A003622 -id:A003622 - cp's OEIS frontend

A319967 a(n) = A003145(A003144(n)) where A003144 and A003145 are positions of '1' and '2' in the tribonacci word A092782.

2, 9, 15, 22, 26, 33, 39, 46, 53, 59, 66, 70, 77, 83, 90, 96, 103, 107, 114, 120, 127, 134, 140, 147, 151, 158, 164, 171, 175, 182, 188, 195, 202, 208, 215, 219, 226, 232, 239, 245, 252, 256, 263, 269, 276, 283, 289, 296, 300, 307, 313, 320, 327, 333, 340, 344

Offset: 1

N. J. A. Sloane, Oct 05 2018

By analogy with the Wythoff compound sequences A003622 etc., the nine compounds of A003144, A003145, A003146 might be called the tribonacci compound sequences. They are A278040, A278041, and A319966-A319972.

This sequence gives the positions of the word bac in the tribonacci word t = abacabaa..., fixed point of the morphism a->ab, b->ac, c->a. This follows from the fact that the word ac is always preceded in t by the letter b, and the formula BA = C-2, where A := A003144, B := A003145, C := A003146. - Michel Dekking, Apr 09 2019

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Elena Barcucci, Luc Belanger and Srecko Brlek, On tribonacci sequences, Fib. Q., 42 (2004), 314-320. Compare page 318.
L. Carlitz, R. Scoville and V. E. Hoggatt, Jr., Fibonacci representations of higher order, Fib. Quart., 10 (1972), 43-69, Theorem 13.

Cf. A003144, A003145, A003146, A003622, A278039, A278040, A278041, and A319966-A319972.

Cf. A092782 (ternary tribonacci word).

a(n+1) = B(A(n)) = B(A(n) + 1) - 2 = A(n) + B(n) + n + 1, for n >= 0, where B = A278039 and A = A278040. For a proof see the W. Lang link in A278040, Proposition 9, eq. (51). - Wolfdieter Lang, Dec 13 2018

More terms from Rémy Sigrist, Oct 16 2018

A348853 Delete any least significant 0's from the Zeckendorf representation of n, leaving its "odd" part.

Original entry on oeis.org

1, 1, 1, 4, 1, 6, 4, 1, 9, 6, 4, 12, 1, 14, 9, 6, 17, 4, 19, 12, 1, 22, 14, 9, 25, 6, 27, 17, 4, 30, 19, 12, 33, 1, 35, 22, 14, 38, 9, 40, 25, 6, 43, 27, 17, 46, 4, 48, 30, 19, 51, 12, 53, 33, 1, 56, 35, 22, 59, 14, 61, 38, 9, 64, 40, 25, 67, 6, 69, 43, 27, 72

Offset: 1

Kevin Ryde, Nov 14 2021

Terms are odd Zeckendorfs A003622 and the fixed points are where n is odd already so that a(n) = n iff n is in A003622.
A139764(n) is the least significant "10..00" part of n so Zeckendorf multiplication n = A101646(a(n), A139764(n)).
The equivalent delete least significant 0's in binary is A000265 so that conversion to Fibbinary (A003714) and back gives a(n) = A022290(A000265(A003714(n))).
a(n) = 1 iff n is a Fibonacci number >= 1 (A000045) since they are Zeckendorf 100..00.
a(n) = 4 iff n is a Lucas number >= 4 (A000032) since they are Zeckendorf 10100..00 which reduces to 101.
In the Wythoff array A035513, a(n) is the term in column 0 of the row containing n, and hence the formula below using row number A019586 to select which of the odds (column 0) is a(n).

			n    = 81 = Zeckendorf 101001000.
a(n) = 19 = Zeckendorf 101001.

Cf. A189920 (Zeckendorf digits), A003622 (odds), A003849 (final digit), A005206, A319433 (shift down).
Cf. A000045 (Fibonacci), A000032 (Lucas).
Cf. A035513 (Wythoff array), A019586 (row number).
Cf. A003714 (Fibbinary), A022290 (its inverse).
In other bases: A000265 (binary), A004151 (decimal).
Cf. A001622, A101646, A139764.

my(phi=quadgen(5)); a(n) = my(q,r); while([q,r]=divrem(n+2,phi); r<1, n=q-1); n;

a(n) = n if A003849(n)=1, otherwise a(n) = a(A005206(n)) = a(A319433(n)).
a(n) = A003622(A019586(n) + 1).
Sum_{k=1..n} a(k) ~ n^2/(2*phi), where phi is the golden ratio (A001622). - Amiram Eldar, Feb 17 2024

A035339 5th column of Wythoff array.

