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 1-8 of 8 results.

A016949 a(n) = (6*n + 3)^5.

Original entry on oeis.org

243, 59049, 759375, 4084101, 14348907, 39135393, 90224199, 184528125, 345025251, 601692057, 992436543, 1564031349, 2373046875, 3486784401, 4984209207, 6956883693, 9509900499, 12762815625, 16850581551, 21924480357, 28153056843, 35723051649, 44840334375, 55730836701
Offset: 0

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Author

Keywords

Crossrefs

Subsequence of A000584.

Programs

  • Magma
    [(6*n+3)^5: n in [0..50]]; // Vincenzo Librandi, May 05 2011
  • Mathematica
    a[n_] := (6*n + 3)^5; Array[a, 50, 0] (* Amiram Eldar, Mar 30 2022 *)

Formula

From Amiram Eldar, Mar 30 2022: (Start)
a(n) = A016945(n)^5.
a(n) = 3^5*A016757(n).
Sum_{n>=0} 1/a(n) = 31*zeta(5)/7776.
Sum_{n>=0} (-1)^n/a(n) = 5*Pi^5/373248. (End)

A016762 a(n) = (2*n + 1)^10.

Original entry on oeis.org

1, 59049, 9765625, 282475249, 3486784401, 25937424601, 137858491849, 576650390625, 2015993900449, 6131066257801, 16679880978201, 41426511213649, 95367431640625, 205891132094649, 420707233300201, 819628286980801, 1531578985264449, 2758547353515625, 4808584372417849
Offset: 0

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Crossrefs

Programs

Formula

a(n) = A016757(n)^2. - Michel Marcus, Dec 27 2016
From G. C. Greubel, Dec 27 2016: (Start)
G.f.: (1 +59038*x +9116141*x^2 +178300904*x^3 +906923282*x^4 + 1527092468*x^5 +906923282*x^6 +178300904*x^7 +9116141*x^8 +59038*x^9 + x^10)/(1-x)^11.
E.g.f.: (1 +59048*x +4823764*x^2 +42225920*x^3 +100635040*x^4 + 93590784*x^5 +40322688*x^6 +8724480*x^7 +963840*x^8 +51200*x^9 + 1024*x^10)*exp(x). (End)
Sum_{n>=0} 1/a(n) = 31*Pi^10/2903040. - Amiram Eldar, Oct 11 2020

A002594 a(n) = n^2*(16*n^4-20*n^2+7)/3.

Original entry on oeis.org

1, 244, 3369, 20176, 79225, 240276, 611569, 1370944, 2790801, 5266900, 9351001, 15787344, 25552969, 39901876, 60413025, 89042176, 128177569, 180699444, 250043401, 340267600, 456123801, 603132244, 787660369, 1017005376, 1299480625, 1644505876, 2062701369, 2565985744, 3167677801, 3882602100, 4727198401, 5719634944, 6879925569, 8230050676
Offset: 1

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Comments

It is possible that the Croxton and Crowden reference gives a better explanation than the simple formula in the new definition.
Sum of the fifth powers of the first n odd numbers. - Michel Marcus, Dec 01 2015

References

  • F. E. Croxton and D. J. Cowden, Applied General Statistics. 2nd ed., Prentice-Hall, Englewood Cliffs, NJ, 1955, p. 742.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Crossrefs

Programs

  • Magma
    [n^2/3 * (16*n^4 - 20*n^2 + 7): n in [1..40]]; // Vincenzo Librandi, Sep 07 2011
  • Maple
    A002594:=-(z+1)*(z**4+236*z**3+1446*z**2+236*z+1)/(z-1)**7; # Simon Plouffe in his 1992 dissertation

Formula

G.f.: x*(1+x)*(1+236*x+1446*x^2+236*x^3+x^4)/(1-x)^7. [Simon Plouffe]

Extensions

The old definition was wrong, entry revised by N. J. A. Sloane, Jun 10 2012

A017333 a(n) = (10*n + 5)^5.

Original entry on oeis.org

3125, 759375, 9765625, 52521875, 184528125, 503284375, 1160290625, 2373046875, 4437053125, 7737809375, 12762815625, 20113571875, 30517578125, 44840334375, 64097340625, 89466096875, 122298103125, 164130859375, 216699865625, 281950621875, 362050628125, 459401384375
Offset: 0

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Keywords

Crossrefs

Cf. A272914 (first comment). [Bruno Berselli, May 26 2016]

Programs

  • Magma
    [(10*n+5)^5: n in [0..25]]; // Vincenzo Librandi, Aug 02 2011
  • Mathematica
    (10*Range[0,20]+5)^5 (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{3125,759375,9765625,52521875,184528125,503284375},20] (* Harvey P. Dale, May 15 2018 *)

Formula

G.f.: 3125*(x+1)*(x^4+236*x^3+1446*x^2+236*x+1)/(x-1)^6. - Colin Barker, Nov 14 2012
From Amiram Eldar, Apr 18 2023: (Start)
a(n) = A017329(n)^5.
a(n) = 5^5 * A016757(n).
Sum_{n>=0} 1/a(n) = 31*zeta(5)/100000.
Sum_{n>=0} (-1)^n/a(n) = Pi^5/960000. (End)

A016745 a(n) = (2*n)^5.

