A105067 a(n) = Sum_{j=0..11} n^j.

1, 12, 4095, 265720, 5592405, 61035156, 435356467, 2306881200, 9817068105, 35303692060, 111111111111, 313842837672, 810554586205, 1941507093540, 4361070182715, 9267595563616, 18764998447377, 36413889826860

Offset: 0

Author

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

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Crossrefs

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).

Programs

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

Mathematica

1+Sum[Range[0,20]^j, {j,1,11}] (* G. C. Greubel, Apr 13 2019 *)

PARI

a(n)=polcyclo(11,n)+n^11 \\ Charles R Greathouse IV, Sep 03 2011

Sage

[sum(n^j for j in (0..11)) for n in (0..20)] # G. C. Greubel, Apr 13 2019

Formula

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

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

Extensions

Signature changed by Georg Fischer, Apr 13 2019