cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A160906 Row sums of A159841.

Original entry on oeis.org

1, 5, 29, 176, 1093, 6885, 43796, 280600, 1807781, 11698223, 75973189, 494889092, 3231947596, 21153123932, 138712176296, 911137377456, 5993760282021, 39481335979779, 260377117268087, 1719026098532296, 11360252318843933, 75141910203168229, 497431016774189912
Offset: 0

Views

Author

R. J. Mathar, May 29 2009

Keywords

Links

Crossrefs

Cf. A075273, A159841.
Cf. A005809, A006256, A183160, A226751.
Cf. A000302, A386811, A386812.
Cf. A244038, A383326.

Programs

  • Maple
    A160906 := proc(n) add( A159841(n,k), k=0..n) ; end:
seq(A160906(n), n=0..20) ;
  • Mathematica
    Table[Sum[Binomial[3*n+1, 2*n+k+1], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 25 2017 *)
  • PARI
    a(n) = sum(k=0, n, binomial(3*n+1, 2*n+k+1)); \\ Michel Marcus, Oct 31 2017
  • Sage
    a = lambda n: binomial(3*n+1,n)*hypergeometric([1,-n],[2*n+2],-1)
[simplify(a(n)) for n in range(21)] # Peter Luschny, May 19 2015

Formula

a(n) = Sum_{k=0..n} A159841(n,k).
Conjecture: a(2n+1) = A075273(3n).
a(n) = C(3*n+1,n)*Hyper2F1([1,-n],[2*n+2],-1). - Peter Luschny, May 19 2015
Conjecture: 2*n*(2*n-1)*(5*n-4)*a(n) +(-295*n^3+451*n^2-130*n-24)*a(n-1) +24*(5*n+1)*(3*n-4)*(3*n-2)*a(n-2) = 0. - R. J. Mathar, Jul 20 2016
a(n) = [x^n] 1/((1 - 2*x)*(1 - x)^(2*n+1)). - Ilya Gutkovskiy, Oct 25 2017
a(n) ~ 3^(3*n + 3/2) / (sqrt(Pi*n) * 2^(2*n + 1)). - Vaclav Kotesovec, Oct 25 2017
a(n) = Sum_{k=0..n} 2^(n-k) * binomial(2*n+k,k). - Seiichi Manyama, Aug 03 2025
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(3*n+1,k) * binomial(3*n-k,n-k). - Seiichi Manyama, Aug 07 2025
G.f.: g^2/((2-g) * (3-2*g)) where g = 1+x*g^3 is the g.f. of A001764. - Seiichi Manyama, Aug 12 2025
G.f.: B(x)^2/(1 + (B(x)-1)/3), where B(x) is the g.f. of A005809. - Seiichi Manyama, Aug 15 2025
G.f.: 1/(1 - x*g*(6-g)) where g = 1+x*g^3 is the g.f. of A001764. - Seiichi Manyama, Aug 16 2025

A381812 Number of moves required to reach a position with the maximum number of heads in the game of blet with 2*n coins.

Original entry on oeis.org

1, 1, 2, 5, 3, 6, 11, 7, 10, 17, 11, 16, 25, 15, 22, 33, 21, 28, 41, 27, 34
Offset: 2

Views

Author

Pontus von Brömssen, Mar 08 2025

Keywords

Comments

See A075273 or Rodriguez Villegas, Sadun, and Voloch (2002) for the definition of blet.
Rodriguez Villegas, Sadun, and Voloch (2002) prove that the maximum number of heads achievable is A047206(n).

Examples

			For n = 5, a(5) = 5 moves are needed to go from HTHTHTHTHT to a sequence with the maximum number A047206(5) = 8 of heads: HTHTHTHTHT -> THHTHTHTHH -> THHTHTTHTH -> THHHTHTHTH -> THHHHTHHTH -> HHHHHTHHHT.

Links

Crossrefs

Cf. A047206, A075273, A381813, A381814.

Programs

  • Python
    # see linked program

Extensions

a(17)-a(18) from Michael S. Branicky, Mar 11 2025
a(19)-a(22) from Bert Dobbelaere, Mar 15 2025

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

Views

Author

Pontus von Brömssen, Mar 08 2025

Keywords

Comments

See A381813 for the definition of the blet graph.

Examples

			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}.

Links

Crossrefs

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

Programs

  • Python
    # see linked program

Formula

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

Extensions

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

Views

Author

N. J. A. Sloane, Oct 12 2002

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

Cf. A047206, A075273, A381812.

Programs

  • Python
    # see linked program

Extensions

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

Views

Author

Pontus von Brömssen, Mar 08 2025

Keywords

Comments

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.

Examples

			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}.

Links

Crossrefs

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

Programs

  • Python
    # see linked program

Extensions

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
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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