cp's OEIS Frontend

A060890 a(n) = n^8 + 1.

%I A060890 #35 Dec 20 2024 10:12:39
%S A060890 1,2,257,6562,65537,390626,1679617,5764802,16777217,43046722,
%T A060890 100000001,214358882,429981697,815730722,1475789057,2562890626,
%U A060890 4294967297,6975757442,11019960577,16983563042,25600000001,37822859362,54875873537,78310985282,110075314177,152587890626
%N A060890 a(n) = n^8 + 1.
%C A060890 a(n) = Phi_16(n) where Phi_k(x) is the k-th cyclotomic polynomial.
%H A060890 Harry J. Smith, <a href="/A060890/b060890.txt">Table of n, a(n) for n = 0..1000</a>
%H A060890 <a href="/index/Cy#CyclotomicPolynomialsValuesAtX">Index to values of cyclotomic polynomials of integer argument</a>
%H A060890 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,1).
%F A060890 a(0)=1, a(1)=2, a(2)=257, a(3)=6562, a(4)=65537, a(5)=390626, a(6)=1679617, a(7)=5764802, a(8)=16777217, a(n)=9*a(n-1)-36*a(n-2)+84*a(n-3)-126*a(n-4)+ 126*a(n-5)-84*a(n-6)+36*a(n-7)-9*a(n-8)+a(n-9). - _Harvey P. Dale_, Mar 12 2013
%F A060890 Sum_{n>=0} 1/a(n) = 1/2 + Pi*((sqrt(2 + sqrt(2)) * sin(sqrt(2 + sqrt(2))*Pi) + sqrt(2 - sqrt(2)) * sinh(sqrt(2 - sqrt(2))*Pi)) / (cosh(sqrt(2 - sqrt(2))*Pi) - cos(sqrt(2 + sqrt(2))*Pi)) + (sqrt(2 - sqrt(2)) * sin(sqrt(2 - sqrt(2))*Pi) + sqrt(2 + sqrt(2)) * sinh(sqrt(2 + sqrt(2))*Pi)) / (cosh(sqrt(2 + sqrt(2))*Pi) - cos(sqrt(2 - sqrt(2))*Pi))) / 8 = 1.5040621333147995112929... . - _Vaclav Kotesovec_, Feb 14 2015
%F A060890 Sum_{n>=0} (-1)^n/a(n) = 1/2 + Pi*((sqrt(2 - sqrt(2)) * sin(sqrt(2 - sqrt(2))*Pi/2) - sqrt(2 + sqrt(2)) * sinh(sqrt(2 + sqrt(2))*Pi/2)) / (cos(sqrt(2 - sqrt(2))*Pi/2) + cosh(sqrt(2 + sqrt(2))*Pi/2)) - (sqrt(2 - sqrt(2)) * sin(sqrt(2 - sqrt(2))*Pi/2) + sqrt(2 + sqrt(2)) * sinh(sqrt(2 + sqrt(2))*Pi/2)) / (cos(sqrt(2 - sqrt(2))*Pi/2) - cosh(sqrt(2 + sqrt(2))*Pi/2)) + (sqrt(2 + sqrt(2)) * sin(sqrt(2 + sqrt(2))*Pi/2) - sqrt(2 - sqrt(2)) * sinh(sqrt(2 - sqrt(2))*Pi/2)) / (cos(sqrt(2 + sqrt(2))*Pi/2) + cosh(sqrt(2 - sqrt(2))*Pi/2)) - (sqrt(2 + sqrt(2)) * sin(sqrt(2 + sqrt(2))*Pi/2) + sqrt(2 - sqrt(2)) * sinh(sqrt(2 - sqrt(2))*Pi/2)) / (cos(sqrt(2 + sqrt(2))*Pi/2) - cosh(sqrt(2 - sqrt(2))*Pi/2)))/16 = 0.5037518217314416642671664241... . - _Vaclav Kotesovec_, Feb 14 2015
%F A060890 G.f.: (1-7*x+275*x^2+4237*x^3+15689*x^4+15563*x^5+4321*x^6+239*x^7+2*x^8)/ (1-x)^9. - _Colin Barker_, Apr 21 2012
%p A060890 A060890 := proc(n)
%p A060890         numtheory[cyclotomic](16,n) ;
%p A060890 end proc:
%p A060890 seq(A060890(n),n=0..20) ; # _R. J. Mathar_, Feb 11 2014
%t A060890 Table[n^8+1,{n,0,40}] (* _Vladimir Joseph Stephan Orlovsky_, Apr 15 2011 *)
%t A060890 LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{1,2,257,6562,65537,390626,1679617,5764802,16777217},40] (* _Harvey P. Dale_, Mar 12 2013 *)
%o A060890 (PARI) a(n) = n^8 + 1 \\ _Harry J. Smith_, Jul 14 2009
%Y A060890 Cf. A002522, A001093, A002523, A002561, A002604.
%K A060890 nonn,easy
%O A060890 0,2
%A A060890 _N. J. A. Sloane_, May 05 2001