A345291 -id:A345291 - cp's OEIS frontend

A345290 a(n) is obtained by replacing 2^k in binary expansion of n with Fibonacci(-k-2).

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

Offset: 0

Rémy Sigrist, Jun 13 2021

This sequence is a variant of A022290; here we consider Fibonacci numbers with negative indices (A039834), there Fibonacci numbers with positive indices (A000045).

After the initial 0, the sequence alternates runs of positive terms and runs of negative terms, the k-th run having 2^(k-1) terms.

			For n = 3:
- 3 = 2^1 + 2^0,
- so a(3) = A039834(2+1) + A039834(2+0) = 2 - 1 = 1.

Rémy Sigrist, Table of n, a(n) for n = 0..8191

Cf. A000045, A000695, A020989, A022290, A039834, A062880, A063694, A063695, A072197, A080675, A136412, A345291, A345292.

a(n) = { my (v=0, e); while (n, n-=2^e=valuation(n, 2); v+=fibonacci(-2-e)); v }

a(n) = A022290(A063695(n)) - A022290(A063694(n)).
a(n) = A022290(n) iff n belongs to A062880.
a(n) = -A022290(n) iff n belongs to A000695.
a(n) = 0 iff n = 0.
a(n) = 1 iff n belongs to A072197.
a(n) = 2 iff n belongs to A080675.
a(n) = -1 iff n belongs to A020989.
a(n) = -2 iff n belongs to A136412.

A345292 a(n) is the least k >= 0 such that A345290(k) = -n.

Original entry on oeis.org

0, 1, 7, 4, 5, 31, 18, 19, 16, 17, 23, 20, 21, 127, 74, 75, 72, 73, 79, 66, 67, 64, 65, 71, 68, 69, 95, 82, 83, 80, 81, 87, 84, 85, 511, 298, 299, 296, 297, 303, 290, 291, 288, 289, 295, 292, 293, 319, 266, 267, 264, 265, 271, 258, 259, 256, 257, 263, 260, 261

Offset: 0

Rémy Sigrist, Jun 13 2021

The binary plot of the sequence has interesting features (see Illustration in Links section).

			We have:
  n         | 0   1  2  3   4   5   6   7  8  9  10  11  12  13  14  15  16  17
  ----------+------------------------------------------------------------------
  A345290(n)| 0  -1  2  1  -3  -4  -1  -2  5  4   7   6   2   1   4   3  -8  -9
- so a(0) = 0,
     a(1) = 1,
     a(3) = 4,
     a(4) = 5,
     a(2) = 7,
     a(8) = 16,
     a(9) = 17.

Rémy Sigrist, Table of n, a(n) for n = 0..6764
Rémy Sigrist, Binary plot of the first Fibonacci(13) = 233 terms
Rémy Sigrist, PARI program for A345292

Cf. A345290, A345291.

PARI
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See Links section.
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cp's OEIS Frontend

A345290 a(n) is obtained by replacing 2^k in binary expansion of n with Fibonacci(-k-2).

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