A003714 -id:A003714 - cp's OEIS frontend

A300890 Lexicographically earliest sequence of distinct positive terms such that, for any n > 0, 2*a(n) + a(n+1) is a Fibbinary number (A003714).

1, 2, 4, 8, 5, 6, 9, 3, 10, 12, 13, 7, 18, 28, 16, 32, 17, 30, 14, 36, 56, 20, 24, 21, 22, 25, 15, 11, 19, 26, 29, 23, 27, 31, 66, 33, 62, 37, 54, 38, 52, 34, 60, 40, 48, 41, 46, 44, 42, 45, 39, 50, 49, 35, 58, 53, 43, 47, 51, 59, 138, 61, 134, 57, 55, 146

Offset: 1

Rémy Sigrist, Mar 14 2018

The scatterplot of the sequence has fractal features.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Scatterplot of the first 250000 terms
Rémy Sigrist, C++ program for A300890

Cf. A003714.

A379175 Irregular triangle T(n, k), n >= 0, k = 1..ceiling(2^(A007895(n)-1)); the n-th row lists the nonnegative integers m such that A184617(m) = A003714(n).

Original entry on oeis.org

0, 1, 2, 4, 3, 5, 8, 7, 9, 6, 10, 16, 15, 17, 14, 18, 12, 20, 11, 13, 19, 21, 32, 31, 33, 30, 34, 28, 36, 27, 29, 35, 37, 24, 40, 23, 25, 39, 41, 22, 26, 38, 42, 64, 63, 65, 62, 66, 60, 68, 59, 61, 67, 69, 56, 72, 55, 57, 71, 73, 54, 58, 70, 74, 48, 80, 47, 49, 79, 81

Offset: 0

Rémy Sigrist, Dec 17 2024

Also the nonnegative terms of A379147, in order of appearance.
This sequence is a permutation of the nonnegative integers with inverse A379176.
This sequence shares graphical features with A368225.

			Triangle T(n, k) begins:
  n   n-th row
  --  --------------
   0  0
   1  1
   2  2
   3  4
   4  3, 5
   5  8
   6  7, 9
   7  6, 10
   8  16
   9  15, 17
  10  14, 18
  11  12, 20
  12  11, 13, 19, 21
  13  32
  14  31, 33
  15  30, 34

Rémy Sigrist, Table of n, a(n) for n = 0..10922 (rows for n = 0..986 flattened)
Index entries for sequences related to Zeckendorf expansion of n
Index entries for sequences that are permutations of the natural numbers

Cf. A003714, A007895, A184617, A368225, A379147, A379176 (inverse).

tozeck(n) = { for (i=0, oo, if (n<=fibonacci(2+i), my (v=0, f); forstep (j=i, 0, -1, if (n>=f=fibonacci(2+j), n-=f; v+=2^j;); if (n==0, return (v););););); }
row(n) = { my (z = tozeck(n), r = [0], b); while (z, z -= b = 2^valuation(z, 2); r = concat([v - b | v <- r], [v + b | v <- r]);); return (select(v -> v >= 0, r)); }

T(n, ceiling(2^(A007895(n)-1))) = A003714(n).

A056017 Permutation of nonnegative integers formed by ranking fibbinary numbers (A003714) as if they were representatives of the circular binary sequences with forbidden -11- subsequence.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 6, 8, 11, 9, 10, 12, 13, 18, 14, 15, 19, 16, 20, 17, 21, 29, 22, 23, 30, 24, 31, 25, 26, 32, 27, 28, 33, 34, 47, 35, 36, 48, 37, 49, 38, 39, 50, 40, 41, 51, 42, 52, 43, 44, 53, 45, 54, 46, 55, 76, 56, 57, 77, 58, 78, 59, 60, 79, 61, 62, 80, 63, 81, 64, 65

Offset: 0

Antti Karttunen, Jun 08 2000

nonn

Function CircBinSeqNo11Rank gives the position of any 11-free binary sequence in this sequence, where each block consists of Lucas(n-2) sequences of length n: (either the leftmost or the rightmost digit is 1, but not both).

In this permutation the Fibonacci numbers themselves (A000045) are fixed.

			0; 01,10; 100; 0101,1000,1010; 01001,10000,10010,10100; 010001,010101,100000,100010,100100,101000,101010; etc.

Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A056018. For fibbinary function see A048679, interpret_as_zeckendorf_expansion given in A048680.

CircBinSeqNo11Rank := n -> fibonacci(floor_log_2(n)+1-((-1)^n)) + interpret_as_zeckendorf_expansion(floor(n/(3-((-1)^n))));

[seq(CycBinSeqNo11Rank(fibbinary(j)), j=0..233)];

a[0] = 0, a[n] = CircBinSeqNo11Rank(fibbinary(n)) for n >= 1.

A106028 Minimal number of editing steps (delete, insert or substitute) to transform the binary representation of n into that of A003714(n), the n-th fibbinary number.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, May 05 2005

nonn

A014417(n) = A007088(A003714(n)).

Michael Gilleland, Levenshtein Distance [It has been suggested that this algorithm gives incorrect results sometimes. - _N. J. A. Sloane_]
Eric Weisstein's World of Mathematics, Zeckendorf Representation
Index entries for sequences related to binary expansion of n

Cf. A035517, A000045, A072649, A070939.

a(n) = LevenshteinDistance(A014417(n), A007088(n)).

A276488 Numbers k that divide A003714(k).

Original entry on oeis.org

1, 2, 8, 84, 130, 288, 856, 1034, 1230, 2568, 11200, 36565, 41509, 81536, 212586, 409541, 516368, 521248, 984064, 1771007, 5835276, 6230310, 11951289, 12298144, 16284017, 32568034, 32655554, 39382944, 69585548, 69656573, 101118154, 309345058, 384373140, 1165237916, 2623448060

Offset: 1

Altug Alkan, Sep 05 2016

nonn

Corresponding Fibbinary numbers are 1, 2, 16, 336, 650, 2304, 10272, ...

Next term > 2^32. - Joerg Arndt, Sep 05 2016

			84 = Fibonacci(10) + Fibonacci(8) + Fibonacci(6) and 84 divides A003714(84) = 2^8 + 2^6 + 2^4 = 336.

Cf. A003714, A357073.

Terms 2568 and beyond from Joerg Arndt, Sep 05 2016

A277331 a(n) = A253563(A003714(n)).

Original entry on oeis.org

1, 2, 4, 8, 6, 16, 12, 18, 32, 24, 36, 54, 30, 64, 48, 72, 108, 60, 162, 90, 150, 128, 96, 144, 216, 120, 324, 180, 300, 486, 270, 450, 750, 210, 256, 192, 288, 432, 240, 648, 360, 600, 972, 540, 900, 1500, 420, 1458, 810, 1350, 2250, 630, 3750, 1050, 1470, 512, 384, 576, 864, 480, 1296, 720, 1200, 1944, 1080, 1800, 3000, 840

Offset: 0

Antti Karttunen, Oct 12 2016

nonn

After the initial terms 1, 2 and 4, all other terms can be inductively generated by applying any finite composition-combination of A253560 and A253550 to 4, but with such a restriction that A253550 may not be applied twice in succession.

A permutation of A055932.

Antti Karttunen, Table of n, a(n) for n = 0..10946

Cf. A003714, A055932 (same sequence sorted into ascending order), A253550, A253560, A253563, A122111.

Cf. also A277006, A277332.

(define (A277331 n) (A253563 (A003714 n)))

a(n) = A253563(A003714(n)).

a(n) = A122111(A277006(n)).

A294359 a(n) = [x^n] F(x)^(-(n+1)^2) such that F(x) = F(x^2) + x*F(x^4), where F(x) = Sum_{n>=0} x^A003714(n) and A003714 is the Fibbinary numbers.

Original entry on oeis.org

1, -4, 36, -544, 12000, -353016, 13024690, -578027008, 29965705056, -1776380879600, 118487748235604, -8781184406967264, 715759620936227036, -63634560244855290488, 6127715132571003255000, -635341671628285381320704, 70567080867797749860480968, -8358996420744136578157248864, 1051888164647093035820630830470, -140135781917815169726696222119200, 19704058040921706609228103696785954

Offset: 0

Paul D. Hanna, Nov 03 2017

sign

It is conjectured that all terms are even after the initial '1'.

Fibbinary numbers are integers whose binary representation contains no consecutive ones (see A003714 for definition); it is unexpected that the characteristic function F(x) of the Fibbinary numbers would have only even coefficients of x^n in the negative square powers F(x)^(-(n+1)^2), as described by this sequence.

