A095096 -id:A095096 - cp's OEIS frontend

A337287 Numbers k such that both k and k+1 are in A095096.

6, 9, 10, 14, 15, 22, 23, 35, 36, 46, 53, 56, 57, 67, 74, 75, 82, 85, 86, 90, 91, 101, 108, 109, 116, 119, 120, 129, 132, 133, 137, 138, 145, 146, 156, 163, 164, 171, 174, 175, 184, 187, 188, 192, 193, 205, 208, 209, 213, 214, 221, 222, 234, 235, 245, 252, 253, 260, 263, 264, 273

Offset: 1

N. J. A. Sloane, Sep 12 2020

From Michel Dekking, Oct 09 2020: (Start)
Let s_Z = A095076 be the parity of the sum of digits function of the Zeckendorf representation. Shutov's main result is that the number of times that s_Z(k) mod 2 = 0 AND s_Z(k+1) mod 2 = 0 in [0,n] divided by n tends to sqrt(5)/10.
It is possible to derive this result in a few lines by using the representation of s_Z as a morphic sequence, as given in the Comments of A095076.
To this end one considers the 2-block substitution sigma^[2] of the Zeckendorf morphism
      sigma:  1->12, 2->4, 3->1, 4->43.
There are 10 words of length 2 occurring in the fixed points of this morphism. These are 11, 12, 14, 21, 24, 31, 34, 41, 43 and 44. Since the sigma^[2]-images of both 12 and 14 are 12,24, and this is also the case for the pair 41 and 43, one can reduce the number of letters to 8.
Coding the words of length 2 in lexicographic order this gives sigma^[2] on the alphabet {1,2,...,7,8} as
        sigma^[2]:  1->23, 2->24, 3->7, 4->8, 5->1, 6->2, 7->75, 8->76.
The letter-to-letter map lambda mapping the fixed point of sigma^[2] to the sequence s_Z = A095076 is given by lambda(1)=0, lambda(2)=1, lambda(3)=0, lambda(4)=1 (see A095076).
We see that lambda(11) = lambda(31) = 00, and these are the only words of length 2 mapping to 00. It follows that the frequency of 00 in s_Z is equal to the sum of the frequencies of 1 and 5 in the fixed point starting with 2 of the morphism sigma^[2]. It is well known that these frequencies are given by the normalized eigenvector corresponding to the Perron-Frobenius eigenvalue of the incidence matrix of the morphism sigma^[2].
An eigenvalue calculation then gives the value sqrt(5)/10 from above.
Final remark: the same result has been derived for the base-phi expansion of the natural numbers, and the limit is the same.
(End)

Anton Shutov, On the sum of digits of the Zeckendorf representations of two consecutive numbers, Fib. Q., 58:3 (2020), 203-207.

Amiram Eldar, Table of n, a(n) for n = 1..10000
F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
Michel Dekking, The sum of digits function of the base phi expansion of the natural numbers,arXiv:1911.10705 [math.NT], 2019.

Cf. A020899, A095076, A095096, A337288, A337289, A337290, A337634, A337635, A337636, A337637.

SequencePosition[Mod[DigitCount[Select[Range[0, 3000], BitAnd[#, 2 #] == 0 &], 2, 1], 2], {0, 0}][[;; , 1]] - 1 (* Amiram Eldar, Feb 05 2023 *)

A337289 Numbers k such that k+1 is in A095096 and k is in A020899.

Original entry on oeis.org

3, 5, 8, 13, 17, 21, 25, 28, 32, 34, 38, 41, 45, 50, 52, 55, 59, 62, 66, 71, 73, 79, 81, 84, 89, 93, 96, 100, 105, 107, 113, 115, 118, 122, 126, 128, 131, 136, 140, 144, 148, 151, 155, 160, 162, 168, 170, 173, 177, 181, 183, 186, 191, 195, 198, 202, 204, 207, 212, 216, 220, 224, 227

Offset: 1

N. J. A. Sloane, Sep 12 2020

Anton Shutov, On the sum of digits of the Zeckendorf representations of two consecutive numbers, Fib. Q., 58:3 (2020), 203-207.

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

Cf. A020899, A095096, A337287, A337287, A337290, A337634, A337635, A337636, A337637.