Original entry on oeis.org

8, 29, 42, 63, 84, 97, 118, 131, 152, 173, 186, 207, 228, 241, 262, 275, 296, 317, 330, 351, 364, 385, 406, 419, 440, 461, 474, 495, 508, 529, 550, 563, 584, 605, 618, 639, 652, 673, 694, 707, 728, 741, 762, 783, 796, 817, 838, 851, 872, 885, 906, 927, 940, 961

Offset: 0

N. J. A. Sloane and J. H. Conway

nonn

The asymptotic density of this sequence is 1/phi^6 = A094214^6 = 0.05572809... . - Amiram Eldar, Mar 24 2025

Seiichi Manyama, Table of n, a(n) for n = 0..10000
John H. Conway and N. J. A. Sloane, Notes on the Para-Fibonacci and related sequences.
N. J. A. Sloane, Classic Sequences.

Column k of A035513: A003622 (k=1), A035336 (k=2), A035337 (k=3), A035338 (k=4), this sequence (k=5), A035340 (k=6).

Cf. A094214.

t:= (1+sqrt(5))/2: [ seq(8*floor((n+1)*t)+5*n,n=0..80) ];

a[n_] := 8 * Floor[n * GoldenRatio] + 5*(n-1); Array[a, 100] (* Amiram Eldar, Mar 24 2025 *)

A035340 6th column of Wythoff array.

Original entry on oeis.org

13, 47, 68, 102, 136, 157, 191, 212, 246, 280, 301, 335, 369, 390, 424, 445, 479, 513, 534, 568, 589, 623, 657, 678, 712, 746, 767, 801, 822, 856, 890, 911, 945, 979, 1000, 1034, 1055, 1089, 1123, 1144, 1178, 1199, 1233, 1267, 1288, 1322, 1356, 1377, 1411, 1432

Offset: 0

N. J. A. Sloane and J. H. Conway

nonn

The asymptotic density of this sequence is 1/phi^7 = A094214^7 = 0.03444185... . - Amiram Eldar, Mar 24 2025

Seiichi Manyama, Table of n, a(n) for n = 0..10000
John H. Conway and N. J. A. Sloane, Notes on the Para-Fibonacci and related sequences.
N. J. A. Sloane, Classic Sequences.

Column k of A035513: A003622 (k=1), A035336 (k=2), A035337 (k=3), A035338 (k=4), A035339 (k=5), this sequence (k=6).

Cf. A094214.

t:= (1+sqrt(5))/2: [ seq(13*floor((n+1)*t)+8*n,n=0..80) ];

a[n_] := 13 * Floor[n * GoldenRatio] + 8*(n-1); Array[a, 100] (* Amiram Eldar, Mar 24 2025 *)

A051656 Sum_{i=0..n} (C(n,i) mod 2)Fibonacci(2i).

Original entry on oeis.org

0, 1, 3, 12, 21, 77, 168, 609, 987, 3572, 7755, 28059, 47376, 171409, 372099, 1346268, 2178309, 7881197, 17108664, 61899729, 104512485, 378129724, 820851717, 2969869413, 4809706272, 17401680769, 37775923491, 136674575148

Offset: 0

Antti Karttunen, Nov 30 1999

Positions in the first column (A003622) of Wythoff array of the terms which have their Zeckendorf Expansion patterned as row[2n+1] in Pascal's Triangle computed modulo 2 (A047999)

Proof in preparation, to be published (see A048757).

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Cf. A048757, A047999, A035513, A038183, A051256. First row of A050609, First column of A050610.
a(n) = A019586[A048757[n]]. A048757[n] = SS(Athis_sequence[n])+1, where SSx means the second Fibonacci Successor of x (= x's Z.E. shifted left twice).
Cf. A001906.

a051656 = sum . zipWith (*) a001906_list . a047999_row
-- Reinhard Zumkeller, Feb 27 2011

Table[Sum[Mod[Binomial[n,i],2]Fibonacci[2i],{i,0,n}],{n,0,30}] (* Harvey P. Dale, Oct 30 2011 *)

a(n)=sum(i=0,n,if(!bitand(i,n-i),fibonacci(2*i))) \\ Charles R Greathouse IV, Jan 04 2013

a(n) = sum_{i=0..n} (C(2n, 2i) mod 2)*F(2*i) = FL(n)product_{i=0..inf} L(2^i)^bit(n, i) where L is n-th Lucas number (A000032) and FL is defined as in A048757: FL(n) = n-th fibonacci number if n even, n-th Lucas number if n odd.

A095081 Fibodd primes, i.e., primes p whose Zeckendorf-expansion A014417(p) ends with one.