			Given the characteristic function of the Fibbinary numbers (A003714):
F(x) = 1 + x + x^2 + x^4 + x^5 + x^8 + x^9 + x^10 + x^16 + x^17 + x^18 + x^20 + x^21 + x^32 + x^33 + x^34 + x^36 + x^37 + x^40 + x^41 + x^42 + x^64 + x^65 + x^66 + x^68 + x^69 + x^72 + x^73 + x^74 + x^80 +...+ x^A003714(n) +...
such that F(x) = F(x^2) + x*F(x^4),
then this sequence equals the coefficients of x^n in F(x)^(-(n+1)^2).
ILLUSTRATION OF TERMS.
The table of coefficients of x^k in F(x)^(-n^2) begins:
n=1: [1, -1, 0, 1, -2, 1, 2, -4, 2, 3, -8, 7, 4, -16, 16, 2, -30, ...];
n=2: [1, -4, 6, 0, -19, 40, -26, -56, 166, -160, -110, 560, -705, ...];
n=3: [1, -9, 36, -75, 36, 279, -942, 1278, 531, -5956, 11700, ...];
n=4: [1, -16, 120, -544, 1548, -2192, -2720, 23936, -63426, 67984, ...];
n=5: [1, -25, 300, -2275, 12000, -45005, 112450, -116350, -441375, ...];
n=6: [1, -36, 630, -7104, 57573, -353016, 1668774, -5996664, ...];
n=7: [1, -49, 1176, -18375, 209426, -1846859, 13024690, -74680760, ...];
n=8: [1, -64, 2016, -41600, 631216, -7491392, 72180992, -578027008, ...]; ...
in which the main diagonal forms this sequence.
RELATED SEQUENCES.
Terms (-1)^n * a(n)/(n+1) begin:
[1, 2, 12, 136, 2400, 58836, 1860670, 72253376, 3329522784, 177638087960, ...].
Sequence A294475(n) = (-1)^n * a(n)/(n+1)^2 and begins:
[1, 1, 4, 34, 480, 9806, 265810, 9031672, 369946976, 17763808796, ...].

Paul D. Hanna, Table of n, a(n) for n = 0..600

Cf. A085357, A003714, A294475, A292877.

terms = 21; selfibb = Select[Range[terms], BitAnd[#, 2*#] == 0&]; lenfibb = Length[selfibb]; fibb[0] = 0; fibb[n_] := selfibb[[n]]; F[x_] = Sum[x^fibb[n], {n, 0, lenfibb}]; a[n_] := SeriesCoefficient[F[x]^(-(n + 1)^2), {x, 0, n}]; Array[a, terms, 0] (* Jean-François Alcover, Nov 04 2017 *)

a(n) = (-1)^n * n^2 * A294475(n).

A300868 Lexicographically earliest sequence of distinct positive terms such that, for any n > 0, n * a(n) is a Fibbinary number (A003714).

Original entry on oeis.org

1, 2, 3, 4, 8, 6, 12, 5, 9, 13, 15, 7, 10, 19, 11, 16, 17, 18, 14, 26, 32, 24, 23, 22, 41, 20, 38, 39, 36, 43, 44, 21, 33, 34, 59, 29, 64, 27, 28, 52, 25, 49, 30, 31, 53, 46, 47, 48, 42, 82, 51, 40, 45, 76, 96, 78, 72, 73, 35, 71, 84, 75, 86, 37, 65, 66, 70

Offset: 1

Rémy Sigrist, Mar 14 2018

This sequence is a self-inverse permutation of the natural numbers.

			The first terms, alongside the binary representation of n * a(n), are:
  n  a(n)    binary(n * a(n))
  -- ----    ----------------
   1    1             1
   2    2           100
   3    3          1001
   4    4         10000
   5    8        101000
   6    6        100100
   7   12       1010100
   8    5        101000
   9    9       1010001
  10   13      10000010
  11   15      10100101
  12    7       1010100
  13   10      10000010
  14   19     100001010
  15   11      10100101
  16   16     100000000
  17   17     100100001
  18   18     101000100
  19   14     100001010
  20   26    1000001000

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers
Rémy Sigrist, Colored logarithmic scatterplot of the first 2^15 terms (where the color is function of A070939(n * a(n)))
Rémy Sigrist, C++ program for A300868

Cf. A003714, A070939.

