A060890 a(n) = 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, 78310985282, 110075314177, 152587890626
Offset: 0
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).
Programs
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Maple
A060890 := proc(n) numtheory[cyclotomic](16,n) ; end proc: seq(A060890(n),n=0..20) ; # R. J. Mathar, Feb 11 2014
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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 *) -
PARI
a(n) = n^8 + 1 \\ Harry J. Smith, Jul 14 2009
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
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
Comments