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A053699 a(n) = n^4 + n^3 + n^2 + n + 1.

%I A053699 #67 Aug 05 2025 12:23:47
%S A053699 1,5,31,121,341,781,1555,2801,4681,7381,11111,16105,22621,30941,41371,
%T A053699 54241,69905,88741,111151,137561,168421,204205,245411,292561,346201,
%U A053699 406901,475255,551881,637421,732541,837931,954305,1082401,1222981,1376831,1544761,1727605
%N A053699 a(n) = n^4 + n^3 + n^2 + n + 1.
%C A053699 a(n) = 11111 in base n.
%C A053699 a(n) = Phi_5(n), where Phi_k is the k-th cyclotomic polynomial.
%H A053699 Vincenzo Librandi, <a href="/A053699/b053699.txt">Table of n, a(n) for n = 0..1000</a>
%H A053699 Carlos M. da Fonseca and Anthony G. Shannon, <a href="https://doi.org/10.7546/nntdm.2024.30.3.491-498">A formal operator involving Fermatian numbers</a>, Notes Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 491-498.
%H A053699 Michael Penn, <a href="https://www.youtube.com/watch?v=AtGG8aU8vG0">What base makes me a perfect square??</a>, YouTube video, 2022.
%H A053699 <a href="/index/Cy#CyclotomicPolynomialsValuesAtX">Index to values of cyclotomic polynomials of integer argument</a>.
%H A053699 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A053699 a(n) = n^4 + n^3 + n^2 + n + 1 = (n^5-1)/(n-1).
%F A053699 G.f.: (1 + 16*x^2 + 6*x^3 + x^4)/(1-x)^5. - _Colin Barker_, Jan 10 2012
%F A053699 E.g.f.: exp(x)*(1 + 4*x + 11*x^2 + 7*x^3 + x^4). - _Stefano Spezia_, Oct 03 2024
%F A053699 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
%p A053699 A053699 := proc(n)
%p A053699         numtheory[cyclotomic](5,n) ;
%p A053699 end proc:
%p A053699 seq(A053699(n),n=0..20) ; # _R. J. Mathar_, Feb 07 2014
%t A053699 f[n_]:=((1+n+n^2+n^3+n^4));Table[f[n],{n,0,6!}] (* _Vladimir Joseph Stephan Orlovsky_, Mar 03 2010 *)
%t A053699 Join[{1},Table[Total[n^Range[0,4]],{n,40}]] (* _Harvey P. Dale_, Feb 02 2014 *)
%o A053699 (Magma) [n^4+n^3+n^2+n+1: n in [0..50]]; // _Vincenzo Librandi_, May 01 2011
%o A053699 (PARI) a(n)=polcyclo(5,n) \\ _Charles R Greathouse IV_, Jul 19 2011
%o A053699 (Maxima) A053699(n):=n^4 + n^3 + n^2 + n + 1$
%o A053699 makelist(A053699(n),n,0,30); /* _Martin Ettl_, Nov 07 2012 */
%Y A053699 5th row of the array A055129.
%Y A053699 Cf. A104878.
%K A053699 nonn,easy
%O A053699 0,2
%A A053699 _Henry Bottomley_, Mar 23 2000