A060896 - OEIS

A060896

a(n) = n^12 - n^6 + 1.

%I #29 Dec 20 2024 10:13:46

%S 1,1,4033,530713,16773121,244125001,2176735681,13841169553,

%T 68719214593,282429005041,999999000001,3138426605161,8916097462273,

%U 23298080295673,56693904845761,129746326500001,281474959933441,582622213092193,1156831347413953,2213314872020281,4095999936000001

%N a(n) = n^12 - n^6 + 1.

%C a(n) = Phi_36(n) where Phi_k is the k-th cyclotomic polynomial.

%H Harry J. Smith, <a href="/A060896/b060896.txt">Table of n, a(n) for n = 0..1000</a>

%H Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha103.htm">Factorizations of many number sequences</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.: -(x^12 +4020*x^11 +478362*x^10 +10188140*x^9 +66317319*x^8 +162512496*x^7 +162514212*x^6 +66316032*x^5 +10188855*x^4 +478076*x^3 +4098*x^2 -12*x +1)/(x -1)^13. - _Colin Barker_, Oct 29 2012

%F a(0)=1, a(1)=1, a(2)=4033, a(3)=530713, a(4)=16773121, a(5)=244125001, a(6)=2176735681, a(7)=13841169553, a(8)=68719214593, a(9)=282429005041, a(10)=999999000001, a(11)=3138426605161, a(12)=8916097462273, a(n)=13*a(n-1)- 78*a(n-2)+ 286*a(n-3)- 715*a(n-4)+ 1287*a(n-5)- 1716*a(n-6)+ 1716*a(n-7)- 1287*a(n-8)+ 715*a(n-9)- 286*a(n-10)+ 78*a(n-11)- 13*a(n-12)+ a(n-13). - _Harvey P. Dale_, Dec 16 2013

%p A060896 := proc(n)

%p numtheory[cyclotomic](36,n) ;

%p end proc:

%p seq(A060896(n),n=0..20) ; # _R. J. Mathar_, Feb 11 2014

%t Table[n^12-n^6+1,{n,0,30}] (* or *) LinearRecurrence[{13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1},{1,1,4033,530713,16773121,244125001,2176735681,13841169553,68719214593,282429005041,999999000001,3138426605161,8916097462273},30] (* _Harvey P. Dale_, Dec 16 2013 *)

%o (PARI) a(n) = { n^12 - n^6 + 1 } \\ _Harry J. Smith_, Jul 19 2009

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_, May 05 2001