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A268413 a(n) = Sum_{k = 0..n} (-1)^k*14^k. 1
1, -13, 183, -2561, 35855, -501969, 7027567, -98385937, 1377403119, -19283643665, 269971011311, -3779594158353, 52914318216943, -740800455037201, 10371206370520815, -145196889187291409, 2032756448622079727, -28458590280709116177, 398420263929927626479 (list; graph; refs; listen; history; text; internal format)
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
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. 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
KEYWORD
sign,easy
AUTHOR
Ilya Gutkovskiy, Feb 04 2016
STATUS
approved

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Last modified February 21 19:15 EST 2024. Contains 370237 sequences. (Running on oeis4.)