SequencePosition[Mod[DigitCount[Select[Range[0, 3000], BitAnd[#, 2 #] == 0 &], 2, 1], 2], {1, 0}][[;; , 1]] - 1 (* Amiram Eldar, Feb 05 2023 *)

A337634 Number of numbers k <= n such that both k and k+1 are in A095096.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 3, 3, 3, 3, 4, 5, 5, 5, 5, 5, 5, 5, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 15, 16, 16, 16, 16, 16, 16, 16, 17

Offset: 0

N. J. A. Sloane, Sep 12 2020

Anton Shutov, On the sum of digits of the Zeckendorf representations of two consecutive numbers, Fib. Q., 58:3 (2020), 203-207.

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

Cf. A020899, A095096, A337287, A337288, A337289, A337290, A337635, A337636, A337637.

s = SequencePosition[Mod[DigitCount[Select[Range[0, 400], BitAnd[#, 2 #] == 0 &], 2, 1], 2], {0, 0}][[;; , 1]] - 1; t = Table[0, {s[[-1]] + 1}]; t[[s + 1]] = 1; Accumulate[t] (* Amiram Eldar, Feb 05 2023 *)

Offset corrected by Amiram Eldar, Feb 05 2023

A337635 Number of numbers k <= n such that k is in A095096 and k+1 is in A020899.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Sep 12 2020

Anton Shutov, On the sum of digits of the Zeckendorf representations of two consecutive numbers, Fib. Q., 58:3 (2020), 203-207.

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

Cf. A020899, A095096, A337287, A337288, A337289, A337290, A337634, A337636, A337637.

s = SequencePosition[Mod[DigitCount[Select[Range[0, 400], BitAnd[#, 2 #] == 0 &], 2, 1], 2], {0, 1}][[;; , 1]] - 1; t = Table[0, {s[[-1]] + 1}]; t[[s + 1]] = 1; Accumulate[t] (* Amiram Eldar, Feb 05 2023 *)

Offset corrected by Amiram Eldar, Feb 05 2023

A337636 Number of numbers k <= n such that k+1 is in A095096 and k is in A020899.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Sep 12 2020

Anton Shutov, On the sum of digits of the Zeckendorf representations of two consecutive numbers, Fib. Q., 58:3 (2020), 203-207.

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

Cf. A020899, A095096, A337287, A337288, A337289, A337290, A337634, A337635, A337637.

s = SequencePosition[Mod[DigitCount[Select[Range[0, 400], BitAnd[#, 2 #] == 0 &], 2, 1], 2], {1, 0}][[;; , 1]] - 1; t = Table[0, {s[[-1]] + 1}]; t[[s + 1]] = 1; Accumulate[t] (* Amiram Eldar, Feb 05 2023 *)

Offset corrected by Amiram Eldar, Feb 05 2023

A337288 Numbers k such that k is in A095096 and k+1 is in A020899.

Original entry on oeis.org

0, 4, 7, 11, 16, 18, 24, 26, 29, 33, 37, 39, 42, 47, 51, 54, 58, 60, 63, 68, 72, 76, 80, 83, 87, 92, 94, 97, 102, 106, 110, 114, 117, 121, 123, 127, 130, 134, 139, 141, 147, 149, 152, 157, 161, 165, 169, 172, 176, 178, 182, 185, 189, 194, 196, 199, 203, 206, 210, 215, 217, 223, 225

Offset: 1

N. J. A. Sloane, Sep 12 2020

Anton Shutov, On the sum of digits of the Zeckendorf representations of two consecutive numbers, Fib. Q., 58:3 (2020), 203-207.

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

Cf. A020899, A095096, A337287, A337289, A337290, A337634, A337635, A337636, A337637.

SequencePosition[Mod[DigitCount[Select[Range[0, 3000], BitAnd[#, 2 #] == 0 &], 2, 1], 2], {0, 1}][[;; , 1]] - 1 (* Amiram Eldar, Feb 05 2023 *)

A337290 Numbers k such that both k and k+1 are in A020899.

Original entry on oeis.org

1, 2, 12, 19, 20, 27, 30, 31, 40, 43, 44, 48, 49, 61, 64, 65, 69, 70, 77, 78, 88, 95, 98, 99, 103, 104, 111, 112, 124, 125, 135, 142, 143, 150, 153, 154, 158, 159, 166, 167, 179, 180, 190, 197, 200, 201, 211, 218, 219, 226, 229, 230, 239, 242, 243, 247, 248, 255, 256, 268, 269, 279

Offset: 1

N. J. A. Sloane, Sep 12 2020

Anton Shutov, On the sum of digits of the Zeckendorf representations of two consecutive numbers, Fib. Q., 58:3 (2020), 203-207.

