A060890 n^8 + 1.

1, 2, 257, 6562, 65537, 390626, 1679617, 5764802, 16777217, 43046722, 100000001, 214358882, 429981697, 815730722, 1475789057, 2562890626, 4294967297, 6975757442, 11019960577, 16983563042, 25600000001, 37822859362, 54875873537

Offset

0,2

Comments

a(n) = Phi_16(n) where Phi_k(x) is the k-th cyclotomic polynomial.

Links

Harry J. Smith, Table of n, a(n) for n = 0..1000

Index to values of cyclotomic polynomials of integer argument

Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Formula

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

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

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

Example

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

Maple

A060890 := proc(n)
        numtheory[cyclotomic](16, n) ;
end proc:
seq(A060890(n), n=0..20) ; # R. J. Mathar, Feb 11 2014

Mathematica

Table[n^8+1, {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Apr 15 2011 *)
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 *)

Prog

(PARI) for (n=0, 1000, write("b060890.txt", n, " ", n^8 + 1)) \\ Harry J. Smith, Jul 14 2009

Crossrefs

Cf. A002522, A001093, A002523, A002561, A002604.

Sequence in context: A078168 A003380 A259310 * A294275 A085316 A006686

Adjacent sequences: A060887 A060888 A060889 * A060891 A060892 A060893

Keyword

nonn,easy

Author

N. J. A. Sloane, May 05 2001

Status

approved