A006209 -id:A006209 - cp's OEIS frontend

A006206 Number of aperiodic binary necklaces of length n with no subsequence 00, excluding the necklace "0".

1, 1, 1, 1, 2, 2, 4, 5, 8, 11, 18, 25, 40, 58, 90, 135, 210, 316, 492, 750, 1164, 1791, 2786, 4305, 6710, 10420, 16264, 25350, 39650, 61967, 97108, 152145, 238818, 374955, 589520, 927200, 1459960, 2299854, 3626200, 5720274, 9030450, 14263078

Offset: 1

N. J. A. Sloane and Frank Ruskey

Bau-Sen Du (1985/1989)'s Table 1 has this sequence, denoted A_{n,1}, as the second column. - Jonathan Vos Post, Jun 18 2007

			Necklaces are: 1, 10, 110, 1110; 11110, 11010, 111110, 111010, ...

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 499.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

James Spahlinger, Table of n, a(n) for n = 1..5000
Ricardo Gómez Aíza, Symbolic dynamical scales: modes, orbitals, and transversals, arXiv:2009.02669 [math.DS], 2020.
Joerg Arndt, Matters Computational (The Fxtbook), p. 710.
Kam Cheong Au, Evaluation of one-dimensional polylogarithmic integral, with applications to infinite series, arXiv:2007.03957 [math.NT], 2020. See 2nd line of Table 1 (p. 6).
Michael Baake, Joachim Hermisson, and Peter Pleasants, The torus parametrization of quasiperiodic LI-classes, J. Phys. A 30(9) (1997), 3029-3056.
Latham Boyle and Paul J. Steinhardt, Self-Similar One-Dimensional Quasilattices, arXiv:1608.08220 [math-ph], 2016.
D. J. Broadhurst, On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory, arXiv:hep-th/9604128, 1996.
D. J. Broadhurst and D. Kreimer, Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops, arXiv:hep-th/9609128, 1996.
D. J. Broadhurst and D. Kreimer, Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops, Phys. Lett B., 393 (1997), 403-412.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. 3 (2000), #00.1.5.
M. Conder, S. Du, R. Nedela, and M. Skoviera, Regular maps with nilpotent automorphism group, Journal of Algebraic Combinatorics, 44(4) (2016), 863-874. ["... We note that the sequence h_n above agrees in all but the first term with the sequence A006206 in ..."]
Bau-Sen Du, The Minimal Number of Periodic Orbits of Periods Guaranteed in Sharkovskii's Theorem, Bull. Austral. Math. Soc. 31 (1985), 89-103. Corrigendum: 32 (1985), 159.
Bau-Sen Du, The Minimal Number of Periodic Orbits of Periods Guaranteed in Sharkovskii's Theorem, arXiv:0706.2297 [math.DS], 2007.
Bau-Sen Du, A Simple Method Which Generates Infinitely Many Congruence Identities, Fib. Quart. 27 (1989), 116-124.
Bau-Sen Du, A Simple Method Which Generates Infinitely Many Congruence Identities, arXiv:0706.2421 [math.NT], 2007.
Larry Ericksen, Primality Testing and Prime Constellations, Šiauliai Mathematical Seminar, 3(11), 2008; mentions this sequence.
R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, arXiv:0903.2514 [math.NT], 2009-2011; sequence gamma_{1,j}^(A).
A. Pakapongpun and T. Ward, Functorial Orbit counting, J. Integer Seqs. 12 (2009), #09.2.4; example 21.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs. 4 (2001), #01.2.1.
Robert Schneider, Andrew V. Sills, and Hunter Waldron, On the q-factorization of power series, arXiv:2501.18744 [math.CO], 2025. See p. 6.
Index entries for sequences related to Lyndon words

Cf. A006207 (A_{n,2}), A006208 (A_{n,3}), A006209 (A_{n,4}), A130628 (A_{n,5}), A208092 (A_{n,6}), A006210 (D_{n,2}), A006211 (D_{n,3}), A094392.
Cf. A001461 (partial sums), A000045, A008683, A027750.
Cf. A125951 and A113788 for similar sequences.

