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

%I A268413 #14 Sep 08 2022 08:46:15
%S A268413 1,-13,183,-2561,35855,-501969,7027567,-98385937,1377403119,
%T A268413 -19283643665,269971011311,-3779594158353,52914318216943,
%U A268413 -740800455037201,10371206370520815,-145196889187291409,2032756448622079727,-28458590280709116177,398420263929927626479
%N A268413 a(n) = Sum_{k = 0..n} (-1)^k*14^k.
%C A268413 Alternating sum of powers of 14.
%C A268413 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 A268413 G. C. Greubel, <a href="/A268413/b268413.txt">Table of n, a(n) for n = 0..870</a>
%H A268413 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (-13,14).
%F A268413 G.f.: 1/(1 + 13*x - 14*x^2).
%F A268413 a(n) = ((-1)^n*14^(n + 1) + 1)/15.
%F A268413 a(n) = 1 - 14*a(n - 1) for n>0 and a(0)=1.
%F A268413 a(n) = Sum_{k = 0..n} A033999(k)*A001023(k).
%F A268413 Lim_{n -> infinity} a(n)/a(n + 1) = - 1/14.
%t A268413 Table[((-1)^n 14^(n + 1) + 1)/15, {n, 0, 18}]
%t A268413 LinearRecurrence[{-13, 14}, {1, -13}, 19]
%t A268413 Table[Sum[(-1)^k*14^k, {k, 0, n}], {n, 0, 18}]
%o A268413 (PARI) x='x+O('x^30); Vec(1/(1 + 13*x - 14*x^2)) \\ _G. C. Greubel_, May 26 2018
%o A268413 (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 A268413 Cf. A001023, A033999.
%Y A268413 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 A268413 sign,easy
%O A268413 0,2
%A A268413 _Ilya Gutkovskiy_, Feb 04 2016