Tristan Cam - cp's OEIS frontend

A386962 Number of equivalence classes of connected 3-regular graphs on 2n unlabeled nodes up to local complementation.

0, 1, 2, 4, 15, 60

Offset: 1

Tristan Cam, Aug 11 2025

Number of equivalences classes of 3-regular graphs on 2n nodes up to a sequence of local complementation or isomorphism, also called orbits for the local equivalence relation.
a(n) is necessarily less than:
  A005638(n) (number of non-isomorphic, not necessarily connected 3-regular graphs);
  A002851(n) (number of non-isomophic connected 3-regular graphs);
  A090899(n) (number of local equivalence classes of connected graphs); and
  A156800(n) (number of equivalence classes for connected graphs up to pivots and isomorphism).
This is relevant in the study of optimal quantum circuit synthesis for graph state preparation.

			There are only two 3-regular graphs with 6 nodes and they are not equivalent up to a sequence of local complementation, thus a(3) = 2.

Niels Bohr Institute Center for Hybrid Quantum Networks, graph_table (github)
Tristan Cam, Cyril Gavoille, Yvan Le Borgne, and Simon Martiel, Universal Graph Theory Operations for Graph State Preparation

Cf. A002851, A005638, A090899, A156800.

A385629 Number of equivalence classes of connected 4-regular graphs on n unlabeled nodes up to local complementation.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 6, 13, 56, 261

Offset: 1

Tristan Cam, Aug 09 2025

Number of equivalences classes of 4-regular graphs on n nodes up to a sequence of local complementation or isomorphism.
a(n) is necessarily less than:
  A033301(n) (number of non-isomorphic, not necessarily connected 4-regular graphs);
  A006820(n) (number of non-isomophic connected 4-regular graphs);
  A090899(n) (number of local equivalence classes of connected graphs); and
  A156800(n) (number of equivalence classes for connected graphs up to pivots and isomorphism).
This is relevant in the study of optimal quantum circuit synthesis for graph state preparation.

			There are only two 4-regular graphs with 7 nodes and they are not equivalent up to a sequence of local complementation, thus a(7) = 2.

Niels Bohr Institute Center for Hybrid Quantum Networks, graph_table (github)
Tristan Cam, Cyril Gavoille, Yvan Le Borgne, and Simon Martiel, Universal Graph Theory Operations for Graph State Preparation

Cf. A006820, A033301, A090899, A156800.

A327614 Number of transfers of marbles between four sets until the first repetition.

Original entry on oeis.org

4, 5, 10, 11, 12, 12, 12, 10, 15, 17, 12, 12, 12, 15, 17, 16, 16, 15, 19, 17, 17, 15, 15, 19, 22, 17, 16, 15, 19, 22, 21, 19, 19, 24, 26, 21, 19, 19, 24, 26, 21, 19, 19, 24, 26, 21, 19, 23, 28, 26, 21, 19, 23, 28, 26, 21, 19, 23, 28, 26, 21, 19, 23, 28

Offset: 1

Tristan Cam, Sep 19 2019

nonn

There are initially n marbles in each of the four sets. In the first turn, half of the marbles of set A are transferred to set B, rounding to the upper integer when halving. In the second turn, half of the marbles of set B are transferred to set C, following the same rule. The game goes on back on following the pattern (A to B), (B to C), (C to D), (D to A) etc. until we reach a distribution already encountered.
a(n) is then the number of steps until the first repetition occurs.
The indexes of the maximal values are 1, 2, 3, 4, 5, 9, 10, 19, 25, 34, 35, 49, 105, 194, 330, 334, 480, 1553, 1780, 2834, 2870, 4079, ...

			For n = 2, (SetA ; SetB ; SetC ; SetD):
(2 ; 2 ; 2 ; 2), ceiling(2/2)=1 marble get transferred from SetA to SetB,
(1 ; 3 ; 2 ; 2), ceiling(3/2)=2 marbles get transferred from SetB to SetC,
(1 ; 1 ; 4 ; 2), ceiling(4/2)=2 marbles get transferred from SetC to SetD,
(1 ; 1 ; 2 ; 4), ceiling(4/2)=2 marbles get transferred from SetD to SetA,
(3 ; 1 ; 2 ; 2), ceiling(3/2)=2 marbles get transferred from SetA to SetB,
(1 ; 3 ; 2 ; 2), this is a repetition, it took 5 steps to get there, so a(2) = 5.
For n = 4, (SetA ; SetB ; SetC ; SetD):
(4 ; 4 ; 4 ; 4), (2 ; 6 ; 4 ; 4), (2 ; 3 ; 7 ;4), (2 ; 3 ; 3 ; 8), (6 ; 3 ; 3 ; 4), (3 ; 6 ; 3 ; 4), (3 ; 3 ; 6 ; 4), (3 ; 3 ; 3 ; 7), (7 ; 3 ; 3 ; 3), (3 ; 7 ; 3 ; 3), (3 ; 3 ; 7 ; 3), (3 ; 3 ; 3 ; 7) which is a repetition, so a(4) = 11.

