A270204 - OEIS

%I #15 Sep 08 2022 08:46:16

%S 1,1,3277,478297,15790321,234750601,2117950381,13564461457,

%T 67662254017,278985273841,990099009901,3112703553961,8854610100337,

%U 23161037562937,56406126018061,129172239050401,280379743338241,580613195032417,1153271900252557,2207200789455481

%N a(n) = n^12 - n^10 + n^8 - n^6 + n^4 - n^2 + 1.

%C a(n) = Phi_28(n) where Phi_k(x) is the k-th cyclotomic polynomial.

%H G. C. Greubel, <a href="/A270204/b270204.txt">Table of n, a(n) for n = 0..1000</a>

%H 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CyclotomicPolynomial.html">Cyclotomic Polynomial</a>

%H <a href="/index/Cy#CyclotomicPolynomialsValuesAtX">Index to values of cyclotomic polynomials of integer argument</a>

%H <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (13, -78, 286, -715, 1287, -1716, 1716, -1287, 715, -286, 78, -13, 1).

%F G.f.: (1 - 12*x + 3342*x^2 + 435488*x^3 + 9828495*x^4 + 65845800*x^5 + 163388148*x^6 + 163386432*x^7 + 65847087*x^8 + 9827780*x^9 + 435774*x^10 + 3264*x^11 + x^12)/(1 - x)^13.

%F Sum_{n>=0} 1/a(n) = 2.000307316...

%p a:= n-> add((-n^2)^j, j=0..6):

%p seq(a(n), n=0..20);  # _Alois P. Heinz_, Apr 24 2019

%t Table[n^12 - n^10 + n^8 - n^6 + n^4 - n^2 + 1, {n, 0, 17}]

%t Table[Cyclotomic[28, n], {n, 0, 17}]

%o (PARI) a(n) = polcyclo(28, n); \\ _Altug Alkan_, Mar 13 2016

%o (Magma) [(&+[(-n^2)^j: j in [0..6]]): n in [0..20]]; // _G. C. Greubel_, Apr 24 2019

%o (Sage) [sum((-n^2)^j for j in (0..6)) for n in (0..20)] # _G. C. Greubel_, Apr 24 2019

%o (GAP) List([0..20], n-> Sum([0..6], j-> (-n^2)^j)) # _G. C. Greubel_, Apr 24 2019

%Y Cf. similar sequences of the type Phi_k(n) listed in A269442.

%K nonn,easy

%O 0,3

%A _Ilya Gutkovskiy_, Mar 13 2016