, 41, 45, 48, 50, 51, 52, 54, 58, 61, 63, 64, 65, 69, 71, 72, 73, 76, 77, 78, 80, 81, 83, 87, 90, 92, 93, 94, 97, 98, 99, 101, 102, 104, 108, 110, 111, 112, 114, 115, 117, 121, 122, 123, 125, 129, 130

Offset: 1

Clark Kimberling, Jun 10 2004

nonn

			The first few Lazy Fibonacci representations (as in A095791) are 0 = 0, 1 = 1, 2 = 2, 3 = 2 + 1, 4 = 3 + 1, 5 = 3 + 2, 6 = 3 + 2 + 1, 7 = 5 + 2, 8 = 5 + 2 + 1, so that a(1), a(2), a(3), a(4) and a(5) are 0, 3, 4, 5, 7.

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

Cf. A095791, A095879, A104326.

lazyFib = Select[Range[0, 1000], SequenceCount[IntegerDigits[#, 2], {0, 0}] == 0 &]; binWt[n_] := DigitCount[n, 2, 1]; -1 + Position[binWt /@ lazyFib, ?(EvenQ[#] &)] // Flatten (* _Amiram Eldar, Jan 18 2020 *)

a(1) = 0 inserted by Amiram Eldar, Jan 18 2020

A331084 The number of terms in the negaFibonacci representation of -n (A215023).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jan 08 2020

The Fibonacci numbers F(2*n - 1) are the indices of records of this sequence.

			The negaFibonacci representation of 2 is A215023(2) = 1001, thus a(2) = 1 + 0 + 0 + 1 = 2.

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

Cf. A000045, A215023, A331083.

Cf. A000120, A007895, A007953, A053735, A095791.

ind[n_] := Floor[Log[Abs[n]*Sqrt[5] + 1/2]/Log[GoldenRatio]]; f[1] = 1; f[n_] := If[n > 0, i = ind[n - 1]; If[EvenQ[i], i++]; i, i = ind[-n]; If[OddQ[i], i++]; i]; nf[n_] := Module[{k = n, s = 0}, While[k != 0, i = f[k]; s += 1; k -= Fibonacci[-i]]; s]; a[n_] := nf[-n]; Array[a, 100]

a(A000045(2*n)) = 1.

a(A000045(2*n - 1)) = n.

A356895 a(n) is the length of the maximal tribonacci representation of n (A352103).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Sep 03 2022

			  n  a(n)  A352103(n)
  -  ----  ----------
  0     1           0
  1     1           1
  2     2          10
  3     2          11
  4     3         100
  5     3         101
  6     3         110
  7     3         111
  8     4        1001
  9     4        1010

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A352103, A352104, A356894.

Similar sequences: A070939, A072649, A095791, A278044.

t[1] = 1; t[2] = 2; t[3] = 4; t[n_] := t[n] = t[n - 1] + t[n - 2] + t[n - 3]; trib[n_] := Module[{s = {}, m = n, k}, While[m > 0, k = 1; While[t[k] <= m, k++]; k--; AppendTo[s, k]; m -= t[k]; k = 1]; IntegerDigits[Total[2^(s - 1)], 2]]; a[n_] := Module[{v = trib[n]}, nv = Length[v]; i = 1; While[i <= nv - 3, If[v[[i ;; i + 3]] == {1, 0, 0, 0}, v[[i ;; i + 3]] = {0, 1, 1, 1}; If[i > 3, i -= 4]]; i++]; i = Position[v, _?(# > 0 &)]; If[i == {}, 1, Length[v[[i[[1, 1]] ;; -1]]]]]; Array[a, 100, 0]

a(n) = A352104(n) + A356894(n).

a(n) ~ log(n)/log(c), where c is the tribonacci constant (A058265).

A360259 a(0) = 0, and for any n > 0, let k > 0 be as small as possible and such that F(2) + ... + F(1+k) >= n (where F(m) denotes A000045(m), the m-th Fibonacci number); a(n) = k + a(F(2) + ... + F(1+k) - n).

Original entry on oeis.org

0, 1, 3, 2, 6, 4, 3, 10, 6, 7, 5, 4, 15, 8, 9, 11, 7, 8, 6, 5, 21, 10, 11, 13, 12, 16, 9, 10, 12, 8, 9, 7, 6, 28, 12, 13, 15, 14, 18, 16, 15, 22, 11, 12, 14, 13, 17, 10, 11, 13, 9, 10, 8, 7, 36, 14, 15, 17, 16, 20, 18, 17, 24, 20, 21, 19, 18, 29, 13, 14, 16

Offset: 0

Rémy Sigrist, Jan 31 2023

See A095791 for the corresponding k's.

