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A269527 a(n) = n^20 + n^15 + n^10 + n^5 + 1.

%I A269527 #24 Mar 11 2025 11:41:04
%S A269527 1,5,1082401,3501192601,1100586419201,95397958987501,3656628685521505,
%T A269527 79797014141614001,1152956690052710401,12157871353675866901,
%U A269527 100001000010000100001,672754172206667010505,3833775399530967321601,19005014960911672392701,83668410993669614565601
%N A269527 a(n) = n^20 + n^15 + n^10 + n^5 + 1.
%C A269527 a(n) = Phi_25(n) where Phi_k(x) is the k-th cyclotomic polynomial.
%H A269527 G. C. Greubel, <a href="/A269527/b269527.txt">Table of n, a(n) for n = 0..1000</a>
%H A269527 OEIS Wiki, <a href="https://oeis.org/wiki/Cyclotomic Polynomials at x=n, n! and sigma(n)">Cyclotomic Polynomials at x=n, n! and sigma(n)</a>
%H A269527 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CyclotomicPolynomial.html">Cyclotomic Polynomial</a>
%H A269527 <a href="/index/Cy#CyclotomicPolynomialsValuesAtX">Index to values of cyclotomic polynomials of integer argument</a>
%H A269527 <a href="/index/Rec#order_21">Index entries for linear recurrences with constant coefficients</a>, signature (21, -210, 1330, -5985, 20349, -54264, 116280, -203490, 293930, -352716, 352716, -293930, 203490, -116280, 54264, -20349, 5985, -1330, 210, -21, 1).
%F A269527 G.f.: (1 - 16*x + 1082506*x^2 + 3478461900*x^3 + 1027288678125*x^4 + 73019455046736*x^5 + 1879744586779368*x^6 + 21598535807594352*x^7 + 124747954270666290*x^8 + 388588494914595280*x^9 + 679562623971548876*x^10 + 679561811581541512*x^11 + 388588026599578578*x^12 + 124748409900243120*x^13 + 21598656801538440*x^14 + 1879672759361136*x^15 + 72997577685261*x^16 + 1025730080064*x^17 + 3451141690*x^18 + 1016780*x^19 + x^20)/(1 - x)^21.
%F A269527 Sum_{n>=0} 1/a(n) = 1.200000924158557891...
%p A269527 a:= n-> add(n^(5*j), j=0..4):
%p A269527 seq(a(n), n=0..18);  # _Alois P. Heinz_, Apr 24 2019
%t A269527 Table[Cyclotomic[25, n], {n, 0, 12}]
%t A269527 Join[{1},Table[Total[n^(5*Range[0,4])],{n,20}]] (* _Harvey P. Dale_, Sep 01 2016 *)
%o A269527 (Magma) [n^20+n^15+n^10+n^5+1: n in [0..20]]; // _Vincenzo Librandi_, Feb 29 2016
%o A269527 (PARI) a(n) = polcyclo(25, n); \\ _Michel Marcus_, Feb 29 2016
%o A269527 (Sage) [sum(n^(5*j) for j in (0..4)) for n in (0..20)] # _G. C. Greubel_, Apr 24 2019
%o A269527 (GAP) List([0..20], n-> Sum([0..4], j-> n^(5*j))); # _G. C. Greubel_, Apr 24 2019
%Y A269527 Cf. similar sequences of the type Phi_k(n) listed in A269442.
%K A269527 nonn,easy
%O A269527 0,2
%A A269527 _Ilya Gutkovskiy_, Feb 29 2016