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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

References

• S. Bilotta, E. Pergola, R. Pinzani, and 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
• Stefano Bilotta, Elisa Pergola, Renzo Pinzani, and Simone Rinaldi, Recurrence relations versus succession rules, arXiv preprint arXiv:1301.2967 [cs.DM], 2013.
• Sela Fried, Even-up words and their variants, arXiv:2505.14196 [math.CO], 2025. See p. 8.
• 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).

Crossrefs

Partial sums of A007070.

Cf. A000129, A007052.

Programs

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 *)

PARI

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

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

E.g.f.: exp(2*x)*(2*cosh(sqrt(2)*x) + sqrt(2)*sinh(sqrt(2)*x)) - cosh(x) - sinh(x). - Stefano Spezia, May 20 2024

Extensions

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