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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A201865 Expansion of 1/((1-3*x)*(1+7*x)).

Original entry on oeis.org

1, -4, 37, -232, 1705, -11692, 82573, -575824, 4037329, -28241620, 197750389, -1384075576, 9689060473, -67821828988, 474757585885, -3323288752288, 23263064312737, -162841321048996, 1139889634763461, -7979226281082760, 55854587454363721, -390982101720192844
Offset: 0

Views

Author

Bruno Berselli, Dec 07 2011

Keywords

Links

Crossrefs

Cf. A014986, A083296, A083300.

Programs

  • Magma
    m:=22; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-3*x)*(1+7*x))));
  • Mathematica
    CoefficientList[Series[1/((1-3*x)*(1+7*x)), {x, 0, 22}], x]
  • Maxima
    makelist(coeff(taylor(1/((1-3*x)*(1+7*x)), x, 0, n), x, n), n, 0, 21);
  • PARI
    Vec(1/((1-3*x)*(1+7*x))+O(x^22))

Formula

G.f.: 1/((1-3*x)*(1+7*x)).
a(n) = (3^(n+1)+7*(-7)^n)/10.
a(n) = -4*a(n-1)+21*a(n-2) with n>0, a(-1)=0, a(0)=1.
a(n)-a(n-1) = A083300(n)*(-1)^n.
a(n)+5*a(n-1) = A083296(n) with a(-1)=0.

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

Original entry on oeis.org

1, -13, 183, -2561, 35855, -501969, 7027567, -98385937, 1377403119, -19283643665, 269971011311, -3779594158353, 52914318216943, -740800455037201, 10371206370520815, -145196889187291409, 2032756448622079727, -28458590280709116177, 398420263929927626479
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 04 2016

Keywords

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

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).

Programs

  • 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
  • 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}]
  • PARI
    x='x+O('x^30); Vec(1/(1 + 13*x - 14*x^2)) \\ G. C. Greubel, May 26 2018

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.
Previous Showing 11-12 of 12 results.

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