Original entry on oeis.org

17, 19, 43, 53, 59, 61, 67, 101, 103, 127, 137, 163, 179, 197, 211, 229, 239, 263, 271, 281, 307, 313, 331, 347, 349, 373, 383, 389, 433, 449, 457, 467, 491, 499, 509, 569, 577, 593, 601, 619, 643, 653, 661, 677, 739, 773, 787, 797, 821, 823

Offset: 1

Antti Karttunen, Jun 01 2004

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..10000
A. Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

Intersection of A000040 and A003622. Union of A095086 and A095089. Cf. A095061, A095080.

r = Map[Fibonacci, Range[2, 12]]; Select[Prime@ Range@ 144, Last@ Flatten@ Map[Position[r, #] &, Abs@ Differences@ NestWhileList[Function[k, k - SelectFirst[Reverse@ r, # < k &]], # + 1, # > 1 &]] == 1 &] (* Michael De Vlieger, Mar 27 2016, Version 10 *)

genit(maxx)={for(n=1,maxx,q=(n-1)+(n+sqrtint(5*n^2))\2; if(isprime(q), print1(q,",")));} \\ Bill McEachen, Mar 26 2016

from sympy import fibonacci, primerange
def a(n):
    k=0
    x=0
    while n>0:
        k=0
        while fibonacci(k)<=n: k+=1
        x+=10**(k - 3)
        n-=fibonacci(k - 1)
    return x
def ok(n):
    return str(a(n))[-1]=="1"
print([n for n in primerange(1, 1001) if ok(n)]) # Indranil Ghosh, Jun 07 2017

A095098 Fib001 numbers: those k for which the Zeckendorf expansion A014417(k) ends with two zeros and a final one.

Original entry on oeis.org

6, 9, 14, 19, 22, 27, 30, 35, 40, 43, 48, 53, 56, 61, 64, 69, 74, 77, 82, 85, 90, 95, 98, 103, 108, 111, 116, 119, 124, 129, 132, 137, 142, 145, 150, 153, 158, 163, 166, 171, 174, 179, 184, 187, 192, 197, 200, 205, 208, 213, 218, 221, 226, 229, 234, 239, 242

Offset: 1

Antti Karttunen, Jun 01 2004

The asymptotic density of this sequence is sqrt(5)-2. - Amiram Eldar, Mar 21 2022

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

Cf. A014417, A095086 (fib001 primes).

Set-wise difference of A003622 - A134860.

a[n_] = 2 Floor[(n + 1) GoldenRatio^2] - n - 3;
a /@ Range[100] (* Jean-François Alcover, Oct 28 2019, after Vladeta Jovovic *)

from sympy import fibonacci
def a(n):
    k=0
    x=0
    while n>0:
        k=0
        while fibonacci(k)<=n: k+=1
        x+=10**(k - 3)
        n-=fibonacci(k - 1)
    return x
def ok(n): return str(a(n))[-3:]=="001"
print([n for n in range(1, 501) if ok(n)]) # Indranil Ghosh, Jun 08 2017

a(n) = 2*floor((n+1)*phi^2)-n-3, where phi = (1+sqrt(5))/2. - Vladeta Jovovic, Jul 05 2004

A102616 Nonprime numbers of order 3.

Original entry on oeis.org

1, 14, 16, 22, 24, 25, 30, 33, 35, 36, 39, 44, 46, 48, 50, 51, 54, 55, 56, 62, 64, 66, 68, 69, 70, 75, 76, 77, 80, 85, 86, 87, 90, 92, 93, 94, 96, 100, 102, 104, 105, 108, 111, 115, 116, 117, 118, 120, 122, 123, 124, 126, 130, 132, 134, 136, 138, 142, 144, 145, 148, 150

Offset: 1

Cino Hilliard, Jan 31 2005

nonn

nps(n,1) -> list nonprime(n) or the sequence of nonprime numbers. nps(n,2) -> list nonprime(nonprime(n)) or nps of order 2. nps(n,3) -> list nonprime(nonprime(nonprime(n))) or npcs of order 3 ..... The order is the number of nestings - 1.

			Nonprime(2) = 4.
Nonprime(4) = 8.
Nonprime(8) = 14, the 2nd entry.

Cf. A018252, A102615.

Let A = primes A000040, B = nonprimes A018252. The 2-level compounds are AA = A006450, AB = A007821, BA = A078782, BB = A102615. The 3-level compounds AAA, AAB, ..., BBB are A038580, A049078, A270792, A102617, A270794, A270796, A102216.

