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.

User: Kelvin Voskuijl

Kelvin Voskuijl's wiki page.

Kelvin Voskuijl has authored 42 sequences. Here are the ten most recent ones:

A386405 Decimal expansion of Product_{k>=0} (1 + 1/k!^3).

Original entry on oeis.org

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

Views

Author

Kelvin Voskuijl, Aug 20 2025

Keywords

Examples

			4.52116298976821848102535993606283488697241501987188319...

Crossrefs

Cf. A000442, A238695, A384510.

Programs

  • PARI
    prodinf(k=0, 1 + 1/k!^3)

A387175 Continued fraction expansion of Product_{k>=0} (1+1/k!).

Original entry on oeis.org

7, 2, 1, 2, 1, 11, 1, 1, 2, 1, 15, 3, 1, 2, 1, 11, 1, 1, 1, 9, 1, 1, 3, 5, 9, 1, 6, 1, 1, 1, 44, 2, 1, 26, 1, 5, 1, 6, 19, 1, 1, 8, 2, 1, 8, 1, 2, 25, 9, 1, 4, 94, 1, 1, 25, 6, 2, 2, 1, 2, 4, 1, 2, 25, 28, 1, 2, 1, 2, 2, 1, 1, 4, 12, 1, 1591, 7, 1, 4, 4, 22, 1, 1, 4, 9, 1, 2, 2, 1, 2, 3, 1
Offset: 1

Views

Author

Kelvin Voskuijl, Aug 20 2025

Keywords

Examples

			7 + 1/(2 + 1/(1 + 1/(...))) = Product_{k>=0} (1+1/k!).

Crossrefs

Cf. A238695 (decimal expansion).

Programs

  • Mathematica
    terms= 92; ContinuedFraction[Product[1+1/k!,{k,0,3*terms}],terms] (* Stefano Spezia, Aug 21 2025 *)

A384510 Decimal expansion of Product_{k>=0} (1 + 1/k!^2).

Original entry on oeis.org

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

Views

Author

Kelvin Voskuijl, Aug 14 2025

Keywords

Examples

			5.1481781945902396880952694880498895078630362906948...

Crossrefs

Cf. A001044, A238695.

Programs

  • PARI
    prodinf(k=0, 1 + 1/k!^2)

A386990 Decimal expansion of Sum_{k>=0} 2/(k!*(k! + 1)).

Original entry on oeis.org

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

Views

Author

Kelvin Voskuijl, Aug 12 2025

Keywords

Comments

Sum of reciprocals of A055555 (triangular numbers of factorials).

Examples

			2.3844273879714288211644804923804481846149857064...

Crossrefs

Cf. A000217, A070910 (of n!^2), A055555, A091131 (of n!).

Programs

  • Maple
    evalf(sum(2/(n!*(n!+1)),n=0..infinity), 120);  # Alois P. Heinz, Aug 13 2025
  • PARI
    suminf(k=0, 2/(k!*(k!+1)))
  • PARI
    sumpos(k=0,1/binomial(k!+1,2)) \\ Charles R Greathouse IV, Aug 19 2025

Formula

Equals Sum_{k>=0} 1/A055555(k).

A386934 Decimal expansion of BesselI(1, 2*sqrt(2))/(sqrt(2)*BesselI(2, 2*sqrt(2))).

Original entry on oeis.org

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

Views

Author

Kelvin Voskuijl, Aug 09 2025

Keywords

Examples

			1.2892652359849177261502470461093801583744352570790...

Links

Crossrefs

Cf. A145051 (continued fraction), A363679, A386710.

Programs

  • Mathematica
    RealDigits[BesselI[1, 2 Sqrt[2]]/(Sqrt[2] BesselI[2, 2 Sqrt[2]]), 10, 100][[1]]
  • PARI
    besseli(1, 2*sqrt(2))/(sqrt(2)*besseli(2, 2*sqrt(2)))

Formula

Equals ( Sum_{k >= 0} 2^n/((1 + n)n!^2) )/( Sum_{k >= 1} 2^k/((k-1)!*(k+1)!) ).
Equals A363679/A386710.

A386927 Continued fraction of BesselI(2, 2 * sqrt(2)).

