A105312 - OEIS

%I #38 Sep 08 2022 08:45:17

%S 1,16,65535,21523360,1431655765,38146972656,564221981491,

%T 5538821761600,40210710958665,231627523606480,1111111111111111,

%U 4594972986357216,16807659899548765,55451384098598320,167534872139182395,469172025408063616,1229782938247303441

%N a(n) = Sum_{j=0..15} n^j.

%H Vincenzo Librandi, <a href="/A105312/b105312.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_16">Index entries for linear recurrences with constant coefficients</a>, signature (16,-120,560,-1820,4368,-8008,11440,-12870,11440,-8008,4368,-1820,560,-120,16,-1).

%F a(n) = n^15 + n^14 + n^13 + n^12 + n^11 + n^10 + n^9 + n^8 + n^7 + n^6 + n^5 + n^4 + n^3 + n^2 + n^1 + 1.

%F G.f.: (21845*x^14 + 10412160*x^13 + 689427979*x^12 + 12966588160*x^11 + 93207091581*x^10 + 296077418240*x^9 + 446019954555*x^8 + 326065923072*x^7 + 113735241015*x^6 + 17786608768*x^5 + 1095139065*x^4 + 20476160*x^3 + 65399*x^2 +1 )/(x-1)^16. - _Colin Barker_, Nov 04 2012

%p a:= n-> add(n^k, k=0..15):

%p seq(a(n), n=0..20);  # _Alois P. Heinz_, Nov 04 2012

%t Prepend[Table[Total[n^Range[0,15]],{n,20}],1]  (* _Harvey P. Dale_, Jan 19 2011 *)

%o (Magma) [(&+[n^j: j in [0..15]]): n in [0..20]]; // _Vincenzo Librandi_, May 01 2011 (modified by _G. C. Greubel_, Apr 14 2019)

%o (PARI) vector(20, n, n--; sum(j=0,15, n^j)) \\ _G. C. Greubel_, Apr 14 2019

%o (Sage) [sum(n^j for j in (0..15)) for n in (0..20)] # _G. C. Greubel_, Apr 14 2019

%Y Cf. similar sequences of the type a(n) = Sum_{j=0..m} n^j are: 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), A105067 (m=11), A060887 (m=12), A104376 (m=13), A104682 (m=14), this sequence (m=15), A269442 (m=16), A269446 (m=18).

%K nonn,easy

%O 0,2

%A Douglas Winston (douglas.winston(AT)srupc.com), Apr 30 2005

%E More terms from _Harvey P. Dale_, Jan 19 2011

%E Name changed by _G. C. Greubel_, Apr 14 2019