Cf. A327565 (two sets), A327613 (three sets).

A327613 Number of transfers of marbles between three sets until the first repetition.

Original entry on oeis.org

3, 4, 8, 6, 6, 6, 8, 9, 9, 8, 11, 9, 9, 8, 11, 13, 12, 9, 11, 13, 12, 9, 11, 13, 12, 11, 14, 13, 12, 11, 14, 13, 12, 11, 14, 13, 12, 11, 14, 16, 12, 11, 14, 16, 12, 11, 14, 13, 12, 14, 14, 13, 12, 14, 14, 16, 12, 14, 14, 16, 12, 14, 14, 16, 14, 14, 14, 16, 14, 14

Offset: 1

Tristan Cam, Sep 19 2019

nonn

There are initially n marbles in each of the three sets. In the first turn, half of the marbles of set A are transferred to set B, rounding to the upper integer when halving. In the second turn, half of the marbles of set B are transferred to set C, following the same rule. The game goes on back on following the pattern (A to B), (B to C), (C to A) etc. until we reach a distribution already encountered.
a(n) is then the number of steps until the first repetition occurs.
The indexes of the maximal values are 1, 2, 3, 8, 11, 16, 27, 40, 83, 176, 179, 528, 907, 1256, 2379, 3408, ...

			For n = 2, (SetA ; SetB ; SetC):
(2 ; 2 ; 2), ceiling(2/2)=1 marble get transferred from SetA to SetB,
(1 ; 3 ; 2), ceiling(3/2)=2 marbles get transferred from SetB to SetC,
(1 ; 1 ; 4), ceiling(4/2)=2 marbles get transferred from SetC to SetA,
(3 ; 1 ; 2), ceiling(3/2)=2 marbles get transferred from SetA to SetB,
(1 ; 3 ; 2), this is a repetition, it took 4 steps to get there, so a(2) = 4.
For n = 4, (SetA ; SetB ; SetC):
(4 ; 4 ; 4), (2 ; 6 ; 4), (2 ; 3 ; 7), (6 ; 3 ; 3), (3 ; 6 ; 3), (3 ; 3 ; 6), (6 ; 3 ; 3) which is a repetition, so a(4) = 6.

Cf. A327565 (two sets), A327614 (four sets).

A327565 Number of transfers of marbles between two sets until the first repetition.

Original entry on oeis.org

2, 3, 4, 3, 5, 4, 4, 5, 6, 4, 5, 6, 5, 5, 6, 5, 7, 6, 5, 7, 6, 5, 7, 6, 6, 7, 6, 6, 7, 6, 6, 7, 8, 6, 7, 8, 6, 7, 8, 6, 7, 8, 6, 7, 8, 6, 7, 8, 7, 7, 8, 7, 7, 8, 7, 7, 8, 7, 7, 8, 7, 7, 8, 7, 9, 8, 7, 9, 8, 7, 9, 8, 7, 9, 8, 7, 9, 8, 7, 9

Offset: 1

Tristan Cam, Sep 17 2019

nonn

There are initially n marbles in both sets. In the first turn, half of the marbles of set A are transferred to set B, rounding to the upper integer when halving. In the second turn, half of the marbles of set B are transferred back to set A, following the same rule. The game goes on back and forth until we reach a distribution already encountered.
a(n) is then the number of steps until the first repetition occurs.
First occurrence of a(n) = m > 1 in this sequence: 1, 2, 3, 5, 9, 17, 33, 65, 129, 257, 513, 1025, 2049...
Conjecture: for m > 2, the first occurrence of a(n) = m is for n = 2^(m-3) + 1.

			For n = 3, (SetA ; SetB):
(3 ; 3), ceiling(3/2)=2 marbles get transferred,
(1 ; 5), ceiling(5/2)=3 marbles get transferred,
(4 ; 2), ceiling(4/2)=2 marbles get transferred,
(2 ; 4), ceiling(4/2)=2 marbles get transferred,
(4 ; 2), this is a repetition, it took 4 steps to get there, so a(3) = 4.
For n = 9, (SetA ; SetB):
(9 ; 9), (4 ; 14), (11 ; 7), (5 ; 13), (12 ; 6), (6 ; 12), (12 ; 6) which is a repetition, so a(9) = 6.

