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.

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A005257 Number of weighted voting procedures.

Original entry on oeis.org

2, 3, 5, 9, 17, 33, 64, 126, 249, 495, 984, 1962, 3913, 7815, 15608, 31194, 62346, 124650, 249216, 498348, 996531, 1992897, 3985464, 7970598, 15940542, 31880430, 63759552, 127517796, 255032987, 510063369, 1020121528, 2040237846, 4080465294, 8160920190
Offset: 1

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Author

N. J. A. Sloane

Keywords

Comments

Appears to satisfy a(1)=2, a(2)=3, a(3)=5 and, for n>3, a(n)=3a(n-1)-2a(n-2) if n is even and a(n)=a(n-1)+2a(n-2)-a([(n-1)/2]-1) if n is odd. - John W. Layman, Jan 10 2000.

References

  • T. V. Narayana, Recent progress and unsolved problems in dominance theory, pp. 68-78 of Combinatorial mathematics (Canberra 1977), Lect. Notes Math. Vol. 686, 1978.
  • T. V. Narayana, Lattice Path Combinatorics with Statistical Applications. Univ. Toronto Press, 1979, pp. 100-101.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Programs

  • Mathematica
    a={2, 3, 5}; For[i=4, i<35, i++, If[EvenQ[i], a=Append[a, 3 a[[i-1]]-2a[[i-2]]], a=Append[a, a[[i-1]]+2a[[i-2]]-a[[(i-1)/2-1]]]]]; a

Formula

Empirical g.f.: x*(x^16 +2*x^14 -12*x^13 +8*x^12 -6*x^11 +5*x^10 -3*x^7 +x^6 +9*x^5 -6*x^4 +3*x^3 -2*x^2 -3*x +2) / ((x -1)*(2*x -1)*(x^2 +1)*(2*x^2 -1)*(x^4 +1)*(2*x^4 -1)). - Colin Barker, Mar 16 2015

Extensions

More terms from Vit Planocka (planocka(AT)mistral.cz), Sep 20 2002

A058301 Number of solutions to c(0)F(0) + ... + c(n)F(n) = 0, where c(i) = +-1 for i >= 0, number of (+1)'s >= number of (-1)'s, F(i) = A000045(i) = Fibonacci numbers.

Original entry on oeis.org

1, 0, 2, 3, 0, 6, 4, 0, 8, 11, 0, 22, 16, 0, 32, 42, 0, 84, 64, 0, 128, 165, 0, 330, 256, 0, 512, 654, 0, 1308, 1024, 0, 2048, 2605, 0, 5210, 4096, 0, 8192, 10398, 0, 20796, 16384, 0, 32768, 41550, 0, 83100, 65536, 0, 131072, 166116, 0, 332232, 262144, 0
Offset: 0

Views

Author

Naohiro Nomoto, Dec 08 2000

Keywords

Examples

			a(3) = 3 because +0+1+1-2 = -0+1+1-2 = +0-1-1+2 = 0;
a(5) = 6 because +0+1-1-2-3+5 = +0-1+1-2-3+5 = +0+1-1+2+3-5 = -0+1-1+2+3-5 = +0-1+1+2+3-5 = -0-1+1+2+3-5 = 0.

Crossrefs

Cf. A000045, A002083, A047270, A047238.

Formula

a(3n+1) = 0, a(A047270(n)) = A002083(n+5), a(A047238(n)) = 2^n.

Extensions

More terms from Sean A. Irvine, Aug 02 2022

A242730 Decimal expansion of the Conway-Guy constant, a constant associated with Erdős' sum-distinct set constant.

Original entry on oeis.org

2, 3, 5, 1, 2, 5, 2, 8, 4, 8, 1, 1, 1, 7, 4, 8, 6, 5, 6, 3, 5, 5, 8, 8, 1, 7, 4, 3, 9, 1, 8, 7, 9, 0, 0, 9, 8, 8, 0, 9, 8, 4, 5, 1, 9, 5, 8, 3, 5, 2, 9, 5, 9, 8, 5, 8, 2, 6, 7, 7, 9, 6, 6, 3, 1, 6, 7, 0, 3, 1, 9, 0, 5, 1, 3, 7, 1, 3, 2, 4, 8, 7, 4, 7, 1, 4, 0, 9, 0, 0, 0, 0, 1, 5, 0, 0, 3, 1, 1, 7
Offset: 0

Views

Author

Jean-François Alcover, May 21 2014

Keywords

Examples

			0.23512528481117486563558817439187900988...

References

  • J. H. Conway and R. K. Guy, “Sets of Natural Numbers with Distinct Sums,” Notices Amer. Math. Soc., vol. 15, 1968.
  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.28, p. 189.

Links

Crossrefs

Cf. A002083, A005255.

Programs

  • Mathematica
    digits = 100; Clear[v, s]; v[n_] := v[n] = 2*v[n-1] - v[n-1 - Floor[1/2 + Sqrt[2*(n-1)]]]; v[0] = 0; v[1] = 1; s[k_] := s[k] = v[k]/2^k // N[#, digits + 5] &; s[dk = 250]; s[k = 2*dk]; While[RealDigits[s[k], 10, digits] != RealDigits[s[k - dk], 10, digits],  Print["k = ", k]; k = k + dk]; RealDigits[s[k], 10, digits] // First
Previous Showing 21-23 of 23 results.

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

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