A381812 -id:A381812 - cp's OEIS frontend

A047206 Numbers that are congruent to {1, 3, 4} mod 5.

1, 3, 4, 6, 8, 9, 11, 13, 14, 16, 18, 19, 21, 23, 24, 26, 28, 29, 31, 33, 34, 36, 38, 39, 41, 43, 44, 46, 48, 49, 51, 53, 54, 56, 58, 59, 61, 63, 64, 66, 68, 69, 71, 73, 74, 76, 78, 79, 81, 83, 84, 86, 88, 89, 91, 93, 94, 96, 98, 99, 101, 103, 104, 106, 108

Offset: 1

N. J. A. Sloane

a(n) is the maximum number of heads achievable in the game of blet with 2*n coins. See A075274 and A381812. - Pontus von Brömssen, Mar 09 2025

Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A001622, A075274, A381812.

[ n : n in [1..150] | n mod 5 in [1, 3, 4] ]; // Vincenzo Librandi, Mar 31 2011

A047206:=n->(15*n-6-3*cos(2*n*Pi/3)-sqrt(3)*sin(2*n*Pi/3))/9: seq(A047206(n), n=1..100); # Wesley Ivan Hurt, Jun 14 2016

Select[Range[0, 200], MemberQ[{1, 3, 4}, Mod[#, 5]] &] (* Vladimir Joseph Stephan Orlovsky, Feb 12 2012 *)

a(n)=(5*n-1)\3 \\ Charles R Greathouse IV, Jul 01 2013

G.f.: x*(1+2*x+x^2+x^3)/((1-x)^2*(1+x+x^2)).
a(n) = a(n-1) + a(n-3) - a(n-4) for n>4.
a(n) = 1+(5*n)/3-(i*sqrt(3) * (-1/2+(i*sqrt(3))/2)^n)/9+(i*sqrt(3)* (-1/2-(i*sqrt(3))/2)^n)/9. - Stephen Crowley, Feb 11 2007
a(n) = floor((5*n-1)/3). - Gary Detlefs, May 14 2011
From Wesley Ivan Hurt, Jun 14 2016: (Start)
a(n) = (15*n-6-3*cos(2*n*Pi/3)-sqrt(3)*sin(2*n*Pi/3))/9.
a(3k) = 5k-1, a(3k-1) = 5k-2, a(3k-2) = 5k-4. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt((5-sqrt(5))/2)*Pi/5 + log(phi)/sqrt(5) - log(2)/5, where phi is the golden ratio (A001622). - Amiram Eldar, Apr 16 2023
E.g.f.: (9 + 3*exp(x)*(5*x - 2) - exp(-x/2)*(3*cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2)))/9. - Stefano Spezia, Jun 22 2024

A381814 Size of the largest component of the blet graph for n coins.

Original entry on oeis.org

2, 5, 20, 8, 56, 56, 74, 180, 660, 220, 2288, 2002, 2942, 7280, 24752, 8568, 93024, 77520, 120920, 298452, 1009470, 346104, 3845600, 3289000, 5067974, 12432420, 42921450, 14307150, 161280600, 140244000, 215188426, 524512560, 1835793960

Offset: 3

Pontus von Brömssen, Mar 08 2025

See A381813 for the definition of the blet graph.

			For n = 4, the blet graph has 2 components of maximum size a(4) = 5: {TTTH, THTT, THHH, HTHT, HHTH} and {TTHT, THTH, HTTT, HTHH, HHHT}.

Michael S. Branicky, Python program for OEIS A381813 and A381814

Cf. A075273, A381812, A381813 (number of components of size at least 2).

Python
```
# see linked program
```

a(2*n) >= A075273(n) (the size of the component containing the vertex (HT)^n).

a(24)-a(28) from Michael S. Branicky, Mar 08 2025
a(29)-a(30) from Michael S. Branicky, Mar 12 2025
a(31)-a(35) from Bert Dobbelaere, Mar 16 2025

A075274 Number of reachable arrangements of coins in the game Blet starting with 2n coins that achieve the maximum number of heads.

Original entry on oeis.org

2, 3, 2, 5, 4, 7, 16, 9, 25, 55, 24, 91, 196, 70, 336, 714, 216

Offset: 2

N. J. A. Sloane, Oct 12 2002

Rodriguez Villegas, Sadun, and Voloch (2002) prove that the maximum number of heads achievable is A047206(n). - Pontus von Brömssen, Mar 08 2025

			For 4 coins, starting from HTHT we can reach THTT, HHTH, TTTH, THHH, and no others. Of these, two arrangements HHTH and THHH achieve the maximum of 3 heads, so a(2) = 2.

Michael S. Branicky, Python program for OEIS A075274
F. Rodriguez Villegas, L. Sadun and J. F. Voloch, Blet: a mathematical puzzle, Amer. Math. Monthly, 109 No.8 (2002), 729-740.

Cf. A047206, A075273, A381812.

Python
```
# see linked program
```

Offset corrected by Michel Marcus, Sep 30 2017

a(16)-a(18) from and name clarified by Sean A. Irvine, Feb 14 2025

A381813 Number of connected components, not counting isolated vertices, of the blet graph for n coins.

Original entry on oeis.org

3, 2, 1, 7, 2, 5, 8, 8, 6, 50, 12, 30, 61, 62, 47, 417, 102, 303, 682, 696, 532, 4904, 1250, 3854, 8911, 9218, 7147, 66735, 17298, 53965, 126348, 131740, 103080

Offset: 3

Pontus von Brömssen, Mar 08 2025

The blet graph for n coins has one vertex for each binary heads/tails-sequence of length n. Two vertices are connected by an edge if there is a legal move between them in the game of blet, i.e., if one can be obtained from the other by replacing one occurrence of a triple THT with HTH. The binary sequences are circularly connected, so such a triple is allowed to start at one of the last two elements of the sequence and continue from the beginning.
The number of isolated vertices is A007039(n).
A075273(n) is the size of the component containing (HT)^n in the blet graph for 2*n coins.

			For n = 4, the blet graph has A007039(4) = 6 isolated vertices TTTT, TTHH, THHT, HTTH, HHTT, HHHH, and a(4) = 2 components of size at least 2: {TTTH, THTT, THHH, HTHT, HHTH} and {TTHT, THTH, HTTT, HTHH, HHHT}.

Michael S. Branicky, Python program for OEIS A381813 and A381814

Cf. A007039, A075273, A381812, A381814 (size of the largest component).

Python
```
# see linked program
```

a(24)-a(28) from Michael S. Branicky, Mar 08 2025
a(29)-a(30) from Michael S. Branicky, Mar 12 2025
a(31)-a(35) from Bert Dobbelaere, Mar 16 2025

cp's OEIS Frontend

A047206 Numbers that are congruent to {1, 3, 4} mod 5.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A381814 Size of the largest component of the blet graph for n coins.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python

Formula

Extensions

A075274 Number of reachable arrangements of coins in the game Blet starting with 2n coins that achieve the maximum number of heads.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python

Extensions

A381813 Number of connected components, not counting isolated vertices, of the blet graph for n coins.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python

Extensions