A319432 The first differences (A129761) of the tribonacci representation numbers (A003714 or A014417) consists of runs of 1's separated by the terms of the present sequence.

Original entry on oeis.org

2, 3, 6, 2, 11, 2, 3, 22, 2, 3, 6, 2, 43, 2, 3, 6, 2, 11, 2, 3, 86, 2, 3, 6, 2, 11, 2, 3, 22, 2, 3, 6, 2, 171, 2, 3, 6, 2, 11, 2, 3, 22, 2, 3, 6, 2, 43, 2, 3, 6, 2, 11, 2, 3, 342, 2, 3, 6, 2, 11, 2, 3, 22, 2, 3, 6, 2, 43, 2, 3, 6, 2, 11, 2, 3, 86, 2, 3, 6, 2, 11, 2, 3, 22, 2, 3, 6, 2, 683, 2

Offset: 1

N. J. A. Sloane, Sep 30 2018

nonn

The runs of 1's in A129761 have lengths that apparently are given by the Fibonacci word A005614 (with a slight change at the start). The present sequence shows the terms greater than 1. The structure of this sequence is (partially) analyzed in the comments in A129761.

Rémy Sigrist, Table of n, a(n) for n = 1..25000

Cf. A003714, A005614, A014417, A129761,

p=0; for (n=1, 2052, if (bitand(n,2*n)==0, if (n-p>1, print1 (n-p ", ")); p=n)) \\ Rémy Sigrist, Oct 07 2018

A357073 For n >= 1, a(n) = A003714(n) mod n.

Original entry on oeis.org

0, 0, 1, 1, 3, 3, 3, 0, 8, 8, 9, 9, 6, 5, 4, 4, 3, 4, 3, 2, 1, 21, 20, 20, 19, 20, 19, 18, 22, 21, 20, 20, 19, 26, 24, 22, 21, 19, 19, 17, 15, 18, 16, 14, 13, 11, 19, 17, 15, 14, 12, 12, 10, 8, 36, 33, 30, 28, 25, 24, 21, 18, 20, 17, 14, 12, 9, 16, 13, 10, 8, 5

Offset: 1

Ctibor O. Zizka, Sep 10 2022

nonn

a(n) = 0 see A276488.

			a(5) = A003714(5) mod 5 = 8 mod 5 = 3.

Cf. A003714, A276488.

fib = Select[Range[300], BitAnd[#, 2*#] == 0 &]; Mod[fib, Range[Length[fib]]] (* Amiram Eldar, Sep 10 2022 *)

def A357073(n):
    tlist, s, m = [1, 2], 0, n
    while (t:=tlist[-1]+tlist[-2]) <= n:
        tlist.append(t)
    for d in tlist[::-1]:
        s = (s<<1)%n
        if d <= m:
            s = (s+1)%n
            m -= d
    return s # Chai Wah Wu, Sep 11 2022

a(n) = A003714(n) mod n.

cp's OEIS Frontend

A300890 Lexicographically earliest sequence of distinct positive terms such that, for any n > 0, 2*a(n) + a(n+1) is a Fibbinary number (A003714).

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A379175 Irregular triangle T(n, k), n >= 0, k = 1..ceiling(2^(A007895(n)-1)); the n-th row lists the nonnegative integers m such that A184617(m) = A003714(n).

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A056017 Permutation of nonnegative integers formed by ranking fibbinary numbers (A003714) as if they were representatives of the circular binary sequences with forbidden -11- subsequence.

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A106028 Minimal number of editing steps (delete, insert or substitute) to transform the binary representation of n into that of A003714(n), the n-th fibbinary number.

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A276488 Numbers k that divide A003714(k).

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A277331 a(n) = A253563(A003714(n)).

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Scheme

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A294359 a(n) = [x^n] F(x)^(-(n+1)^2) such that F(x) = F(x^2) + x*F(x^4), where F(x) = Sum_{n>=0} x^A003714(n) and A003714 is the Fibbinary numbers.

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A300868 Lexicographically earliest sequence of distinct positive terms such that, for any n > 0, n * a(n) is a Fibbinary number (A003714).

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A319432 The first differences (A129761) of the tribonacci representation numbers (A003714 or A014417) consists of runs of 1's separated by the terms of the present sequence.

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