A268413 a(n) = Sum_{k = 0..n} (-1)^k*14^k.

1, -13, 183, -2561, 35855, -501969, 7027567, -98385937, 1377403119, -19283643665, 269971011311, -3779594158353, 52914318216943, -740800455037201, 10371206370520815, -145196889187291409, 2032756448622079727, -28458590280709116177, 398420263929927626479

Offset

0,2

Comments

Alternating sum of powers of 14.

More generally, the ordinary generating function for the Sum_{k = 0..n} (-1)^k*m^k is 1/(1 + (m - 1)*x - m*x^2). Also, Sum_{k = 0..n} (-1)^k*m^k = ((-1)^n*m^(n + 1) + 1)/(m + 1).

Links

G. C. Greubel, Table of n, a(n) for n = 0..870

Index entries for linear recurrences with constant coefficients, signature (-13,14).

Formula

G.f.: 1/(1 + 13*x - 14*x^2).

a(n) = ((-1)^n*14^(n + 1) + 1)/15.

a(n) = 1 - 14*a(n - 1) for n>0 and a(0)=1.

a(n) = Sum_{k = 0..n} A033999(k)*A001023(k).

Lim_{n -> infinity} a(n)/a(n + 1) = - 1/14.

Mathematica

Table[((-1)^n 14^(n + 1) + 1)/15, {n, 0, 18}]
LinearRecurrence[{-13, 14}, {1, -13}, 19]
Table[Sum[(-1)^k*14^k, {k, 0, n}], {n, 0, 18}]

Prog

(PARI) x='x+O('x^30); Vec(1/(1 + 13*x - 14*x^2)) \\ G. C. Greubel, May 26 2018
(Magma) I:=[1, -19]; [n le 2 select I[n] else -13*Self(n-1) +14*Self(n-2): n in [1..30]]; // G. C. Greubel, May 26 2018

Crossrefs

Cf. A001023, A033999.

Cf. similar sequences of the type Sum_{k=0..n} (-1)^k*m^k: A059841 (m=1), A077925 (m=2), A014983 (m=3), A014985 (m=4), A014986 (m=5), A014987 (m=6), A014989 (m=7), A014990 (m=8), A014991 (m=9), A014992 (m=10), A014993 (m=11), A014994 (m=12), A015000 (m=13), this sequence (m=14), A239284 (m=15).

Sequence in context: A158548 A285399 A297581 * A274345 A227503 A091540

Adjacent sequences: A268410 A268411 A268412 * A268414 A268415 A268416

Keyword

sign,easy

Author

Ilya Gutkovskiy, Feb 04 2016

Status

approved