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.

A027620 a(n) = n + (n+1)^2 + (n+2)^3.

Original entry on oeis.org

9, 32, 75, 144, 245, 384, 567, 800, 1089, 1440, 1859, 2352, 2925, 3584, 4335, 5184, 6137, 7200, 8379, 9680, 11109, 12672, 14375, 16224, 18225, 20384, 22707, 25200, 27869, 30720, 33759, 36992, 40425, 44064, 47915, 51984, 56277, 60800
Offset: 0

Views

Author

Patrick De Geest

Keywords

Comments

Numbers n > 0 such that x^3 + 2*x^2 + n factors over the integers. - James R. Buddenhagen, Apr 19 2005

Links

Crossrefs

Cf. A000027, A027621, A027622, A028387.

Programs

Formula

a(n) = (n+1)*(n+3)^2. - Zerinvary Lajos, Sep 24 2006, corrected Dec 21 2010
G.f.: (9 - 4*x + x^2)/(1 - x)^4. - R. J. Mathar, Dec 21 2010
a(n) = coefficient of x^3 in the Maclaurin expansion of -1/((n+3)*x^2 + (n+3)*x + 1). - Francesco Daddi, Aug 04 2011
E.g.f.: (9 + 23*x + 10*x^2 + x^3)*exp(x). - G. C. Greubel, Aug 05 2022

A027622 a(n) = n + (n+1)^2 + (n+2)^3 + (n+3)^4 + (n+4)^5.

Original entry on oeis.org

1114, 3413, 8476, 18247, 35414, 63529, 107128, 171851, 264562, 393469, 568244, 800143, 1102126, 1488977, 1977424, 2586259, 3336458, 4251301, 5356492, 6680279, 8253574, 10110073, 12286376, 14822107, 17760034, 21146189, 25029988, 29464351, 34505822, 40214689
Offset: 0

Views

Author

Patrick De Geest

Keywords

Links

Crossrefs

Cf. A000027, A027620, A027621, A028387.

Programs

  • Magma
    [n+(n+1)^2+(n+2)^3+(n+3)^4+(n+4)^5: n in [0..30]]; // Vincenzo Librandi, Dec 28 2010
  • Maple
    seq( add((n+j)^(j+1), j=0..4), n=0..30); # G. C. Greubel, Aug 05 2022
  • Mathematica
    Table[n +(n+1)^2 +(n+2)^3 +(n+3)^4 +(n+4)^5, {n, 0, 29}] (* Alonso del Arte, Nov 22 2016 *)
Table[ReleaseHold@ Total@ MapIndexed[#1^First@ #2 &, Rest@ FactorList[ Pochhammer[Hold@ n, 5]][[All, 1]]], {n, 0, 29}] (* or *)
CoefficientList[Series[(1114 -3271x +4708x^2 -3694x^3 +1522x^4 -259x^5)/(1-x)^6, {x, 0, 29}], x] (* Michael De Vlieger, Dec 05 2016 *)
Table[Total[Table[(n+k)^(k+1),{k,0,4}]],{n,0,30}] (* or *) LinearRecurrence[{6,-15,20,-15,6,-1}, {1114,3413,8476,18247,35414,63529}, 30] (* Harvey P. Dale, Aug 04 2022 *)
  • PARI
    Vec((1114-3271*x+4708*x^2-3694*x^3+1522*x^4-259*x^5) / (1-x)^6 + O(x^30)) \\ Colin Barker, Dec 05 2016
  • SageMath
    [sum((n+j)^(j+1) for j in (0..4)) for n in (0..30)] # G. C. Greubel, Aug 05 2022

Formula

From Colin Barker, Dec 05 2016: (Start)
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) for n>5.
G.f.: (1114-3271*x+4708*x^2-3694*x^3+1522*x^4-259*x^5) / (1-x)^6.
(End)
E.g.f.: (1114 +2299*x +1382*x^2 +324*x^3 +31*x^4 +x^5)*exp(x). - G. C. Greubel, Aug 05 2022

A061224 a(n) = n^2 + (n + 1)^3 + (n + 2)^4 + (n + 3)^5.