a006206 n = sum (map f $ a027750_row n) `div` n where
   f d = a008683 (n `div` d) * (a000045 (d - 1) + a000045 (d + 1))
-- Reinhard Zumkeller, Jun 01 2013

with(numtheory): with(combinat):
A006206 := proc(n) local sum; sum := 0; for d in divisors(n) do sum := sum + mobius(n/d)*(fibonacci(d+1)+fibonacci(d-1)) end do; sum/n; end proc:

a[n_] := Total[(MoebiusMu[n/#]*(Fibonacci[#+1] + Fibonacci[#-1]) & ) /@ Divisors[n]]/n;
(* or *) a[n_] := Sum[LucasL[k]*MoebiusMu[n/k], {k, Divisors[n]}]/n; Table[a[n], {n,100}] (* Jean-François Alcover, Jul 19 2011, after given formulas *)

a(n)=if(n<1,0,sumdiv(n,d,moebius(n/d)*(fibonacci(d-1)+fibonacci(d+1)))/n)

z = PowerSeriesRing(ZZ, 'z').gen().O(30)
r = (1 - (z + z**2))
F = -z*r.derivative()/r
[sum(moebius(n//d)*F[d] for d in divisors(n))//n for n in range(1, 24)] # F. Chapoton, Apr 24 2020

Euler transform is Fibonacci(n+1): 1/((1 - x) * (1 - x^2) * (1 - x^3) * (1 - x^4) * (1 - x^5)^2 * (1 - x^6)^2 * ...) = 1/(Product_{n >= 1} (1 - x^n)^a(n)) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 8*x^5 + ...
Coefficients of power series of natural logarithm of the infinite product Product_{n>=1} (1 - x^n - x^(2*n))^(-mu(n)/n), where mu(n) is the Moebius function. This is related to Fibonacci sequence since 1/(1 - x^n - x^(2*n)) expands to a power series whose terms are Fibonacci numbers.
a(n) = (1/n) * Sum_{d|n} mu(n/d) * (Fibonacci(d-1) + Fibonacci(d+1)) = (1/n) * Sum_{d|n} mu(n/d) * Lucas(d). Hence Lucas(n) = Sum_{d|n} d*a(d).
a(n) = round((1/n) * Sum_{d|n} mu(d)*phi^(n/d)), n > 2. - David Broadhurst [Formula corrected by Jason Yuen, Dec 29 2024]
G.f.: Sum_{n >= 1} -mu(n) * log(1 - x^n - x^(2*n))/n.
a(n) = (1/n) * Sum_{d|n} mu(d) * A001610(n/d - 1), n > 1. - R. J. Mathar, Mar 07 2009
For n > 2, a(n) = A060280(n) = A031367(n)/n.

A006208 Generalized Fibonacci numbers A_{n,3}.

Original entry on oeis.org

1, 1, 0, 1, 0, 2, 1, 3, 2, 6, 4, 9, 8, 18, 16, 32, 32, 61, 64, 115, 128, 224, 258, 431, 520, 850, 1050, 1673, 2128, 3328, 4320, 6649, 8788, 13366, 17920, 26957, 36610, 54634, 74932, 111057, 153656, 226514, 315616, 463243, 649334, 949823, 1337984, 1951760

Offset: 1

N. J. A. Sloane

nonn

Bau-Sen Du's [1985/2007] Table 1, p. 6, has this sequence as the 4th column. - Jonathan Vos Post, Jun 18 2007

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Bau-Sen Du, The Minimal Number of Periodic Orbits of Periods Guaranteed in Sharkovskii's Theorem. Bull. Austral. Math. Soc. 31(1985), 89-103. Corrigendum: 32 (1985), 159.
Bau-Sen Du, A Simple Method Which Generates Infinitely Many Congruence Identities, Fib. Quart. 27 (1989), 116-124.

