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.

A137357 a(n) = Sum_{k <= n/2 } binomial(n-2k, 3k+1).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 12, 23, 44, 80, 138, 230, 379, 629, 1060, 1810, 3109, 5336, 9120, 15521, 26349, 44713, 75949, 129177, 219918, 374521, 637699, 1085401, 1846804, 3141826, 5344988, 9093989, 15474230, 26332515, 44810670, 76253683, 129755543, 220790480
Offset: 0

Views

Author

Don Knuth, Apr 11 2008

Keywords

References

  • D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.4.

Links

Crossrefs

Cf. A137356, A137358, A137359, A137360, A137361, A136444.

Programs

  • Mathematica
    LinearRecurrence[{3, -3, 1, 0, 1}, Range[0, 4], 50] (* Paolo Xausa, Mar 17 2024 *)

Formula

G.f.: x*(x-1) / (x^5+x^3-3*x^2+3*x-1). - Colin Barker, Jan 23 2013

A137358 a(n) = Sum_{k <= n/2 } binomial(n-2k, 3k+2).

Original entry on oeis.org

0, 0, 1, 3, 6, 10, 15, 22, 34, 57, 101, 181, 319, 549, 928, 1557, 2617, 4427, 7536, 12872, 21992, 37513, 63862, 108575, 184524, 313701, 533619, 908140, 1545839, 2631240, 4478044, 7619870, 12964858, 22058847, 37533077, 63865592, 108676262, 184929945, 314685488
Offset: 0

Views

Author

Don Knuth, Apr 11 2008

Keywords

References

  • D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.4.

Links

Crossrefs

Cf. A137356, A137357, A137359, A137360, A137361, A136444.

Programs

  • Mathematica
    Table[Sum[Binomial[n-2k,3k+2],{k,0,Floor[n/2]}],{n,0,50}] (* or *) LinearRecurrence[{3,-3,1,0,1},{0,0,1,3,6},50] (* Harvey P. Dale, Nov 06 2012 *)

Formula

a(0)=0, a(1)=0, a(2)=1, a(3)=3, a(4)=6, a(n)=3*a(n-1)-3*a(n-2)+ a(n-3)+ a(n-5). - Harvey P. Dale, Nov 06 2012
G.f.: -x^2/(x^5+x^3-3*x^2+3*x-1). - Colin Barker, Jan 23 2013

A138778 Triangle read by rows: T(n,k)=k*binomial(n-2k,3k) (n>=5, 1<=k<=n/5).

Original entry on oeis.org

1, 4, 10, 20, 35, 56, 2, 84, 14, 120, 56, 165, 168, 220, 420, 286, 924, 3, 364, 1848, 30, 455, 3432, 165, 560, 6006, 660, 680, 10010, 2145, 816, 16016, 6006, 4, 969, 24752, 15015, 52, 1140, 37128, 34320, 364
Offset: 5

Views

Author

Emeric Deutsch, May 10 2008

Keywords

Comments

Row n contains floor(n/5) terms.
Row sums yield A137359.

References

  • D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.4.

Crossrefs

Cf. A137359.

Programs

  • Maple
    T:=proc(n,k) options operator, arrow: k*binomial(n-2*k,3*k) end proc: for n from 5 to 22 do seq(T(n,k),k=1..(1/5)*n) end do; # yields sequence in triangular form
  • Mathematica
    Flatten[Table[k*Binomial[n-2k,3k],{n,5,30},{k,1,n/5}]] (* Harvey P. Dale, Dec 20 2014 *)
Showing 1-3 of 3 results.

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

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