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 1-4 of 4 results.

A022855 a(n) = [ a(n-1)/a(1) ] + [ a(n-1)/a(2) ] + ... + [ a(n-1)/a(n-1) ] for n >= 3, with initial terms 1,1.

Original entry on oeis.org

1, 1, 2, 5, 13, 35, 97, 271, 761, 2143, 6042, 17043, 48081, 135656, 382752, 1079939, 3047074, 8597406, 24257838, 68444231, 193117503, 544886995, 1537415492, 4337865331, 12239421132, 34533905116, 97438480950, 274925686472, 775711324186, 2188693483680, 6175466331563
Offset: 1

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Author

Keywords

Crossrefs

Programs

  • Mathematica
    Clear[a]; a[n_] := a[n] = If[n == 1, 1, Sum[Floor[a[n-1]/a[k]], {k, 1, n-1}]]; Table[a[n], {n, 1, 30}] (* Vaclav Kotesovec, May 22 2019 *)

Formula

a(n) ~ c * d^n, where d = 2.821530916787913161647514028120517886471375614018590071788760268658400644805..., c = 0.06693303886818014323088828992453194331872567432210373455545066012434187... - Vaclav Kotesovec, May 22 2019

Extensions

Name clarified by Robert C. Lyons, Feb 11 2025

A022874 a(n) = [ a(n-1)/a(1) ] + [ a(n-1)/a(2) ] + ... + [ a(n-1)/a(n-1) ] for n >= 3, with initial terms 2,1.

Original entry on oeis.org

2, 1, 1, 2, 6, 19, 60, 194, 628, 2035, 6597, 21393, 69382, 225031, 729865, 2367255, 7678002, 24902998, 80770936, 261974262, 849693202, 2755914018, 8938593456, 28991634898, 94032120191, 304985891949, 989198096368
Offset: 1

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Author

Keywords

Crossrefs

Programs

  • Mathematica
    Clear[a]; a[n_] := a[n] = If[n==1, 2, Sum[Floor[a[n-1]/a[k]], {k, 1, n-1}]]; Table[a[n], {n, 1, 30}] (* Vaclav Kotesovec, May 22 2019 *)

Formula

a(n) ~ c * d^n, where d = 3.243422474587042277724011259031636390127037162673948731178241793178427249639..., c = 0.015782950162188139059124323554918048406538374877378493529663483700401917... - Vaclav Kotesovec, May 22 2019

Extensions

a(26) corrected by Sean A. Irvine, May 22 2019
Name clarified by Robert C. Lyons, Feb 11 2025

A367787 Let b(0) = 1, b(n) = Sum_{k=0..n-1} b(k) / b(n-k-1), then a(n) is the numerator of b(n).

Original entry on oeis.org

1, 1, 2, 7, 44, 3459, 21398845, 204701870532176, 47683439994850565666251869149, 203292005443961363023193564438853229653319486912062841397
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 30 2023

Keywords

Comments

The next term is too large to include.

Examples

			1, 1, 2, 7/2, 44/7, 3459/308, 21398845/1065372, 204701870532176/5699432573835, ...
		

Crossrefs

Cf. A000108, A022857, A022858, A073833, A367788 (denominators).

Programs

  • Mathematica
    b[0] = 1; b[n_] := b[n] = Sum[b[k]/b[n - k - 1], {k, 0, n - 1}]; a[n_] := Numerator[b[n]]; Table[a[n], {n, 0, 9}]

Formula

G.f. for fractions satisfies: 1 / Sum_{n>=0} b(n) * x^n = 1 - x * Sum_{n>=0} x^n / b(n).

A367788 Let b(0) = 1, b(n) = Sum_{k=0..n-1} b(k) / b(n-k-1), then a(n) is the denominator of b(n).

Original entry on oeis.org

1, 1, 1, 2, 7, 308, 1065372, 5699432573835, 742435596532024691458409520, 1770094160863794205114840009375146894748207874734794924
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 30 2023

Keywords

Comments

The next term is too large to include.

Examples

			1, 1, 2, 7/2, 44/7, 3459/308, 21398845/1065372, 204701870532176/5699432573835, ...
		

Crossrefs

Cf. A000108, A022857, A022858, A073834, A367787 (numerators).

Programs

  • Mathematica
    b[0] = 1; b[n_] := b[n] = Sum[b[k]/b[n - k - 1], {k, 0, n - 1}]; a[n_] := Denominator[b[n]]; Table[a[n], {n, 0, 9}]

Formula

G.f. for fractions satisfies: 1 / Sum_{n>=0} b(n) * x^n = 1 - x * Sum_{n>=0} x^n / b(n).
Showing 1-4 of 4 results.