Cf. A006206 (A_{n,1}), A006207 (A_{n,2}), A006209 (A_{n,4}), A130628 (A_{n,5}), A208092 (A_{n,6}), A006210 (D_{n,2}), A006211 (D_{n,3}), A094392.

max = 50;
Do[Do[b1[1][j, n] = 0; b1[2][j, n] = 1; b2[1][j, n] = b2[2][j, n] = 0, {j, n}]; b2[1][n, n] = b2[2][n, n] = 1, {n, max}];
Do[Do[Do[b1[k][j, n] = b1[k-2][1, n] + b1[k-2][j+1, n]; b2[k][j, n] = b2[k - 2][1, n] + b2[k-2][j+1, n], {j, n-1}]; b1[k][n, n] = b1[k-2][1, n] + b1[k-1][n, n]; b2[k][n, n] = b2[k-2][1, n] + b2[k-1][n, n], {n, max}], {k, 3, max}];
phin[n_] := Table[b2[m][n, n] + 2 Sum[If[m+2-2j > 0, b1[m+2-2j][j, n], 0], {j, n}], {m, max}];
MT[s_] := Table[DivisorSum[n, MoebiusMu[#] s[[n/#]]&]/n, {n, Length[s]}];
MT[phin[3]] (* Jean-François Alcover, Oct 01 2018, after Max Alekseyev in A006207 *)

\\ implementation of MT() and phin() is given in A006207
MT(phin(3)) \\ sequence A_{n,3} \\ Max Alekseyev, Feb 23 2012

arxiv URL replaced with non-cached version, and duplicate of a reference removed, by R. J. Mathar, Oct 30 2009

Terms a(32) onward from Max Alekseyev, Feb 23 2012

A094392 Antidiagonals of the tables formed from b(m,2,n,n), which is defined in Du 1989.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 5, 1, 1, 1, 1, 2, 8, 1, 1, 1, 1, 1, 3, 13, 1, 1, 1, 1, 1, 1, 5, 21, 1, 1, 1, 1, 1, 1, 2, 7, 34, 1, 1, 1, 1, 1, 1, 1, 3, 11, 55, 1, 1, 1, 1, 1, 1, 1, 1, 5, 16, 89, 1, 1, 1, 1, 1, 1, 1, 1, 2, 7, 25, 144, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 11, 37, 233, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Amy Robinson (amylou(AT)mchsi.com), Apr 28 2004

			E.g., for m = 5 and n = 2, b(5,2,2,2)= b(3,2,1,2) + b(4,2,2,2)= 2 because of the definition in the reference.
    1   1  1  1  1 1 1 1 1 1 1 1 1 1 1
    1   1  1  1  1 1 1 1 1 1 1 1 1 1 1
    2   1  1  1  1 1 1 1 1 1 1 1 1 1 1
    3   1  1  1  1 1 1 1 1 1 1 1 1 1 1
    5   2  1  1  1 1 1 1 1 1 1 1 1 1 1
    8   3  1  1  1 1 1 1 1 1 1 1 1 1 1
   13   5  2  1  1 1 1 1 1 1 1 1 1 1 1
   21   7  3  1  1 1 1 1 1 1 1 1 1 1 1
   34  11  5  2  1 1 1 1 1 1 1 1 1 1 1
   55  16  7  3  1 1 1 1 1 1 1 1 1 1 1
   89  25 11  5  2 1 1 1 1 1 1 1 1 1 1
  144  37 15  7  3 1 1 1 1 1 1 1 1 1 1
  233  57 23 11  5 2 1 1 1 1 1 1 1 1 1
  377  85 32 15  7 3 1 1 1 1 1 1 1 1 1
  610 130 49 23 11 5 2 1 1 1 1 1 1 1 1

Bau-Sen Du, A Simple Method Which Generates Infinitely Many Congruence Identities, Fib. Quart. 27 (1989), 116-124.
Bau-Sen Du, A Simple Method Which Generates Infinitely Many Congruence Identities, arXiv:0706.2421 [math.NT], 2007.

Cf. A006206 (A_{n,1}), A006207 (A_{n,2}), A006208 (A_{n,3}), A006209 (A_{n,4}), A130628 (A_{n,5}), A208092 (A_{n,6}), A006210 (D_{n,2}), A006211 (D_{n,3}), A094392.

