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.

A307131 Numerator of the expected fraction of occupied places on n-length lattice randomly filled with 2-length segments.

Original entry on oeis.org

1, 2, 5, 4, 37, 52, 349, 338, 11873, 14554, 157567, 466498, 11994551, 41582906, 618626159, 614191052, 7545655031, 92853583996, 1755370057489, 8737266957604, 365468962351379, 2002633668589496, 45904893141293831
Offset: 1

Views

Author

Philipp O. Tsvetkov, Mar 26 2019

Keywords

Comments

The limit of expected fraction of occupied places on n-length lattice randomly filled with 2-length segments at n tends to infinity is equal to 1-1/e^2 (see A219863).

Examples

			0, 1, 2/3, 5/6, 4/5, 37/45, 52/63, 349/420, 338/405, 11873/14175, ...

Links

Crossrefs

Cf. A219863, A231580, A307132 (denominators).

Programs

  • Mathematica
    RecurrenceTable[{f[n] == (2 + 2 (n - 2) f[n - 2] + (n - 1) (n - 2) f[n - 1])/(n (n - 1)),f[0] == 0, f[1] == 0}, f, {n, 2, 100}] // Numerator

Formula

Numerator of f(n), where f(0)=0; f(1)=0 and f(n) = (2 + 2(n-2)f(n-2) + (n-1)(n-2)f(n-1))/(n(n-1)) for n>1.

A307132 Denominator of the expected fraction of occupied places on n-length lattice randomly filled with 2-length segments.

Original entry on oeis.org

1, 3, 6, 5, 45, 63, 420, 405, 14175, 17325, 187110, 552825, 14189175, 49116375, 729729000, 723647925, 8881133625, 109185701625, 2062396586250, 10257709336875, 428772250281375, 2348038513445625, 53791427762572500, 160789593855515625, 16025362854266390625
Offset: 1

Views

Author

Philipp O. Tsvetkov, Mar 26 2019

Keywords

Comments

The limit of expected fraction of occupied places on n-length lattice randomly filled with 2-length segments at n tends to infinity is equal to 1-1/e^2 (see A219863).

Examples

			0, 1, 2/3, 5/6, 4/5, 37/45, 52/63, 349/420, 338/405, 11873/14175, ...

Links

Crossrefs

Cf. A219863, A231580, A307131 (numerators).

Programs

  • Mathematica
    RecurrenceTable[{f[n] == (2 + 2 (n - 2) f[n - 2] + (n - 1) (n - 2) f[n - 1])/(n (n - 1)), f[0] == 0, f[1] == 0}, f, {n, 2, 100}] // Denominator

Formula

Denominator of f(n), where f(0)=0; f(1)=0 and f(n) = (2 + 2(n-2)f(n-2) + (n-1)(n-2)f(n-1))/(n(n-1)) for n>1.

A231634 a(n) is the denominator of the probability that n segments of length 2, each placed randomly on a line segment of length 2n, will completely cover the line segment.

Original entry on oeis.org

1, 3, 15, 105, 2835, 31185, 2027025, 91216125, 10854718875, 206239658625, 7795859096025, 4482618980214375, 72512954091703125, 99850337784275203125, 37643577344671751578125, 8168656283793770092453125, 12518528979807790079765625
Offset: 1

Views

Author

Philipp O. Tsvetkov, Nov 12 2013

Keywords

Comments

Denominators of the probability function defined in A231580.

Examples

			1, 2/3, 7/15, 34/105, 638/2835, 4876/31185, 220217/2027025, 6885458/91216125, 569311642/10854718875, 7515775348/206239658625, 197394815194/7795859096025, ...

Crossrefs

Cf. A231580.

Programs

  • Mathematica
    f[g_List, l_] := f[g, l] = Sum[f[g[[;; n]], l] f[g[[n + 1 ;;]], l], {n, Length[g] - 1}]/(Total[l + g] - 2 l + 1);
f[{}] = f[{}, _] = 1;
f[ConstantArray[0, #], 2] & /@ Range[2, 20] // Denominator
  • PARI
    f=[1]; for(n=2, 25, f=concat(f, sum(k=1, n-1, (f[k]*f[n-k])) / (2*n-3))); f
vector(#f, k, denominator(f[k])) \\ Colin Barker, Jul 24 2014, sequence shifted by 1 index

Formula

Denominator of f(n) where f(0)=1 and f(n) = Sum_{k=0..n-1} f(n)*f(n-k-1)/(2*n-1). - Michael Somos, Mar 01 2014

Extensions

More terms from Colin Barker, Jul 24 2014
Name edited by Jon E. Schoenfield, Nov 13 2018
Showing 1-3 of 3 results.

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