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 1-10 of 10 results.

A087798 a(n) = 9*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 9.

Original entry on oeis.org

2, 9, 83, 756, 6887, 62739, 571538, 5206581, 47430767, 432083484, 3936182123, 35857722591, 326655685442, 2975758891569, 27108485709563, 246952130277636, 2249677658208287, 20494051054152219, 186696137145578258
Offset: 0

Views

Author

Nikolay V. Kosinov, Dmitry V. Poljakov (kosinov(AT)unitron.com.ua), Oct 10 2003

Keywords

Comments

a(n+1)/a(n) converges to (9 + sqrt(85))/2.
For more information about this type of recurrence follow the Khovanova link and see A054413 and A086902. - Johannes W. Meijer, Jun 12 2010

Examples

			a(4) = 9*a(3) + a(2) = 9*756 + 83 = 6887.

Links

Crossrefs

Cf. A014511.

Programs

  • Magma
    I:=[2,9]; [n le 2 select I[n] else 9*Self(n-1)+Self(n-2): n in [1..30]]; // Vincenzo Librandi, Sep 19 2016
  • Mathematica
    RecurrenceTable[{a[0] == 2, a[1] == 9, a[n] == 9 a[n-1] + a[n-2]}, a, {n, 30}] (* Vincenzo Librandi, Sep 19 2016 *)
LinearRecurrence[{9,1}, {2,9}, 30] (* G. C. Greubel, Nov 07 2018 *)
  • PARI
    x='x+O('x^30); Vec((2-9*x)/(1-9*x-x^2)) \\ G. C. Greubel, Nov 07 2018

Formula

a(n) = ((9 + sqrt(85))/2)^n + ((9 - sqrt(85))/2)^n.
G.f.: (2 - 9*x)/(1 - 9*x - x^2). - Philippe Deléham, Nov 02 2008
From Johannes W. Meijer, Jun 12 2010: (Start)
a(2n+1) = 9*A097840(n), a(2n) = A099373(n).
a(3n+1) = A041150(5n), a(3n+2) = A041150(5n+3), a(3n+3) = 2*A041150(5n+4).
Lim_{k->infinity} a(n+k)/a(k) = (A087798(n) + A099371(n)*sqrt(85))/2.
Lim_{n->infinity} A087798(n)/A099371(n) = sqrt(85). (End)

Extensions

More terms from Ray Chandler, Nov 06 2003

A014510 a(n) = floor( Gamma(n+1/2) ).

Original entry on oeis.org

1, 0, 1, 3, 11, 52, 287, 1871, 14034, 119292, 1133278, 11899423, 136843365, 1710542068, 23092317922, 334838609873, 5189998453040, 85634974475162, 1498612053315336, 27724322986333718, 540624298233507504
Offset: 0

Views

Author

Simon Plouffe

Keywords

References

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 255.

Links

Crossrefs

Cf. A014511, A014512, A014513, A014514, A014516, A014517, A014518, A014521, A014522.

Programs

  • Maple
    seq(floor(evalf(GAMMA(n+1/2),100)), n=0..24);
  • Mathematica
    a[n_] := Floor[Gamma[n + 1/2]]; Array[a, 21, 0] (* Amiram Eldar, Sep 20 2019 *)

Formula

a(n)=floor((2n-1)!!/2^n*sqrt(Pi)) where (2n-1)!!=prod(k=1, n, 2*k-1); asymptotically a(n)=n!/sqrt(n)*(1-1/(8n)+1/(128n^2)-5/(1024n^3)+O(1/n^4)) - Benoit Cloitre, Apr 29 2003

A014513 a(n) = floor( Gamma(n+1/4) ).

Original entry on oeis.org

3, 0, 1, 2, 8, 35, 184, 1155, 8376, 69106, 639232, 6552134, 73711509, 902965985, 11964299312, 170491265198, 2599991794272, 42249866656927, 728810199831991, 13300786146933842, 256040133328476465
Offset: 0

Views

Author

Simon Plouffe

Keywords

References

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 255.

Links

Crossrefs

Cf. A014510, A014511, A014512, A014514, A014516, A014517, A014518, A014521, A014522.

Programs

  • Maple
    seq(floor(evalf(GAMMA(n+1/4),100)), n=0..24);
  • Mathematica
    a[n_] := Floor[Gamma[n + 1/4]]; Array[a, 21, 0] (* Amiram Eldar, Sep 20 2019 *)

A014512 a(n) = floor( Gamma(n+2/3) ).

Original entry on oeis.org

1, 0, 1, 4, 14, 68, 389, 2593, 19884, 172328, 1665837, 17768938, 207304278, 2625854188, 35886673906, 526337883959, 8245960182038, 137432669700633, 2427977164711192, 45322240407942257, 891337394689531069
Offset: 0

Views

Author

Simon Plouffe

Keywords

References

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 255.

Links

Crossrefs

Cf. A014510, A014511, A014513, A014514, A014516, A014517, A014518, A014521, A014522.

Programs

  • Maple
    seq(floor(evalf(GAMMA(n+2/3),100)), n=0..24);
  • Mathematica
    a[n_] := Floor[Gamma[n + 2/3]]; Array[a, 21, 0] (* Amiram Eldar, Sep 20 2019 *)

A014514 a(n) = floor( Gamma(n+3/4) ).

Original entry on oeis.org

1, 0, 1, 4, 16, 78, 453, 3057, 23698, 207358, 2021746, 21733773, 255371835, 3255990905, 44769874946, 660355655453, 10400601573396, 174210076354396, 3092228855290534, 57979291036697518, 1145090997974775999, 23760638207976601980, 516793881023491093069
Offset: 0

Views

Author

Simon Plouffe

Keywords

References

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 255.