b := proc(k,i,j,n) option remember; if k = 1 then if i = 1 then return 0; end if; if i = 2 then if j = n then return 1; end if; return 0; end if; end if; if k = 2 then if i = 1 then return 1; end if; if i = 2 then if j = n then return 1; end if; return 0; end if; end if; if j = n then return b(k-2, i, 1, n) + b(k-1, i, n, n); end if; return b(k-2, i, 1, n) + b(k-2, i, j+1, n); end proc; # Chris Deugau (deugaucj(AT)uvic.ca), Dec 19 2005

b[k_, i_, j_, n_] := b[k, i, j, n] = Which[k == 1, Which[i == 1, 0, i == 2 , If[j == n, 1, 0], True, 0], k == 2, Which[i == 1, 1, i == 2, If[j == n, 1, 0], True, 0], j == n, b[k - 2, i, 1, n] + b[k - 1, i, n, n], True, b[k - 2, i, 1, n] + b[k - 2, i, j + 1, n]];
a[m_, n_] := b[m, 2, n, n];
Table[a[m - n + 1, n], {m, 1, 14}, {n, m, 1, -1}] // Flatten (* Jean-François Alcover, Nov 21 2017, adapted from Maple *)

For i=2 and k >= 1 b(k+2, 2, n, n)=b(k, 2, 1, n) + b(k+1, 2, n, n). The remaining portion for the recurrence is defined in Du 1989.

Corrected and extended by Chris Deugau (deugaucj(AT)uvic.ca), Dec 19 2005

Typo 891 -> 89,1 corrected by Jean-François Alcover, Nov 21 2017

A006207 Generalized Fibonacci numbers A_{n,2}.

Original entry on oeis.org

1, 1, 0, 1, 1, 2, 2, 3, 4, 6, 8, 11, 16, 23, 32, 46, 66, 94, 136, 195, 282, 408, 592, 856, 1248, 1814, 2646, 3858, 5644, 8246, 12088, 17706, 25992, 38155, 56102, 82490, 121474, 178902, 263776, 389033, 574304, 848069, 1253344, 1852926, 2741164

Offset: 1

N. J. A. Sloane

nonn

Bau-Sen Du (1985)'s Table 1, p. 6, has this sequence as the third column. - Jonathan Vos Post, Jun 18 2007

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Bau-Sen Du, The Minimal Number of Periodic Orbits of Periods Guaranteed in Sharkovskii's Theorem. Bull. Austral. Math. Soc. 31(1985), 89-103. Corrigendum: 32 (1985), 159.
Bau-Sen Du, A Simple Method Which Generates Infinitely Many Congruence Identities, Fib. Quart., 27 (1989), 116-124.

Cf. A006206 (A_{n,1}), A006208 (A_{n,3}), A006209 (A_{n,4}), A130628 (A_{n,5}), A208092 (A_{n,6}), A006210 (D_{n,2}), A006211 (D_{n,3}), A094392.

max = 100; Clear[b1, b2]; For[n=1, n <= max, n++, For[j=1, j <= n, j++, b1[1][j, n] = 0; b1[2][j, n] = 1; b2[1][j, n] = b2[2][j, n] = 0]; b2[1][n, n] = b2[2][n, n] = 1]; For[k=3, k <= max, k++, For[n=1, n <= max, n++, For[j=1, j <= n-1, j++, b1[k][j, n] = b1[k-2][1, n] + b1[k - 2][j+1, n]; b2[k][j, n] = b2[k-2][1, n] + b2[k-2][j+1, n]; ]; b1[k][n, n] = b1[k-2][1, n] + b1[k-1][n, n]; b2[k][n, n] = b2[k-2][1, n] + b2[k - 1][n, n]]];
phin[n_] := Table[b2[m][n, n] + 2*Sum[If[m + 2 - 2*j > 0, b1[m + 2 - 2*j][j, n], 0], {j, 1, n}], {m, 1, max}];
MT[s_List] := Table[ DivisorSum[n, MoebiusMu[#]*s[[n/#]]&]/n, {n, 1, Length[s]}];
MT[phin[2]] (* Jean-François Alcover, Dec 07 2015, adapted from Max Alekseyev's PARI script *)