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

Cf. A020899, A095096, A337287, A337288, A337289, A337634, A337635, A337636, A337637.

SequencePosition[Mod[DigitCount[Select[Range[0, 3000], BitAnd[#, 2 #] == 0 &], 2, 1], 2], {1, 1}][[;; , 1]] - 1 (* Amiram Eldar, Feb 05 2023 *)

A337637 Number of numbers k <= n such that both k and k+1 are in A020899.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Sep 12 2020

Anton Shutov, On the sum of digits of the Zeckendorf representations of two consecutive numbers, Fib. Q., 58:3 (2020), 203-207.

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

Cf. A020899, A095096, A337287, A337288, A337289, A337290, A337634, A337635, A337636.

s = SequencePosition[Mod[DigitCount[Select[Range[0, 400], BitAnd[#, 2 #] == 0 &], 2, 1], 2], {1, 1}][[;; , 1]] - 1; t = Table[0, {s[[-1]] + 1}]; t[[s + 1]] = 1; Accumulate[t] (* Amiram Eldar, Feb 05 2023 *)

Offset corrected by Amiram Eldar, Feb 05 2023

A095084 Fibevil primes, i.e., primes p whose Zeckendorf-expansion A014417(p) contains an even number of 1-fibits.

Original entry on oeis.org

7, 11, 23, 29, 37, 47, 53, 67, 83, 97, 101, 109, 127, 137, 139, 149, 157, 163, 193, 199, 223, 241, 263, 271, 277, 281, 283, 311, 317, 331, 337, 359, 373, 379, 389, 397, 409, 421, 439, 443, 461, 499, 503, 521, 547, 557, 563, 577, 593, 601, 607

Offset: 1

Antti Karttunen, Jun 01 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Antti Karttunen and John Moyer, C-program for computing the initial terms of this sequence.

Intersection of A000040 and A095096.

Cf. A014417, A095083, A095064.

Select[Flatten[Position[Mod[DigitCount[Select[Range[0, 6000], BitAnd[#, 2 #] == 0 &], 2, 1], 2], 0]] - 1, PrimeQ] (* Amiram Eldar, Feb 07 2023 *)

A095111 One minus the parity of 1-fibits in Zeckendorf expansion A014417(n).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1

Offset: 0

Antti Karttunen, Jun 01 2004

nonn

Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.

Amiram Eldar, Table of n, a(n) for n = 0..10000
Leonard Rozendaal, Pisano word, tesselation, plane-filling fractal, Preprint, 2017.
Index entries for characteristic functions.

Characteristic function of A095096.
Run counts are given by A095276.
Cf. A000045, A003714, A010059, A014417, A095076, A105774.

1 - Mod[DigitCount[Select[Range[0, 540], BitAnd[#, 2 #] == 0 &], 2, 1], 2] (* Amiram Eldar, Feb 05 2023 *)

def ok(n): return 1 if n==0 else n*(2*n & n == 0)
print([1 - bin(n)[2:].count("1")%2 for n in range(1001) if ok(n)]) # Indranil Ghosh, Jun 08 2017

a(n) = A010059(A003714(n)).
a(n) = 1 - A095076(n).
a(n) = a'(n+1) where a'(1) = 1 and if n >= 2 with F(k) < n <= F(k+1), a'(n)=1-a'(n-F(k)), where F(k) = A000045(k). E.g., F(5) = 5 < 6 <= F(6) = 8, thus a'(6) = 1 - a'(1) = 0 and a(5) = 0. - Benoit Cloitre, May 10 2005

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A337287 Numbers k such that both k and k+1 are in A095096.

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A337289 Numbers k such that k+1 is in A095096 and k is in A020899.

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A337634 Number of numbers k <= n such that both k and k+1 are in A095096.

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A337635 Number of numbers k <= n such that k is in A095096 and k+1 is in A020899.

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A337636 Number of numbers k <= n such that k+1 is in A095096 and k is in A020899.

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A337288 Numbers k such that k is in A095096 and k+1 is in A020899.

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Mathematica

A337290 Numbers k such that both k and k+1 are in A020899.

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Mathematica

A337637 Number of numbers k <= n such that both k and k+1 are in A020899.

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Mathematica

Extensions

A095084 Fibevil primes, i.e., primes p whose Zeckendorf-expansion A014417(p) contains an even number of 1-fibits.

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