Links

Crossrefs

Cf. A014510, A014511, A014512, A014513, A014516, A014517, A014518, A014521, A014522.

Programs

  • Magma
    [Floor(Gamma(n + 3/4)): n in [0..25]]; // Vincenzo Librandi, Aug 01 2025
  • Maple
    seq(floor(evalf(GAMMA(n+3/4),100)), n=0..24);
  • Mathematica
    a[n_] := Floor[Gamma[n + 3/4]]; Array[a, 21, 0] (* Amiram Eldar, Sep 20 2019 *)

A014516 Nearest integer to Gamma(n + 1/3).

Original entry on oeis.org

3, 1, 1, 3, 9, 40, 214, 1355, 9940, 82834, 773118, 7988888, 90540730, 1116669003, 14888920041, 213407853923, 3272253760147, 53446811415729, 926411397872639, 16984208960998390, 328361373245968868, 6676681256001366975, 142435866794695828799, 3181067691748206843170, 74224912807458159673978
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 255.

Links

Crossrefs

Cf. A014510, A014511, A014512, A014513, A014514, A014517, A014518, A014521, A014522.

Programs

  • Maple
    [ seq(round(evalf(GAMMA(n+1/3),100)), n=0..24) ];
  • Mathematica
    a[n_] := Round[Gamma[n + 1/3]]; Array[a, 25, 0] (* Amiram Eldar, Sep 20 2019 *)
  • PARI
    a(n) = round(gamma(n+1/3)); \\ Michel Marcus, Sep 13 2018

Extensions

Corrected and extended by Robert Israel, Sep 12 2018

A014517 Nearest integer to Gamma(n+2/3).

Original entry on oeis.org

1, 1, 2, 4, 15, 69, 389, 2594, 19884, 172328, 1665838, 17768938, 207304278, 2625854188, 35886673906, 526337883960, 8245960182038, 137432669700634, 2427977164711192, 45322240407942258, 891337394689531069
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 255.

Links

Crossrefs

Cf. A014510, A014511, A014512, A014513, A014514, A014516, A014518, A014521, A014522.

Programs

  • Maple
    [ seq(round(evalf(GAMMA(n+2/3),100)), n=0..24) ];
  • Mathematica
    a[n_] := Round[Gamma[n + 2/3]]; Array[a, 21, 0] (* Amiram Eldar, Sep 20 2019 *)
  • PARI
    a(n) = round(gamma(n + 2/3)); \\ Michel Marcus, Nov 05 2018

Extensions

a(15), a(17), and a(19) corrected by Sean A. Irvine, Nov 04 2018

A014518 Nearest integer to Gamma(n+1/4).

Original entry on oeis.org

4, 1, 1, 3, 8, 35, 185, 1155, 8377, 69106, 639233, 6552134, 73711509, 902965986, 11964299312, 170491265198, 2599991794272, 42249866656927, 728810199831991, 13300786146933842, 256040133328476466
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 255.

Links

Crossrefs

Cf. A014510, A014511, A014512, A014513, A014514, A014516, A014517, A014521, A014522.

Programs

  • Maple
    [ seq(round(evalf(GAMMA(n+1/4),100)), n=0..24) ];
  • Mathematica
    a[n_] := Round[Gamma[n + 1/4]]; Array[a, 21, 0] (* Amiram Eldar, Sep 20 2019 *)
  • PARI
    a(n) = round(gamma(n + 1/4)); \\ Michel Marcus, Nov 05 2018

Extensions

a(20) corrected by Sean A. Irvine, Nov 04 2018

A014521 Nearest integer to Gamma(n+1/2).

Original entry on oeis.org

2, 1, 1, 3, 12, 52, 288, 1871, 14034, 119292, 1133278, 11899423, 136843365, 1710542068, 23092317922, 334838609874, 5189998453040, 85634974475162, 1498612053315336, 27724322986333718, 540624298233507504
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 255.

Links

Crossrefs

Cf. A014510, A014511, A014512, A014513, A014514, A014516, A014517, A014518, A014522.

Programs

  • Maple
    [ seq(round(evalf(GAMMA(n+1/2),100)), n=0..24) ];
  • Mathematica
    Table[Round[Gamma[n + 1/2]], {n, 0, 26}]
  • PARI
    a(n) = round(gamma(n + 1/2)); \\ Michel Marcus, Nov 05 2018

Extensions

a(15) corrected by Sean A. Irvine, Nov 04 2018

A014522 Nearest integer to Gamma(n+3/4).

Original entry on oeis.org

1, 1, 2, 4, 17, 79, 453, 3058, 23698, 207359, 2021746, 21733773, 255371836, 3255990905, 44769874946, 660355655454, 10400601573397, 174210076354396, 3092228855290534, 57979291036697519, 1145090997974775999, 23760638207976601980, 516793881023491093070
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 255.

Links

Crossrefs

Cf. A014510, A014511, A014512, A014513, A014514, A014516, A014517, A014518, A014521.

Programs

  • Maple
    [ seq(round(evalf(GAMMA(n+3/4),100)), n=0..24) ];
  • Mathematica
    a[n_] := Round[Gamma[n + 3/4]]; Array[a, 23, 0] (* Amiram Eldar, Sep 20 2019 *)
  • PARI
    a(n) = round(gamma(n+3/4)) \\ Felix Fröhlich, Nov 05 2018

Extensions

a(15), a(16), and a(19) corrected by Sean A. Irvine, Nov 05 2018
Showing 1-10 of 10 results.

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

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