Cf. A094373; A327613 (three sets), A327614 (four sets).

a(n)={my(v=vector(2*n+1), r=n, f=1, c=0); while(!v[1+r], v[1+r]=1; r=if(f, r-ceil(r/2), r+ceil((2*n-r)/2)); c++; f=!f); c} \\ Andrew Howroyd, Sep 17 2019

For m > 1, first occurrence of a(n) = m is for n = A094373(m-1) (conjectured).

A322604 Factorial expansion of exp(gamma) = Sum_{n>=1} a(n)/n! with a(n) as large as possible.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 2, 4, 7, 5, 6, 5, 12, 1, 12, 9, 0, 7, 4, 14, 10, 17, 2, 14, 23, 4, 2, 2, 16, 2, 10, 18, 23, 26, 26, 26, 24, 1, 17, 26, 18, 12, 0, 15, 42, 34, 39, 33, 20, 18, 40, 43, 12, 47, 51, 10, 50, 35, 14, 23, 16, 1, 55, 41, 34, 29, 14, 41, 35, 60, 53, 45, 61, 35, 49, 73, 13, 13, 57, 59

Offset: 1

Tristan Cam, Dec 20 2018

nonn

Gamma is the Euler-Mascheroni constant (A001620).

			exp(gamma) = 1 + 1/2! + 1/3! + 2/4! + 3/5! + 4/6! + 2/7! + 4/8! + ...

Eric Weisstein's World of Mathematics, Harmonic Expansion
Index entries for factorial base representation

Cf. A073004 (decimal expansion), A094644 (continued fraction), A001620 (Euler-Mascheroni constant).

Digits:=200: a:=n->`if`(n=1,floor(exp(gamma)),floor(factorial(n)*exp(gamma))-n*floor(factorial(n-1)*exp(gamma))): seq(a(n),n=1..100); # Muniru A Asiru, Dec 20 2018

With[{b = Exp[EulerGamma]}, Table[If[n==1, Floor[b], Floor[n!*b] - n*Floor[(n - 1)!*b]], {n, 1, 100}]]

default(realprecision, 250); b = exp(Euler); for(n=1, 80, print1( if(n==1, floor(b), floor(n!*b) - n*floor((n-1)!*b)), ", "))

Sum_{n>=1} a(n)/n! = exp(gamma) = A073004.

A322603 Continued fraction for sinh(gamma).

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 2, 11, 2, 1, 2, 3, 1, 8, 1, 1, 2, 3, 2, 8, 1, 3, 5, 17, 15, 2, 1, 1, 1, 1, 2, 7, 1, 1, 1, 4, 1, 11, 1, 2, 20, 19, 6, 7, 23, 14, 1, 10, 3, 2, 1, 1, 154, 5, 6, 2, 2, 1, 23, 1, 1, 28, 2, 2, 5, 2, 1, 1, 1, 1332, 1, 15, 1, 1, 1, 1, 1, 1, 2, 6, 2, 1, 2, 1, 4, 5, 28, 6, 1

Offset: 0

Tristan Cam, Dec 20 2018

Continued fraction of (exp(gamma)-exp(-gamma))/2 = sinh(gamma) (A147709), where gamma is the Euler-Mascheroni constant (A001620).

See A322602 for the continued fraction of cosh(gamma).

			0 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/(3 + 1/(2 + 1/(11 + ...))))))) = 0.60980646721165640770618044...

Cf. A147709 (decimal expansion), A001620 (Euler-Mascheroni constant), A322602.

with(numtheory): cfrac(sinh(gamma),100,'quotients'); # Muniru A Asiru, Dec 20 2018

ContinuedFraction[ (Exp[EulerGamma] - Exp[ -EulerGamma])/2, 100]

contfrac(sinh(Euler)) \\ Michel Marcus, Dec 21 2018

Offset changed and data corrected by Andrew Howroyd, Jul 05 2024

A322602 Continued fraction for cosh(gamma).

Original entry on oeis.org

1, 5, 1, 5, 4, 1, 5, 1, 1, 2, 9, 1, 1, 8, 1, 16, 1, 2, 1, 2, 1, 1, 1, 4, 27, 2, 1, 1, 1, 2, 1, 8, 1, 3, 5, 1, 1, 1, 1, 1, 16, 2, 1, 4, 1, 2, 62, 1, 8, 12, 1, 4, 1, 4, 3, 1, 1, 4, 1, 3, 20, 1, 2, 2, 106, 1, 13, 2, 7, 2, 1, 2, 4, 7, 1, 2, 1, 1, 2, 11, 1, 1, 2, 24, 1, 2, 2, 1, 1, 12

Offset: 0

Tristan Cam, Dec 20 2018

Continued fraction of (exp(gamma)+exp(-gamma))/2 = cosh(gamma) (A147708), where gamma is the Euler-Mascheroni constant (A001620).

