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A053699
a(n) = n^4 + n^3 + n^2 + n + 1.
30
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
0,2
COMMENTS
a(n) = 11111 in base n.
a(n) = Phi_5(n), where Phi_k is the k-th cyclotomic polynomial.
LINKS
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.
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
MAPLE
A053699 := proc(n)
numtheory[cyclotomic](5, n) ;
end proc:
seq(A053699(n), n=0..20) ; # R. J. Mathar, Feb 07 2014
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 *)
PROG
(Magma) [n^4+n^3+n^2+n+1: n in [0..50]]; // Vincenzo Librandi, May 01 2011
(PARI) a(n)=polcyclo(5, n) \\ Charles R Greathouse IV, Jul 19 2011
(Maxima) A053699(n):=n^4 + n^3 + n^2 + n + 1$
makelist(A053699(n), n, 0, 30); /* Martin Ettl, Nov 07 2012 */
CROSSREFS
5th row of the array A055129.
Cf. A104878.
Sequence in context: A099083 A212523 A096944 * A152122 A260045 A267938
KEYWORD
nonn,easy,changed
AUTHOR
Henry Bottomley, Mar 23 2000
STATUS
approved