b1 = vector(100,k,matrix(100,100)); b2 = vector(100,k,matrix(100,100)); for(n=1,100, for(j=1,n, b1[1][j,n]=0; b1[2][j,n]=1; b2[1][j,n] = b2[2][j,n] = 0); b2[1][n,n] = b2[2][n,n] = 1); for(k=3,100, for(n=1,100, for(j=1,n-1, b1[k][j,n] = b1[k-2][1,n] + b1[k-2][j+1,n]; b2[k][j,n] = b2[k-2][1,n] + b2[k-2][j+1,n]; ); b1[k][n,n] = b1[k-2][1,n] + b1[k-1][n,n]; b2[k][n,n] = b2[k-2][1,n] + b2[k-1][n,n]; )); \\ Computing arrays b(k,1,j,n) and b(k,2,j,n)
{ phin(n) = vector(100,m, b2[m][n,n] + 2*sum(j=1,n, if(m+2-2*j>0, b1[m+2-2*j][j,n]))) } \\ sequence phi_n
{ MT(s) = vector(#s,n,sumdiv(n,d,moebius(d)*s[n/d])/n) } \\ Moebius transform
MT( phin(2) ) \\ sequence A_{n,2}
\\ Max Alekseyev, Feb 23 2012

arxiv URL replaced with non-cached version by R. J. Mathar, Oct 30 2009

Terms a(32) onward from Max Alekseyev, Feb 23 2012

A130628 Related to the minimal number of periodic orbits of periods guaranteed by Sharkovskii's theorem.

Original entry on oeis.org

1, 1, 0, 1, 0, 2, 0, 3, 0, 6, 1, 9, 2, 18, 4, 30, 8, 56, 16, 99, 32, 186, 64, 337, 128, 635, 256, 1177, 512, 2220, 1024, 4176, 2048, 7930, 4098, 15044, 8200, 28738, 16410, 54937, 32848, 105474, 65760, 202845, 131668, 391316, 263680, 756223, 528128

Offset: 1

Jonathan Vos Post, Jun 18 2007

nonn

Bau-Sen Du's [1985/2007] Table 1, p. 6, has this sequence as the 6th column, denoted A_{n,5}.

Bau-Sen Du, The Minimal Number of Periodic Orbits of Periods Guaranteed in Sharkovskii's Theorem, arXiv:0706.2297 [math.DS], 2007; Bull. Austral. Math. Soc. 31(1985), 89-103. Corrigendum: 32 (1985), 159.

Cf. A006206 (A_{n,1}), A006207 (A_{n,2}), A006208 (A_{n,3}), A006209 (A_{n,4}), A208092 (A_{n,6}), A006210 (D_{n,2}), A006211 (D_{n,3}), A094392.

max = 50; Clear[b1, b2];
For[n = 1, n <= max, n++,
For[j = 1, j <= n, j++, b1[1][j, n] = 0; b1[2][j, n] = 1; b2[1][j, n] = b2[2][j, n] = 0]; b2[1][n, n] = b2[2][n, n] = 1];
For[k = 3, k <= max, k++,
For[n = 1, n <= max, n++,
For[j = 1, j <= n-1, j++, b1[k][j, n] = b1[k-2][1, n] + b1[k-2][j+1, n]; b2[k][j, n] = b2[k-2][1, n] + b2[k-2][j+1, n]]; b1[k][n, n] = b1[k-2][1, n] + b1[k-1][n, n]; b2[k][n, n] = b2[k-2][1, n] + b2[k-1][n, n]]];
phin[n_] := Table[b2[m][n, n] + 2 Sum[If[m + 2 - 2j > 0, b1[m + 2 - 2j][j, n], 0], {j, 1, n}], {m, 1, max}];
MT[s_List] := Table[DivisorSum[n, MoebiusMu[#] s[[n/#]] &]/n, {n, 1, Length[s]}];
MT[phin[5]] (* Jean-François Alcover, Nov 06 2018, adapted from Max Alekseyev's PARI script *)

\\ implementation of MT() and phin() is given in A006207
MT(phin(5)) \\ sequence A_{n,5} \\ Max Alekseyev

Terms a(32) onward from Max Alekseyev, Feb 23 2012

cp's OEIS Frontend

A130627 Duplicate of A006209.

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A006206 Number of aperiodic binary necklaces of length n with no subsequence 00, excluding the necklace "0".

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A006208 Generalized Fibonacci numbers A_{n,3}.

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A094392 Antidiagonals of the tables formed from b(m,2,n,n), which is defined in Du 1989.

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A006207 Generalized Fibonacci numbers A_{n,2}.

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A130628 Related to the minimal number of periodic orbits of periods guaranteed by Sharkovskii's theorem.

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A208092 Generalized Fibonacci numbers A_{n,6}.

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