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 1-3 of 3 results.

A160828 a(n) = 4*n^4 + 24*n^3 + 84*n^2 + 144*n + 98.

Original entry on oeis.org

98, 354, 978, 2258, 4578, 8418, 14354, 23058, 35298, 51938, 73938, 102354, 138338, 183138, 238098, 304658, 384354, 478818, 589778, 719058, 868578, 1040354, 1236498, 1459218, 1710818, 1993698, 2310354, 2663378, 3055458, 3489378, 3968018, 4494354
Offset: 0

Views

Author

Al Hakanson (hawkuu(AT)gmail.com), May 27 2009

Keywords

Comments

Sums of 4 consecutive fourth powers.
Subsequence of A217844. - Michel Marcus, Jun 30 2013

Links

Programs

  • Magma
    [4*n^4+24*n^3+84*n^2+144*n+98: n in [0..40]]; // Vincenzo Librandi, Aug 27 2011
  • Maple
    A000583 := proc(n) n^4 ; end: A160828 := proc(n) add(A000583(i),i=n..n+3) ; end: seq(A160828(n),n=0..40) ; # R. J. Mathar, May 29 2009
  • Mathematica
    Table[4n^4+24n^3+84n^2+144n+98,{n,0,40}] (* or *) LinearRecurrence[ {5,-10,10,-5,1},{98,354,978,2258,4578},40] (* Harvey P. Dale, Mar 25 2012 *)
CoefficientList[Series[(18*x^4 -72*x^3 +188*x^2 -136*x +98)/(1-x)^5, {x, 0, 50}], x] (* G. C. Greubel, Apr 30 2018 *)
  • PARI
    x='x+O('x^50); Vec((18*x^4 -72*x^3 +188*x^2 -136*x +98)/(1-x)^5) \\ G. C. Greubel, Apr 30 2018
  • Python
    A160828_list, m = [], [96, 0, 80, 80, 98]
for _ in range(10**2):
    A160828_list.append(m[-1])
    for i in range(4):
        m[i+1] += m[i] # Chai Wah Wu, Jan 23 2016

Formula

a(n) = Sum_{i=0..3} A000583(n+i) = Sum_{j=n..n+3} j^4 = A160827(n) + (n+3)^4.
G.f.: (18*x^4 - 72*x^3 + 188*x^2 - 136*x + 98)/(1-x)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Aug 14 2009
a(0)=98, a(1)=354, a(2)=978, a(3)=2258, a(4)=4578, a(n)=5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Harvey P. Dale, Mar 25 2012
E.g.f.: 2*(49 + 128*x + 92*x^2 + 24*x^3 + 2*x^4)*exp(x). - G. C. Greubel, Apr 30 2018

Extensions

Edited and corrected by R. J. Mathar, May 29 2009

A179995 Generating function A(5,t)(1+t+t^2)/(1-t)^6, where A(5,t) is an Eulerian polynomial.

Original entry on oeis.org

1, 33, 276, 1299, 4392, 11925, 27708, 57351, 108624, 191817, 320100, 509883, 781176, 1157949, 1668492, 2345775, 3227808, 4358001, 5785524, 7565667, 9760200, 12437733, 15674076, 19552599, 24164592, 29609625, 35995908
Offset: 0

Views

Author

Peter Luschny, Aug 05 2010

Keywords

Comments

The Eulerian polynomials A(n,t) are here defined in accordance with the Digital Library of Mathematical Functions, Table 26.14.1.
Sums of 3 consecutive fifth powers: a(n) = (n-1)^5+n^5+(n+1)^5. - Bruno Berselli, Jun 24 2013

Links

Crossrefs

Cf. A158799, A008486, A005918, A027602, A160827 which have generating functions of type A(n, t)(1+t+t^2)/(1-t)^(n+1).

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+27*x+93*x^2+118*x^3+93*x^4+27*x^5+x^6)/(1-x)^6)); // Bruno Berselli, Jun 24 2013
  • Maple
    gfA179995 := proc(t) local i;
add([1,27,93,118,93,27,1][i+1]*t^i,i=0..5)/(1-t)^6 end:
seq(coeff(series(gfA179995(t),t,24),t,j),j=0..16);
  • Mathematica
    Join[{1}, Table[n (3 n^4 + 20 n^2 + 10), {n, 30}]] (* Bruno Berselli, Jun 24 2013 *)
LinearRecurrence[{6, -15, 20, -15, 6, -1}, {1, 33, 276, 1299, 4392, 11925, 27708}, 30] (* Harvey P. Dale, Apr 10 2015 *)

Formula

From Bruno Berselli, Jun 24 2013: (Start)
G.f.: (1 + 27*x + 93*x^2 + 118*x^3 + 93*x^4 + 27*x^5 + x^6) / (1 - x)^6.
a(n) = n*(3*n^4 + 20*n^2 + 10) for n>0, a(0)=1. (End)
a(0)=1, a(1)=33, a(2)=276, a(3)=1299, a(4)=4392, a(5)=11925, a(6)=27708; for n>6, 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). - Harvey P. Dale, Apr 10 2015

A191589 Primes of the form 3*n^4+12*n^2+2, n > 0.

Original entry on oeis.org

17, 353, 7793, 45377, 588737, 1603073, 2131937, 2782097, 23705153, 27488177, 36393857, 142457633, 156688577, 288296417, 423617057, 780627473, 830968337, 938914433, 1254730193, 5724613457, 9150064577, 13500386657, 15247220033
Offset: 1

Views

Author

Rafael Parra Machio, Jun 07 2011

Keywords

Comments

Prime sums of three consecutive fourth powers, since 3*n^4+12*n^2+2 = (n-1)^4+n^4+(n+1)^4.
Primes in A160827.

Examples

			2^4+3^4+4^4 = 353 is prime and therefore in the sequence.

Crossrefs

Cf. A160827.

Programs

  • Magma
    [ p: n in [0..300] | IsPrime(p) where p is n^4+(n+1)^4+(n+2)^4 ];
  • Mathematica
    lst={};Do[If[PrimeQ[p=(n+1)^4+n^4+ (n-1)^4], AppendTo[lst,p]],{n, 100}];lst
lst={};Do[If[PrimeQ[p=3*n^4+12*n^2+2], AppendTo[lst, p]],{n,100}];lst
Showing 1-3 of 3 results.

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