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A269483 a(n) = n^12 - n^11 + n^9 - n^8 + n^6 - n^4 + n^3 - n + 1. 2

%I #28 Sep 08 2022 08:46:15

%S 1,1,2359,368089,12783421,196890121,1822428931,11898664849,

%T 60247241209,251393376241,900900990991,2855262053161,8177824843189,

%U 21515718297529,52663539957211,121132473843361,263947231891441,548461977100129

%N a(n) = n^12 - n^11 + n^9 - n^8 + n^6 - n^4 + n^3 - n + 1.

%C a(n) = Phi_21(n) where Phi_k(x) is the k-th cyclotomic polynomial.

%H G. C. Greubel, <a href="/A269483/b269483.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 + 2424*x^2 + 337214*x^3 + 8182695*x^4 + 58741344*x^5 + 156377856*x^6 + 168607380*x^7 + 73943271*x^8 + 12191420*x^9 + 612600*x^10 + 5406*x^11 + x^12)/(1-x)^13.

%F Sum_{n>=0} 1/a(n) = 2.00042670913...

%t Table[Cyclotomic[21, n], {n, 0, 17}]

%t CoefficientList[Series[(1 -12x +2424x^2 +337214x^3 +8182695x^4 +58741344 x^5 +156377856x^6 +168607380x^7 +73943271x^8 +12191420x^9 + 612600 x^10 +5406x^11 +x^12)/(1-x)^13, {x, 0, 33}], x] (* _Vincenzo Librandi_, Feb 28 2016 *)

%o (Magma) [n^12-n^11+n^9-n^8+n^6-n^4+n^3-n+1: n in [0..20]]; // _Vincenzo Librandi_, Feb 28 2016

%o (Python)

%o A269483_list, m = [], [479001600, -2674425600, 6386688000, -8501915520, 6889478400, -3482100720, 1080164160, -194177280, 17948256, -666714, 5418, 0, 1]

%o for _ in range(10**2):

%o A269483_list.append(m[-1])

%o for i in range(12):

%o m[i+1] += m[i] # _Chai Wah Wu_, Feb 28 2016

%o (PARI) a(n) = polcyclo(21, n); \\ _Michel Marcus_, Feb 29 2016

%o (Sage) [n^12-n^11+n^9-n^8+n^6-n^4+n^3-n+1 for n in (0..20)] # _G. C. Greubel_, Apr 24 2019

%o (GAP) List([0..20], n-> n^12-n^11+n^9-n^8+n^6-n^4+n^3-n+1) # _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_, Feb 27 2016

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Last modified March 29 02:23 EDT 2024. Contains 371264 sequences. (Running on oeis4.)