A053699 a(n) = n^4 + n^3 + n^2 + n + 1.
1, 5, 31, 121, 341, 781, 1555, 2801, 4681, 7381, 11111, 16105, 22621, 30941, 41371, 54241, 69905, 88741, 111151, 137561, 168421, 204205, 245411, 292561, 346201, 406901, 475255, 551881, 637421, 732541, 837931, 954305, 1082401, 1222981, 1376831, 1544761, 1727605
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- Carlos M. da Fonseca and Anthony G. Shannon, A formal operator involving Fermatian numbers, Notes Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 491-498.
- Michael Penn, What base makes me a perfect square??, YouTube video, 2022.
- Index to values of cyclotomic polynomials of integer argument.
- Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
Programs
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Magma
[n^4+n^3+n^2+n+1: n in [0..50]]; // Vincenzo Librandi, May 01 2011
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Maple
A053699 := proc(n) numtheory[cyclotomic](5,n) ; end proc: seq(A053699(n),n=0..20) ; # R. J. Mathar, Feb 07 2014
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Mathematica
f[n_]:=((1+n+n^2+n^3+n^4));Table[f[n],{n,0,6!}] (* Vladimir Joseph Stephan Orlovsky, Mar 03 2010 *) Join[{1},Table[Total[n^Range[0,4]],{n,40}]] (* Harvey P. Dale, Feb 02 2014 *) -
Maxima
A053699(n):=n^4 + n^3 + n^2 + n + 1$ makelist(A053699(n),n,0,30); /* Martin Ettl, Nov 07 2012 */
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PARI
a(n)=polcyclo(5,n) \\ Charles R Greathouse IV, Jul 19 2011
Formula
a(n) = n^4 + n^3 + n^2 + n + 1 = (n^5-1)/(n-1).
G.f.: (1 + 16*x^2 + 6*x^3 + x^4)/(1-x)^5. - Colin Barker, Jan 10 2012
E.g.f.: exp(x)*(1 + 4*x + 11*x^2 + 7*x^3 + x^4). - Stefano Spezia, Oct 03 2024
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Wesley Ivan Hurt, Aug 05 2025
Comments