See A322603 for the continued fraction of sinh(gamma).

			1 + 1/(5 + 1/(1 + 1/(5 + 1/(4 + 1/(1 + 1/(5 + 1/(1 + ...))))))) = 1.17126595077854157753032365...

Cf. A147708 (decimal expansion), A001620 (Euler-Mascheroni constant), A322603.

with(numtheory): cfrac(cosh(gamma),100,'quotients'); # Muniru A Asiru, Dec 20 2018

ContinuedFraction[ (Exp[EulerGamma] + Exp[ -EulerGamma])/2, 100]

contfrac(cosh(Euler)) \\ Michel Marcus, Dec 21 2018

Offset changed by Andrew Howroyd, Jul 07 2024

A322601 Continued fraction for sin(gamma).

Original entry on oeis.org

0, 1, 1, 4, 1, 33, 1, 6, 2, 5, 1, 22, 1, 261, 1, 5, 1, 4, 1, 45, 1, 4, 2, 9, 1, 2, 3, 7, 1, 2, 1, 2, 6, 1, 1, 2, 2, 1, 5, 5, 5, 1, 1, 2, 1, 1, 7, 1, 7, 2, 2, 1, 3, 9, 11, 9, 4, 1, 8, 1, 1, 1, 5, 1, 22, 1, 1, 1, 2, 1, 7, 5, 1, 1, 16, 2, 5, 2, 1, 1, 2, 1, 1, 1, 7, 2, 4, 2, 1, 1, 1, 24

Offset: 0

Tristan Cam, Dec 20 2018

Continued fraction of the imaginary part of exp(i*gamma) = sin(gamma) (A119807), where gamma is the Euler-Mascheroni constant (A001620).

See A322545 for the continued fraction of the real part.

			0 + 1/(1 + 1/(1 + 1/(4 + 1/(1 + 1/(33 + 1/(1 + 1/(6 + ...))))))) = 0.5456928232039927881573565...

Cf. A119807 (decimal expansion), A001620 (Euler-Mascheroni constant), A322545.

with(numtheory): cfrac(sin(gamma),100,'quotients'); # Muniru A Asiru, Dec 20 2018

ContinuedFraction[ Sin[EulerGamma], 100]

contfrac(sin(Euler)) \\ Michel Marcus, Dec 21 2018

Offset changed by Andrew Howroyd, Jul 07 2024

A322545 Continued fraction for cos(gamma).

Original entry on oeis.org

0, 1, 5, 5, 1, 4, 8, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 18, 1, 73, 2, 5, 3, 5, 4, 1, 1, 1, 7, 1, 4, 2, 1, 1, 32, 12, 2, 19, 8, 1, 1, 1, 1, 1, 3, 5, 2, 70, 1, 6, 6, 1, 52, 2, 1, 2, 10, 1, 1, 1, 4, 1, 3, 1, 5, 1, 7, 3, 1, 1, 1, 20, 5, 6, 6, 2, 4, 3, 7, 1, 1, 2, 10, 1, 9, 12, 45, 1, 16, 2

Offset: 0

Tristan Cam, Dec 20 2018

Continued fraction of the real part of exp(i*gamma) = cos(gamma) (A119806), where gamma is the Euler-Mascheroni constant (A001620).

See A322601 for the continued fraction of imaginary part.

			0 + 1/(1 + 1/(5 + 1/(5 + 1/(1 + 1/(4 + 1/(8 + 1/(1 + ...))))))) = 0.8379852878801965399549928...

Cf. A119806 (decimal expansion), A001620 (Euler-Mascheroni constant), A322601.

with(numtheory): cfrac(cos(gamma),100,'quotients'); # Muniru A Asiru, Dec 20 2018

ContinuedFraction[ Cos[EulerGamma], 100]

contfrac(cos(Euler)) \\ Michel Marcus, Dec 21 2018

Offset changed by Andrew Howroyd, Jul 07 2024

cp's OEIS Frontend

User: Tristan Cam

A386962 Number of equivalence classes of connected 3-regular graphs on 2n unlabeled nodes up to local complementation.

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A385629 Number of equivalence classes of connected 4-regular graphs on n unlabeled nodes up to local complementation.

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A327614 Number of transfers of marbles between four sets until the first repetition.

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A327613 Number of transfers of marbles between three sets until the first repetition.

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A327565 Number of transfers of marbles between two sets until the first repetition.

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A322604 Factorial expansion of exp(gamma) = Sum_{n>=1} a(n)/n! with a(n) as large as possible.

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A322603 Continued fraction for sinh(gamma).

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A322602 Continued fraction for cosh(gamma).

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Maple

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A322601 Continued fraction for sin(gamma).