A105312 a(n) = Sum_{j=0..15} n^j.

1, 16, 65535, 21523360, 1431655765, 38146972656, 564221981491, 5538821761600, 40210710958665, 231627523606480, 1111111111111111, 4594972986357216, 16807659899548765, 55451384098598320, 167534872139182395, 469172025408063616, 1229782938247303441

Offset: 0

Author

Douglas Winston (douglas.winston(AT)srupc.com), Apr 30 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 (16,-120,560,-1820,4368,-8008,11440,-12870,11440,-8008,4368,-1820,560,-120,16,-1).

Crossrefs

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).

Programs

Magma

[(&+[n^j: j in [0..15]]): n in [0..20]]; // Vincenzo Librandi, May 01 2011 (modified by G. C. Greubel, Apr 14 2019)

Maple

a:= n-> add(n^k, k=0..15):
seq(a(n), n=0..20);  # Alois P. Heinz, Nov 04 2012

Mathematica

Prepend[Table[Total[n^Range[0,15]],{n,20}],1]  (* Harvey P. Dale, Jan 19 2011 *)

PARI

vector(20, n, n--; sum(j=0,15, n^j)) \\ G. C. Greubel, Apr 14 2019

Sage

[sum(n^j for j in (0..15)) for n in (0..20)] # G. C. Greubel, Apr 14 2019

Formula

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.

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

Extensions

More terms from Harvey P. Dale, Jan 19 2011

Name changed by G. C. Greubel, Apr 14 2019