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.

Previous Showing 31-39 of 39 results.

A367574 Decimal expansion of BesselI(0,2*sqrt(2)).

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, Nov 28 2023

Keywords

Examples

			4.252350879502623825293230824089510302...

Crossrefs

Cf. A000217, A006472, A070910, A072334, A367709, A367710.

Programs

  • Mathematica
    RealDigits[BesselI[0, 2 Sqrt[2]], 10, 100][[1]]

Formula

Equals Sum_{k>=0} 2^k / k!^2.
Equals Sum_{i>=0} (i+1)/Product_{j=1..i} A000217(j). - Davide Rotondo, Feb 25 2025

A367710 Decimal expansion of BesselI(0,2*sqrt(3)).

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, Nov 28 2023

Keywords

Examples

			7.158996536804385093824412306333790648...

Crossrefs

Cf. A070910, A091933, A367574, A367712.

Programs

  • Mathematica
    RealDigits[BesselI[0, 2 Sqrt[3]], 10, 100][[1]]

Formula

Equals Sum_{k>=0} 3^k / k!^2.

A234618 Numbers of undirected cycles in the n-crown graph.

Original entry on oeis.org

1, 28, 586, 16676, 674171, 36729512, 2591431284, 229610080632, 24945009633237, 3259554588092452, 504229440385599358, 91120169013941688700, 19019291896651737256463, 4540685283391286195445008, 1229402290052883559000280168, 374675876836087520170128786864
Offset: 3

Views

Author

Eric W. Weisstein, Dec 28 2013

Keywords

Links

Crossrefs

Cf. A070910, A094047, A137886.

Programs

  • Mathematica
    a[n_] := Sum[Binomial[n, k]*((-1)^k*(k - 1)! + Sum[Sum[(-1)^i*i!*(k - i)!*(k - i - 1)!*Binomial[k, k - j]*Binomial[n - k, j]*Binomial[k - j, i]*Binomial[2*k - i - 1, i]/2, {i, 0, k - 1}], {j, 0, k}]), {k, 2, n}];
Table[a[n], {n, 3, 18}] (* Jean-François Alcover, Oct 02 2017, after Andrew Howroyd *)
RecurrenceTable[{(n - 3) (180 n^5 - 3462 n^4 + 25685 n^3 - 91106 n^2 + 152414 n - 93847) a[n] == (360 n^8 - 8904 n^7 + 93172 n^6 - 538135 n^5 + 1875502 n^4 - 4041070 n^3 + 5268157 n^2 - 3817934 n + 1189124) a[n - 1] - (n - 1) (180 n^9 - 5262 n^8 + 67445 n^7 - 497202 n^6 + 2321291 n^5 - 7107149 n^4 + 14233985 n^3 - 17904305 n^2 + 12741400 n - 3858611) a[n - 2] - (n - 2) (n - 1) (180 n^9 - 5442 n^8 + 71807 n^7 - 543239 n^6 + 2598146 n^5 - 8144697 n^4 + 16705322 n^3 - 21515171 n^2 + 15619923 n - 4754598) a[n - 3] + 2 (n - 3) (n - 2) (n - 1) (540 n^7 - 12585 n^6 + 122039 n^5 - 636205 n^4 + 1920840 n^3 - 3360924 n^2 + 3186108 n - 1302080) a[n - 4] + 2 (n - 4) (n - 3) (n - 2) (n - 1) (540 n^7 - 13806 n^6 + 145494 n^5 - 814365 n^4 + 2591726 n^3 - 4628556 n^2 + 4207415 n - 1449736) a[n - 5] - (n - 5) (n - 4) (n - 3) (n - 2) (n - 1) (1440 n^6 - 28956 n^5 + 230284 n^4 - 915485 n^3 + 1878786 n^2 - 1811640 n + 577483) a[n - 6] + 3 (n - 6) (n - 5) (n - 4) (n - 3) (n - 2) (n - 1) (180 n^5 - 2562 n^4 + 13637 n^3 - 33023 n^2 + 34309 n - 10136) a[n - 7], a[3] == 1, a[4] == 28, a[5] == 586, a[6] == 16676, a[7] == 674171, a[8] == 36729512, a[9] == 2591431284}, a, {n, 3, 20}] (* Eric W. Weisstein, Oct 02 2017 *)