# For Maple code for the prime/nonprime compound sequences (listed in cross-references) see A003622. - N. J. A. Sloane, Mar 30 2016

nonPrime[n_] := FixedPoint[n + PrimePi[ # ] &, n]; Nest[ nonPrime, Range[62], 3] (* Robert G. Wilson v, Feb 04 2005 *)

\\ We perform nesting(s) with a loop.
cics(n,m) = { local(x,y,z); for(x=1,n, z=x; for(y=1,m+1, z=nonprime(z); ); print1(z",") ) }
nonprime(n) = { local(c,x); c=1; x=0; while(c <= n, x++; if(!isprime(x),c++); ); return(x) }

Edited by Robert G. Wilson v, Feb 04 2005

A134570 Array T(n,k) by antidiagonals; T(n,k) = position in row n of k-th occurrence of the Fibonacci number F(2n+1) in A134566.

Original entry on oeis.org

2, 5, 1, 7, 4, 3, 10, 6, 11, 8, 13, 9, 16, 29, 21, 15, 12, 24, 42, 76, 55, 18, 14, 32, 63, 110, 199, 144, 20, 17, 37, 84, 165, 288, 521, 377, 23, 19, 45, 97, 220, 432, 754

Offset: 1

Clark Kimberling, Nov 02 2007

(Row 1) = A001950, the upper Wythoff sequence (Row 2) = (Column 1 of Wythoff array) = A003622 (Row 3) = (Column 3 of Wythoff array) = A035337 (Row 4) = (Column 5 of Wythoff array) = A035339 Except for initial terms, the first two columns of A134570 are bisected Fibonacci and Lucas sequences, A001906 and A002878, resp. Row 1 is the ordered union of all even-numbered columns of the Wythoff array; and A134570 is a permutation of the positive integers.

			Northwest corner:
2 5 7 10 13 15 18 20 23 26
1 4 6 9 12 14
3 11 16 24 32 37
8 29 42 63 84 97
Row 1 consists of numbers k such that 1 is the least m for which {-m*tau}>{k*tau}, where tau=(1+sqrt(5))/2 and {} denotes fractional part.

Cf. A134566, A134571.

A143299 Number of terms in the Zeckendorf representation of every number in row n of the Wythoff array.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Aug 05 2008

nonn

Every number in a row of the Wythoff array has the same number of Zeckendorf summands as the first number in the row; hence A035513(n) is the number of Zeckendorf summands of A003622(n)=n-1+Floor(n*tau), where tau=(1+sqrt(5))/2.

Let M(1) = 1, M(2) = 2 and for n >= 3, M(n) = M(n-1).f(M(n-2)) where f() increments by one and the dot stands for concatenation, then this sequence is 0.M(1).M(2).M(3).M(4)... , see the example. - Joerg Arndt, May 14 2011

			Row 5 of the Wythoff array is (12, 20, 32, ...) and corresponding Zeckendorf representations all have 3 terms:
12 = 1 + 3 + 8,
20 = 2 + 5 + 13,
32 = 3 + 8 + 21, etc.
From _Joerg Arndt_, May 14 2011: (Start)
The sequence as an irregular triangle:
1,        = M(1)
1,        = M(2)
1, 2,     = M(3) = M(2).f(M(1))
1, 2, 2,  = M(4) = M(3).f(M(2))
1, 2, 2, 2, 3,
1, 2, 2, 2, 3, 2, 3, 3,
1, 2, 2, 2, 3, 2, 3, 3, 2, 3, 3, 3, 4,
1, 2, 2, 2, 3, 2, 3, 3, 2, 3, 3, 3, 4, 2, 3, 3, 3, 4, 3, 4, 4,
1, 2, 2, 2, 3, 2, 3, 3, 2, 3, 3, 3, 4, 2, 3, 3, 3, 4, 3, 4, 4, 2, 3, 3, 3, ...
(End)

Paolo Xausa, Table of n, a(n) for n = 1..10946

Cf. A003622, A007895, A035513, A134561.

Flatten[Nest[{Flatten[#], #[[1]] + 1} &, {1, 2}, 9]] (* Paolo Xausa, Jun 17 2024 *)

a(n) = A007895(n-1) + 1. - Paolo Xausa, Jun 17 2024

cp's OEIS Frontend

A319967 a(n) = A003145(A003144(n)) where A003144 and A003145 are positions of '1' and '2' in the tribonacci word A092782.

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A348853 Delete any least significant 0's from the Zeckendorf representation of n, leaving its "odd" part.

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A035339 5th column of Wythoff array.

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A035340 6th column of Wythoff array.

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A051656 Sum_{i=0..n} (C(n,i) mod 2)*Fibonacci(2*i).

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A095081 Fibodd primes, i.e., primes p whose Zeckendorf-expansion A014417(p) ends with one.

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A095098 Fib001 numbers: those k for which the Zeckendorf expansion A014417(k) ends with two zeros and a final one.

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A102616 Nonprime numbers of order 3.

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A051656 Sum_{i=0..n} (C(n,i) mod 2)Fibonacci(2i).