A186446 Expansion of 1/(1 - 7*x + 2*x^2).

1, 7, 47, 315, 2111, 14147, 94807, 635355, 4257871, 28534387, 191224967, 1281505995, 8588092031, 57553632227, 385699241527, 2584787426235, 17322113500591, 116085219651667, 777952310560487, 5213495734620075, 34938565521219551

Offset: 0

Author

Bruno Berselli, Feb 21 2011

Keywords

• nonn
• easy

Comments

The first differences are in A122074.

a(n+1) equals the number of words of length n over {0,1,2,3,4,5,6} avoiding 01 and 02. - Milan Janjic, Dec 17 2015

Links

• Bruno Berselli, Table of n, a(n) for n = 0..800
• Tomislav Doslic, Planar polycyclic graphs and their Tutte polynomials, Journal of Mathematical Chemistry, Volume 51, Issue 6, 2013, pp. 1599-1607.
• Index entries for linear recurrences with constant coefficients, signature (7,-2).

Crossrefs

For similar closed formulas: A015446 [((1+sqrt(41))^(1+n)-(1-sqrt(41))^(1+n))/(2^(1+n)*sqrt(41))], A015525 [((3+sqrt(41))^n-(3-sqrt(41))^n)/(2^n*sqrt(41))], A015537 [((5+sqrt(41))^n-(5-sqrt(41))^n)/(2^n*sqrt(41))], A178869 [((9+sqrt(41))^n-(9-sqrt(41))^n)/(2^n*sqrt(41))].

Same recurrence as in A122074, A003771.

Programs

Magma

m:=21; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1-7*x+2*x^2)));

Magma

I:=[1,7]; [n le 2 select I[n] else 7*Self(n-1)-2*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Aug 19 2013

Mathematica

CoefficientList[Series[1 / (1 - 7 x + 2 x^2), {x, 0, 20}], x] (* Vincenzo Librandi, Aug 19 2013 *)
LinearRecurrence[{7,-2},{1,7},30] (* Harvey P. Dale, Aug 06 2017 *)

PARI

Vec(1/(1-7*x+2*x^2) + O(x^100)) \\ Altug Alkan, Dec 17 2015

Formula

G.f.: 1/(1-7*x+2*x^2).

a(n) = ((7+sqrt(41))^(1+n)-(7-sqrt(41))^(1+n))/(2^(1+n)*sqrt(41)).

a(n) = 7*a(n-1)-2*a(n-2), with a(0)=1, a(1)=7.