A081182 5th binomial transform of (0,1,0,2,0,4,0,8,0,16,...).

0, 1, 10, 77, 540, 3629, 23870, 155233, 1003320, 6462841, 41552050, 266875157, 1713054420, 10992415589, 70523904230, 452413483753, 2902085040240, 18615340276081, 119405446835290, 765901642003037, 4912691142818700

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..300

S. Falcon, Iterated Binomial Transforms of the k-Fibonacci Sequence, British Journal of Mathematics & Computer Science, 4 (22): 2014.

Index entries for linear recurrences with constant coefficients, signature (10,-23).

Formula

a(n) = 10a(n-1) - 23a(n-2), a(0)=0, a(1)=1.

G.f.: x/(1 - 10x + 23x^2).

a(n) = ((5 + sqrt(2))^n - (5 - sqrt(2))^n)/(2*sqrt(2)).

a(n) = Sum_{k=0..n} C(n,2k+1) 2^k*5^(n-2k-1).

E.g.f.: exp(5*x)*sinh(sqrt(2)*x)/sqrt(2). - Ilya Gutkovskiy, Aug 12 2017

Mathematica

Join[{a=0, b=1}, Table[c=10*b-23*a; a=b; b=c, {n, 60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 27 2011 *)
CoefficientList[Series[x / (1 - 10 x + 23 x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 06 2013 *)
LinearRecurrence[{10, -23}, {0, 1}, 30] (* Harvey P. Dale, Jun 06 2021 *)

Prog

(Sage) [lucas_number1(n, 10, 23) for n in range(0, 21)] # Zerinvary Lajos, Apr 26 2009
(Magma) I:=[0, 1]; [n le 2 select I[n] else 10*Self(n-1)-23*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Aug 06 2013

Crossrefs

Binomial transform of A081181.

Cf. A081183.

Sequence in context: A159579 A244720 A081678 * A127536 A016201 A080618

Adjacent sequences: A081179 A081180 A081181 * A081183 A081184 A081185

Keyword

easy,nonn

Author

Paul Barry, Mar 12 2003

Status

approved