Original entry on oeis.org

1, 1, 6, 54, 3, 2, 4, 3, 1, 140, 1, 5, 1, 1, 2, 4, 1, 1, 8, 3, 9, 87, 12, 2, 1, 2, 1, 5, 1, 14, 1, 1, 2, 2, 8, 11, 2, 2, 2, 2, 1, 3, 3, 1, 15, 1, 5, 1, 1, 1, 2, 8, 7, 4, 5, 1, 4, 1, 8, 7, 5, 1, 2, 1, 11, 4, 14, 2, 1, 1, 9, 5, 2, 2, 6, 1, 3, 24, 2, 3, 1, 4, 2, 7, 2, 14, 2, 3, 1, 13, 1, 1, 1, 2, 3, 2, 1, 6, 2
Offset: 1

Views

Author

Kelvin Voskuijl, Aug 08 2025

Keywords

Examples

			1 + 1/(1 + 1/(6 + 1/(...))) = 1.857517...

Crossrefs

Cf. A386710 (decimal expansion).

Programs

  • Mathematica
    ContinuedFraction[BesselI[2, 2 Sqrt[2]] , 100]
  • PARI
    contfrac(besseli(2, 2*sqrt(2))) \\ Amiram Eldar, Aug 08 2025

A386710 Decimal expansion of BesselI(2, 2 * sqrt(2)).

Original entry on oeis.org

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

Views

Author

Kelvin Voskuijl, Jul 30 2025

Keywords

Examples

			1.8575177802292191087705981876531371501390490911357...

Links

Crossrefs

Cf. A386927 (continued fraction).
Cf. A096789 (for factorial squared).
Cf. A006472 (triangular polygorials).

Programs

  • Mathematica
    RealDigits[BesselI[2, 2 * Sqrt[2]],10,100][[1]] (* Stefano Spezia, Aug 02 2025 *)
  • PARI
    besseli(2, 2*sqrt(2)) \\ Amiram Eldar, Aug 02 2025

Formula

Equals Sum_{k >= 1} 2^k/((k-1)!*(k+1)!).

A386550 Indices of hexagonal numbers that are six times another hexagonal number.

Original entry on oeis.org

0, 2, 176, 17222, 1687556, 165363242, 16203910136, 1587817830062, 155589943435916, 15246226638889682, 1493974620667752896, 146394266598800894102, 14345144152061819869076, 1405677732635459546275322, 137742072654122973715112456, 13497317442371415964534745342
Offset: 1

Views

Author

Kelvin Voskuijl, Jul 25 2025

Keywords

Examples

			176 is in this sequence because the 176th hexagonal number (61776) is six times another hexagonal number.

Links

Crossrefs

Cf. A000384, A269551, A385917, A386549.

Formula

a(n) = (3*A269551(n-2) - 7)/4 for n>=2. - Hugo Pfoertner, Jul 26 2025
G.f.: 2*x^2*(1 - 11*x - 2*x^2)/((1 - x)*(1 - 98*x + x^2)). - Stefano Spezia, Jul 27 2025

Extensions

More terms from Jinyuan Wang, Jul 26 2025

A386549 Indices of hexagonal numbers that are one-sixth of another hexagonal number.

Original entry on oeis.org

0, 1, 72, 7031, 688942, 67509261, 6615218612, 648223914691, 63519328421082, 6224245961351321, 609912584884008352, 59765209072671467151, 5856380576536919772422, 573865531291545466230181, 56232965685994918770785292, 5510256771696210494070728411, 539948930660542633500160598962
Offset: 1

Views

Author

Kelvin Voskuijl, Jul 25 2025

Keywords

Examples

			72 is in this sequence because the 72th hexagonal number (10296) is one-sixth of another hexagonal number.

Links

Crossrefs

Cf. A000384, A385917, A386550.

Formula

G.f.: x^2 * (1 - 27*x + 2*x^2) / (1 - 99*x + 99*x^2 - x^3). - Jinyuan Wang, Jul 26 2025

Extensions

More terms from Jinyuan Wang, Jul 26 2025

A385616 Indices of pentagonal numbers that are one-fifth of another one.

Original entry on oeis.org

0, 1, 39, 85705, 4026303, 8886048529, 417455130327, 921323283480793, 43282582818520431, 95524640668969514017
Offset: 1

Views

Author

Kelvin Voskuijl, Jul 04 2025

Keywords

Crossrefs

Cf. A385152.

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