This sequence has similarities with A227192; here we use Fibonacci numbers, there powers of 2.

			The first terms, alongside the corresponding k's, are:
  n      a(n)  k
  -----  ----  ---
      0     0  N/A
      1     1    1
      2     3    2
      3     2    2
      4     6    3
      5     4    3
      6     3    3
      7    10    4
      8     6    4
      9     7    4
     10     5    4
     11     4    4
     12    15    5

Rémy Sigrist, Table of n, a(n) for n = 0..10946

See A095791, A360260 and A360265 for similar sequences.

Cf. A000045, A001911, A227192.

{ t = k = 0; print1 (0); for (n = 1, #a = vector(70), if (n > t, t += fibonacci(1+k++);); print1 (", "a[n] = k+if (t==n, 0, a[t-n]));); }

a(A001911(n)) = n.

A345067 Consider the "Quilt Tiling"; T(n, k) is the area of the tile containing the unit square whose upper right corner has coordinates (n, k); square array T(n, k) read by antidiagonals upwards, n, k > 0.

Original entry on oeis.org

1, 2, 2, 2, 4, 2, 6, 4, 4, 6, 6, 6, 4, 6, 6, 6, 6, 1, 1, 6, 6, 15, 6, 2, 9, 2, 6, 15, 15, 15, 2, 9, 9, 2, 15, 15, 15, 15, 15, 9, 9, 9, 15, 15, 15, 15, 15, 15, 2, 9, 9, 2, 15, 15, 15, 15, 15, 15, 2, 4, 9, 4, 2, 15, 15, 15, 40, 15, 15, 6, 4, 4, 4, 4, 6, 15, 15, 40

Offset: 1

Rémy Sigrist, Jun 06 2021

The "Quilt Tiling" is described in Shectman's paper (see Links section).

All terms belong to A006498.

			Array T(n, k) begins:
  n\k|  1   2   3   4   5   6   7   8   9  10  11
  ---+---+-------+-----------+-------------------+
   1 |  1|  2   2|  6   6   6| 15  15  15  15  15|
     +-----------+           |                   |
   2 |  2|  4   4|  6   6   6| 15  15  15  15  15|
     |   |       +---+-------+                   |
   3 |  2|  4   4|  1|  2   2| 15  15  15  15  15|
     +---+---+---+---+-------+-------+-----------+
   4 |  6   6|  1|  9   9   9|  2   2|  6   6   6|
     |       +---+           +-------+           |
   5 |  6   6|  2|  9   9   9|  4   4|  6   6   6|
     |       |   |           |       +---+-------+
   6 |  6   6|  2|  9   9   9|  4   4|  1|  2   2|
     +-------+---+---+-------+-------+---+-------+
   7 | 15  15  15|  2|  4   4| 25  25  25  25  25|
     |           |   |       |                   |
   8 | 15  15  15|  2|  4   4| 25  25  25  25  25|
     |           +---+---+---+                   |
   9 | 15  15  15|  6   6|  1| 25  25  25  25  25|
     |           |       +---+                   |
  10 | 15  15  15|  6   6|  2| 25  25  25  25  25|
     |           |       |   |                   |
  11 | 15  15  15|  6   6|  2| 25  25  25  25  25|
     +-----------+-------+---+-------------------+

Rémy Sigrist, Table of n, a(n) for n = 1..10153
J. Parker Shectman, A Quilt after Fibonacci, Part 2 of 3: Cohorts, Free Monoids, and Numeration
Rémy Sigrist, Illustration of the connection between the "Quilt Tiling" and the sequences A000201 and A005206
Rémy Sigrist, PARI program for A345067

Cf. A000045, A001654, A005206, A006498, A095791, A005206.

PARI
```
See Links section.
```

T(n, k) = T(k, n).
T(n, n) = A130312(n+1)^2.
T(n, 1) = A001654(A095791(n)+1).
T(n, k) is the square of a Fibonacci number for n = 1+A005206(k+1)..A000201(k).

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A095880 Numbers whose lazy Fibonacci representation has an even number of summands.

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A331084 The number of terms in the negaFibonacci representation of -n (A215023).

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A356895 a(n) is the length of the maximal tribonacci representation of n (A352103).

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A360259 a(0) = 0, and for any n > 0, let k > 0 be as small as possible and such that F(2) + ... + F(1+k) >= n (where F(m) denotes A000045(m), the m-th Fibonacci number); a(n) = k + a(F(2) + ... + F(1+k) - n).

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A345067 Consider the "Quilt Tiling"; T(n, k) is the area of the tile containing the unit square whose upper right corner has coordinates (n, k); square array T(n, k) read by antidiagonals upwards, n, k > 0.

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