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A035344
Expansion of 1/((1 - x)*(1 - 4*x + 2 * x^2)).
7
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, 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
S. Bilotta, E. Pergola, R. Pinzani, and 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)).
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
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
KEYWORD
nonn,easy
EXTENSIONS
a(23) onwards from Andrew Howroyd, Jan 28 2024
STATUS
approved