Original entry on oeis.org

260, 1114, 3412, 8474, 18244, 35410, 63524, 107122, 171844, 264554, 393460, 568234, 800132, 1102114, 1488964, 1977410, 2586244, 3336442, 4251284, 5356474, 6680260, 8253554, 10110052, 12286354, 14822084, 17760010, 21146164, 25029962
Offset: 0

Views

Author

Olivier Gérard, May 31 2001

Keywords

Examples

			a(1) = 1 + 2^3 + 3^4 + 4^5 = 1 + 8 + 81 + 1024 = 1114.

Links

Crossrefs

Cf. A027621.

Programs

  • GAP
    List([0..30],n->n^2+(n+1)^3+(n+2)^4+(n+3)^5); # Muniru A Asiru, Nov 02 2018
  • Magma
    [n^2 + (n + 1)^3 + (n + 2)^4 + (n + 3)^5: n in [0..30]]; // Vincenzo Librandi, Aug 05 2011
  • Maple
    seq(n^2+(n+1)^3+(n+2)^4+(n+3)^5,n=0..30); # Muniru A Asiru, Nov 02 2018
  • Mathematica
    Table[n^2 +(n+1)^3 +(n+2)^4 +(n+3)^5, {n,0,30}] (* G. C. Greubel, Nov 02 2018 *)
CoefficientList[Series[E^x (260 + 854 x + 722 x^2 + 220 x^3 + 26 x^4 + x^5), {x, 0, 50}], x]*Table[k!, {k, 0, 50}] (* Stefano Spezia, Nov 02 2018 *)
Table[260+440 n+298 n^2+99 n^3+16 n^4+n^5,{n,0,30}] (* or *) LinearRecurrence[ {6,-15,20,-15,6,-1},{260,1114,3412,8474,18244,35410},30] (* Harvey P. Dale, Nov 14 2022 *)
  • PARI
    vector(30, n, n--; n^2 +(n+1)^3 +(n+2)^4 +(n+3)^5) \\ G. C. Greubel, Nov 02 2018

Formula

G.f.: 2*(130 - 223*x + 314*x^2 - 244*x^3 + 100*x^4 - 17*x^5)/(1-x)^6. - Bruno Berselli, Aug 05 2011
E.g.f.: exp(x)*(260 + 854*x + 722*x^2 + 220*x^3 + 26*x^4 + x^5). - Stefano Spezia, Nov 02 2018

Extensions

Offset changed from 1 to 0 by Vincenzo Librandi, Aug 05 2011

A061225 Numbers of the form k^3 + (k + 1)^4 + (k + 2)^5 + (k + 3)^6.

Original entry on oeis.org

762, 4356, 16738, 50064, 126114, 280372, 566826, 1063488, 1878634, 3157764, 5091282, 7922896, 11958738, 17577204, 25239514, 35500992, 49023066, 66585988, 89102274, 117630864, 153392002, 197782836, 252393738, 319025344, 399706314
Offset: 0

Views

Author

Olivier Gérard, May 31 2001

Keywords

Examples

			a(1) = 1 + 2^4 + 3^5 + 4^6 = 1 + 16 + 243 + 4096 = 4356.

Links

Crossrefs

Cf. A027621.

Programs

  • Magma
    [ n^3 + (n + 1)^4 + (n + 2)^5 + (n + 3)^6: n in [0..30]]; // Vincenzo Librandi, Aug 05 2011
  • Mathematica
    Table[n^3+(n+1)^4+(n+2)^5+(n+3)^6,{n,0,30}]  (* Harvey P. Dale, Mar 26 2011 *)

Formula

G.f.: 2*(381 - 489*x + 1124*x^2 - 1148*x^3 + 687*x^4 - 227*x^5 + 32*x^6)/(1-x)^7. - Colin Barker, Apr 02 2012

Extensions

Offset changed from 1 to 0 by Vincenzo Librandi, Aug 05 2011
Showing 1-4 of 4 results.

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