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a(n) = Sum_{j=0..11} n^j.
8

%I #34 Sep 08 2022 08:45:17

%S 1,12,4095,265720,5592405,61035156,435356467,2306881200,9817068105,

%T 35303692060,111111111111,313842837672,810554586205,1941507093540,

%U 4361070182715,9267595563616,18764998447377,36413889826860

%N a(n) = Sum_{j=0..11} n^j.

%H Vincenzo Librandi, <a href="/A105067/b105067.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

%F Factorization of the polynomial into irreducible components over integers: n^11 + n^10 + n^9 + n^8 + n^7 + n^6 + n^5 + n^4 + n^3 + n^2 + n + 1 = +- (n + 1) * (n^2 - n + 1) * (n^2 + 1) * (n^2 + n + 1) * (n^4 - n^2 + 1). - _Jonathan Vos Post_, Apr 06 2005

%F G.f.: (1365*x^10 + 116480*x^9 + 1851213*x^8 + 8893248*x^7 + 15593370*x^6 + 10568064*x^5 + 2671890*x^4 + 217152*x^3 + 4017*x^2 + 1)/(x - 1)^12. - _Colin Barker_, Oct 29 2012

%t 1+Sum[Range[0,20]^j, {j,1,11}] (* _G. C. Greubel_, Apr 13 2019 *)

%o (Magma) [n^11 + n^10 + n^9 + n^8 + n^7 + n^6 + n^5 + n^4 + n^3 + n^2 + n + 1: n in [0..20]]; // _Vincenzo Librandi_, May 01 2011

%o (PARI) a(n)=polcyclo(11,n)+n^11 \\ _Charles R Greathouse IV_, Sep 03 2011

%o (Sage) [sum(n^j for j in (0..11)) for n in (0..20)] # _G. C. Greubel_, Apr 13 2019

%Y Cf. similar sequences of the type a(n) = Sum_{j=0..m} n^j: A000027 (m=1), A002061 (m=2), A053698 (m=3), A053699 (m=4), A053700 (m=5), A053716 (m=6), A053717 (m=7), A102909 (m=8), A103623 (m=9), A060885 (m=10), this sequence (m=11), A060887 (m=12), A104376 (m=13), A104682 (m=14), A105312 (m=15), A269442 (m=16), A269446 (m=18).

%K nonn,easy

%O 0,2

%A Douglas Winston (douglas.winston(AT)srupc.com), Apr 05 2005

%E Signature changed by _Georg Fischer_, Apr 13 2019