A035344 Expansion of 1/((1 - x)*(1 - 4*x + 2 * x^2)).

1, 5, 19, 67, 231, 791, 2703, 9231, 31519, 107615, 367423, 1254463, 4283007, 14623103, 49926399, 170459391, 581984767, 1987020287, 6784111615, 23162405887, 79081400319, 270000789503, 921840357375, 3147359850495, 10745758687231, 36688315047935, 125261742817279

Offset

0,2

References

S. Bilotta, E. Pergola, R. Pinzani, S. Rinaldi, Recurrence Relations, Succession Rules, and the Positivity Problem, in Language and Automata Theory and Applications, 9th International Conference, LATA 2015, Nice, France, March 2-6, 2015, Proceedings, Pages 499-510, Lecture Notes Comp. Sci. Vol. 8977.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

S. Bilotta, E. Pergola, R. Pinzani, S. Rinaldi, Recurrence relations versus succession rules, arXiv preprint arXiv:1301.2967 [cs.DM], 2013.

László Németh, Hyperbolic Pascal pyramid, arXiv:1511.02067 [math.CO], 2015 (2nd line of Table 2 is 6*a(n-3)).

László Németh, Pascal pyramid in the space H^2 x R, arXiv:1701.06022 [math.CO], 2017 (2nd line of Table is 2*a(n-3)).

Index entries for linear recurrences with constant coefficients, signature (5,-6,2).

Formula

a(n) = 2*A007052(n)-1. The sequence 0, 0, 1, 5, 19, ... is the binomial transform of the Pell numbers A000129, preceded by an additional 0. a(n) = (1 + 1/sqrt(2))(2 + sqrt(2))^n + (1 - 1/sqrt(2))(2 - sqrt(2))^n - 1. - Paul Barry, Jul 16 2003

a(-1)=0, a(0)=1, a(n) = 4*a(n-1) - 2*a(n-2) + 1. - Miklos Kristof, Mar 09 2005

Maple

a[ -1]:=0:a[0]:=1:for n from 1 to 50 do a[n]:=4*a[n-1]-2*a[n-2]+1 od: seq(a[n], n=0..50); # after Miklos Kristof

Mathematica

Join[{a=1, b=5}, Table[c=4*b-2*a+1; a=b; b=c, {n, 60}]] (* Vladimir Joseph Stephan Orlovsky, Feb 06 2011 *)
CoefficientList[Series[1/((1-x)(1-4x+2x^2)), {x, 0, 30}], x] (* or *) LinearRecurrence[ {5, -6, 2}, {1, 5, 19}, 30] (* Harvey P. Dale, Mar 28 2016 *)

Prog

(PARI) Vec(1/((1-x)*(1-4*x+2*x^2))+O(x^99)) \\ Charles R Greathouse IV, Sep 24 2012

Crossrefs

Partial sums of A007070.

Sequence in context: A121525 A163872 A372884 * A114277 A104496 A001435

Adjacent sequences: A035341 A035342 A035343 * A035345 A035346 A035347

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

a(23) onwards from Andrew Howroyd, Jan 28 2024

Status

approved