Formula

a(n) = Sum_{k=2..n} binomial(n,k) * ( (-1)^k*(k-1)! + Sum_{j=0..k} Sum_{i=0..k-1} (-1)^i*i!*(k-i)!*(k-i-1)!*binomial(k,k-j)*binomial(n-k,j)*binomial(k-j,i)*binomial(2*k-i-1,i)/2 ). - Andrew Howroyd, Feb 24 2016
Recurrence: (n-3)*(180*n^5 - 3462*n^4 + 25685*n^3 - 91106*n^2 + 152414*n - 93847)*a(n) = (360*n^8 - 8904*n^7 + 93172*n^6 - 538135*n^5 + 1875502*n^4 - 4041070*n^3 + 5268157*n^2 - 3817934*n + 1189124)*a(n-1) - (n-1)*(180*n^9 - 5262*n^8 + 67445*n^7 - 497202*n^6 + 2321291*n^5 - 7107149*n^4 + 14233985*n^3 - 17904305*n^2 + 12741400*n - 3858611)*a(n-2) - (n-2)*(n-1)*(180*n^9 - 5442*n^8 + 71807*n^7 - 543239*n^6 + 2598146*n^5 - 8144697*n^4 + 16705322*n^3 - 21515171*n^2 + 15619923*n - 4754598)*a(n-3) + 2*(n-3)*(n-2)*(n-1)*(540*n^7 - 12585*n^6 + 122039*n^5 - 636205*n^4 + 1920840*n^3 - 3360924*n^2 + 3186108*n - 1302080)*a(n-4) + 2*(n-4)*(n-3)*(n-2)*(n-1)*(540*n^7 - 13806*n^6 + 145494*n^5 - 814365*n^4 + 2591726*n^3 - 4628556*n^2 + 4207415*n - 1449736)*a(n-5) - (n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(1440*n^6 - 28956*n^5 + 230284*n^4 - 915485*n^3 + 1878786*n^2 - 1811640*n + 577483)*a(n-6) + 3*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(180*n^5 - 2562*n^4 + 13637*n^3 - 33023*n^2 + 34309*n - 10136)*a(n-7). - Vaclav Kotesovec, Feb 25 2016
a(n) ~ Pi * BesselI(0,2) * n^(2*n) / exp(2*n+2). - Vaclav Kotesovec, Feb 25 2016

Extensions

a(13) from Eric W. Weisstein, Jan 08 2014
a(14) from Eric W. Weisstein, Apr 09 2014
a(15)-a(16) from Andrew Howroyd, Feb 24 2016

A234846 Decimal expansion of Sum_{n>=0} (2n)!/(n!)^3 = Sum_{n>=0} C(2n,n)/n!.

Original entry on oeis.org

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

Views

Author

Richard R. Forberg, Dec 31 2013

Keywords

Examples

			16.843983681258...
Equals 1/1 + 2/1 + 24/8 + 720/216 + 40320/13824 + 3628800/1728000 + ...

Crossrefs

Cf. A000984, A070910.

Programs

Formula

Equals e^2 * BesselI(0,2) = e^2 * Sum_{n>=0} 1/n!^2.
Equals e^4 * Sum_{n>=0} (-1)^n * C(2n,n)/n!. - Amiram Eldar, Nov 06 2020

Extensions

More terms from Jon E. Schoenfield, Mar 21 2021

A351164 Decimal expansion of gamma * BesselI(0,2) + BesselK(0,2).

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, Feb 03 2022

Keywords

Examples

			1.4297062187372083131867465655452809577372778968399203468724...

Crossrefs

Cf. A001008, A001044, A001620, A002805, A070910, A152648, A347952.

Programs

  • Mathematica
    RealDigits[EulerGamma BesselI[0, 2] + BesselK[0, 2], 10, 110] [[1]]

Formula

Equals Sum_{k>=1} H(k) / (k!)^2, where H(k) is the k-th harmonic number.

A367729 Decimal expansion of BesselI(0,2/sqrt(3)).

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, Nov 28 2023

Keywords

Examples

			1.36216163960978990494314362841455...

Crossrefs

Cf. A070910, A092041, A197036, A334381, A367730.

Programs

  • Mathematica
    RealDigits[BesselI[0, 2/Sqrt[3]], 10, 98][[1]]
  • PARI
    besseli(0,2/sqrt(3)) \\ Michel Marcus, Nov 29 2023

Formula

Equals Sum_{k>=0} 1 / (3^k * k!^2).

A070912 Binary expansion of BesselI(0,2).

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1
Offset: 2

Views

Author

Benoit Cloitre, May 20 2002

Keywords

Examples

			10.010001111001...

Crossrefs

Cf. A070910.

Programs

A130818 Decimal expansion of number whose Engel expansion is the sequence of squares, that is, 1, 4, 9, 16,...

Original entry on oeis.org

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

Views

Author

Stephen Casey (hexomino(AT)gmail.com), Jul 17 2007

Keywords

Examples

			1.2795853023360672674372044408115333532858411...

References

  • F. Engel "Entwicklung der Zahlen nach Stammbruechen" Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg. pp. 190-191, 1913.

Links

Crossrefs

Cf. A006784, A064648, A101689.

Programs

Formula

Equal to Sum_{n>=1} 1/n!^2 or BesselI(0,2) - 1. - Gerald McGarvey, Nov 12 2007
Equals A070910 - 1. - R. J. Mathar, Jun 13 2008

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).
Previous Showing 31-39 of 39 results.

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