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 1-3 of 3 results.

A007613 a(n) = (8^n + 2*(-1)^n)/3.

Original entry on oeis.org

1, 2, 22, 170, 1366, 10922, 87382, 699050, 5592406, 44739242, 357913942, 2863311530, 22906492246, 183251937962, 1466015503702, 11728124029610, 93824992236886, 750599937895082, 6004799503160662, 48038396025285290, 384307168202282326, 3074457345618258602
Offset: 0

Views

Author

N. J. A. Sloane, Robert G. Wilson v

Keywords

Comments

Also, the cogrowth sequence of C3 X C3 = ; that is, the number of words of length 3n that reduce to the identity. - Sean A. Irvine, Nov 04 2024

References

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

Links

Crossrefs

Cf. A001045, A006566, A078008, A082311, A139459.

Programs

Formula

a(n) = A078008(3*n). - Paul Barry, Nov 29 2003
From Paul Barry, Mar 24 2004: (Start)
a(n) = (A082311(n) + (-1)^n)/2.
a(n) = (A001045(3*n+1) + (-1)^n)/2. (End)
a(n) = Sum_{k=0..n} binomial(3*n, 3*k). - Paul Barry, Jan 13 2005
a(n) = 8*a(n-1) + 6*(-1)^n. - Paul Curtz, Nov 19 2007
From Colin Barker, Sep 29 2014: (Start)
a(n) = 7*a(n-1) + 8*a(n-2).
G.f.: (1-5*x) / ((1+x)*(1-8*x)). (End)
E.g.f.: (1/3)*(exp(8*x) + 2*exp(-x)). - G. C. Greubel, Apr 23 2023

Extensions

More terms from Colin Barker, Sep 29 2014

A177809 Symmetrical sequence:Binomial(n,5*m).

Original entry on oeis.org

1, 1, 1, 1, 252, 1, 1, 3003, 3003, 1, 1, 15504, 184756, 15504, 1, 1, 53130, 3268760, 3268760, 53130, 1, 1, 142506, 30045015, 155117520, 30045015, 142506, 1, 1, 324632, 183579396, 3247943160, 3247943160, 183579396, 324632, 1, 1, 658008, 847660528, 40225345056
Offset: 0

Views

Author

Roger L. Bagula, Dec 13 2010

Keywords

Comments

Row sums are A070782.
5th in the sequence of sequence Binomial(n,k*m),k=1,2,3,4,5,...

Examples

			{1},
{1, 1},
{1, 252, 1},
{1, 3003, 3003, 1},
{1, 15504, 184756, 15504, 1},
{1, 53130, 3268760, 3268760, 53130, 1},
{1, 142506, 30045015, 155117520, 30045015, 142506, 1},
{1, 324632, 183579396, 3247943160, 3247943160, 183579396, 324632, 1},
{1, 658008, 847660528, 40225345056, 137846528820, 40225345056, 847660528, 658008, 1},
{1, 1221759, 3190187286, 344867425584, 3169870830126, 3169870830126, 344867425584, 3190187286, 1221759, 1},
{1, 2118760, 10272278170, 2250829575120, 47129212243960, 126410606437752, 47129212243960, 2250829575120, 10272278170, 2118760, 1}

Crossrefs

Cf. A086645, A034839, A070775, A177808, A139459, A070782.

Programs

  • Mathematica
    t[n_, m_] = Binomial[n, 5*m];
Table[Table[t[n, m], {m, 0, Floor[n/5]}], {n, 0, 50, 5}];
Flatten[%]

A177810 Triangle binomial(6*n,6*m), 0 <= m <= n, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 924, 1, 1, 18564, 18564, 1, 1, 134596, 2704156, 134596, 1, 1, 593775, 86493225, 86493225, 593775, 1, 1, 1947792, 1251677700, 9075135300, 1251677700, 1947792, 1, 1, 5245786, 11058116888, 353697121050, 353697121050, 11058116888, 5245786, 1
Offset: 0

Views

Author

Roger L. Bagula, Dec 13 2010

Keywords

Comments

Row sums are A070967. k=6 in binomial(k*n,k*m) sequence similar to k=2 in A086645, k=4 in A070775,...

Examples

			1;
1, 1;
1, 924, 1;
1, 18564, 18564, 1;
1, 134596, 2704156, 134596, 1;
1, 593775, 86493225, 86493225, 593775, 1;

Crossrefs

Cf. A034839, A177808, A139459, A070782, A177809, A070967.

Programs

  • Mathematica
    t[n_, m_] := Binomial[n, 6*m]; Flatten@Table[Table[t[n, m], {m, 0, n/6}], {n, 0, 42, 6}]

Formula

Left-right symmetric: binomial(6*n,6*m) = binomial(6*n,6*(n-m)).
Showing 1-3 of 3 results.

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

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