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

%I

%S 1,-13,183,-2561,35855,-501969,7027567,-98385937,1377403119,

%T -19283643665,269971011311,-3779594158353,52914318216943,

%U -740800455037201,10371206370520815,-145196889187291409,2032756448622079727,-28458590280709116177,398420263929927626479

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

%C Alternating sum of powers of 14.

%C 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).

%H G. C. Greubel, <a href="/A268413/b268413.txt">Table of n, a(n) for n = 0..870</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (-13,14).

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

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

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

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

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

%t Table[((-1)^n 14^(n + 1) + 1)/15, {n, 0, 18}]

%t LinearRecurrence[{-13, 14}, {1, -13}, 19]

%t Table[Sum[(-1)^k*14^k, {k, 0, n}], {n, 0, 18}]

%o (PARI) x='x+O('x^30); Vec(1/(1 + 13*x - 14*x^2)) \\ _G. C. Greubel_, May 26 2018

%o (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

%Y Cf. A001023, A033999.

%Y 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).

%K sign,easy

%O 0,2

%A _Ilya Gutkovskiy_, Feb 04 2016

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Last modified January 20 14:40 EST 2019. Contains 319333 sequences. (Running on oeis4.)