Original entry on oeis.org

0, 32, 1024, 7776, 32768, 100000, 248832, 537824, 1048576, 1889568, 3200000, 5153632, 7962624, 11881376, 17210368, 24300000, 33554432, 45435424, 60466176, 79235168, 102400000, 130691232, 164916224, 205962976, 254803968, 312500000, 380204032, 459165024, 550731776
Offset: 0

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Author

Keywords

Crossrefs

Cf. A016757.

Programs

Formula

G.f.: 32*x*(1 + 26*x + 66*x^2 + 26*x^3 + x^4)/(1-x)^6. - Colin Barker, Sep 17 2012
E.g.f.: 32*x*(1 + 15*x + 25*x^2 + 10*x^3 + x^4)*exp(x). - G. C. Greubel, Sep 15 2018
From Amiram Eldar, Oct 10 2020: (Start)
Sum_{n>=1} 1/a(n) = zeta(5)/32.
Sum_{n>=1} (-1)^(n+1)/a(n) = 15*zeta(5)/512. (End)

A016829 a(n) = (4n+2)^5.

Original entry on oeis.org

32, 7776, 100000, 537824, 1889568, 5153632, 11881376, 24300000, 45435424, 79235168, 130691232, 205962976, 312500000, 459165024, 656356768, 916132832, 1252332576, 1680700000, 2219006624, 2887174368, 3707398432, 4704270176, 5904900000, 7339040224, 9039207968, 11040808032
Offset: 0

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Crossrefs

Programs

  • Mathematica
    (4Range[0,20]+2)^5 (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{32,7776,100000,537824,1889568,5153632},20] (* Harvey P. Dale, Aug 31 2011 *)

Formula

From Harvey P. Dale, Aug 31 2011: (Start)
a(0)=32, a(1)=7776, a(2)=100000, a(3)=537824, a(4)=1889568, a(5)=5153632, a(n)=6*a(n-1)-15*a(n-2)+20*a(n-3)-15*a(n-4)+6*a(n-5)- a(n-6).
G.f.: (32*(x+1)*(x*(x*(x*(x+236)+1446)+236)+1))/(x-1)^6. (End)
From Amiram Eldar, Apr 21 2023: (Start)
a(n) = A016825(n)^5.
a(n) = 2^5*A016757(n).
Sum_{n>=0} 1/a(n) = 31*zeta(5)/1024.
Sum_{n>=0} (-1)^n/a(n) = 5*Pi^5/49152. (End)

A017117 a(n) = (8*n + 4)^5.

Original entry on oeis.org

1024, 248832, 3200000, 17210368, 60466176, 164916224, 380204032, 777600000, 1453933568, 2535525376, 4182119424, 6590815232, 10000000000, 14693280768, 21003416576, 29316250624, 40074642432, 53782400000, 71008211968, 92389579776, 118636749824, 150536645632, 188956800000
Offset: 0

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Crossrefs

Programs

  • Magma
    [(8*n+4)^5: n in [0..30] ]; // Vincenzo Librandi, Jul 21 2011
  • Mathematica
    (8*Range[0,20]+4)^5 (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{1024,248832,3200000,17210368,60466176,164916224},20] (* Harvey P. Dale, Nov 24 2012 *)

Formula

a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6); a(0)=1024, a(1)=248832, a(2)=3200000, a(3)=17210368, a(4)=60466176, a(5)=164916224. - Harvey P. Dale, Nov 24 2012
G.f.: 1024*(1+x)*(x^4 + 236*x^3 + 1446*x^2 + 236*x + 1) / (x-1)^6. - R. J. Mathar, May 08 2015
From Amiram Eldar, Apr 25 2023: (Start)
a(n) = A017113(n)^5.
a(n) = 2^5*A016829(n) = 2^10*A016757(n).
Sum_{n>=0} 1/a(n) = 31*zeta(5)/32768.
Sum_{n>=0} (-1)^n/a(n) = 5*Pi^5/1572864. (End)

A165935 a(n) = (-1)^(n-1)*n*(4n^2-5)^2.

Original entry on oeis.org

1, -242, 2883, -13924, 45125, -115926, 255367, -504008, 915849, -1560250, 2523851, -3912492, 5853133, -8495774, 12015375, -16613776, 22521617, -30000258, 39343699, -50880500
Offset: 1

Views

Author

Richard L. Peterson (rl_pete(AT)yahoo.com), Oct 01 2009

Keywords

Comments

These are the partial sums of the alternating series of odd fifth powers beginning with 1. See A016757.

Crossrefs

Cf. A016757.

Programs

  • Mathematica
    Table[(-1)^(n - 1)*n*(4*n^2 - 5)^2, {n, 1, 50}] (* G. C. Greubel, Apr 18 2016 *)
    LinearRecurrence[{-6,-15,-20,-15,-6,-1},{1,-242,2883,-13924,45125,-115926},20] (* Harvey P. Dale, Mar 24 2020 *)
  • PARI
    vector(100, n, (-1)^(n-1)*n*(4*n^2-5)^2) \\ Altug Alkan, Apr 18 2016

Formula

G.f.: x*(1-236*x+1446*x^2-236*x^3+x^4) / (1+x)^6. - R. J. Mathar, Nov 27 2011
E.g.f.: x*(1 - 120*x + 360*x^2 - 160*x^3 + 16*x^4)*exp(-x). - Ilya Gutkovskiy, Apr 17 2016
Showing 1-8 of 8 results.