A153593 a(n) = ((9 + sqrt(2))^n - (9 - sqrt(2))^n)/(2*sqrt(2)).

1, 18, 245, 2988, 34429, 383670, 4186169, 45041112, 480032665, 5082340122, 53559541661, 562566880260, 5895000053461, 61667217421758, 644304909368225, 6725778192309168, 70163919621475249, 731614075994130210

Offset: 1

Author

Al Hakanson (hawkuu(AT)gmail.com), Dec 29 2008

Keywords

• nonn

Comments

Preceded by zero, this is the eighth binomial transform of the Pell sequence A000129. - Sergio Falcon, Sep 21 2009; edited by Klaus Brockhaus, Oct 11 2009

Eighth binomial transform of A048697.

First differences are in A164600.

lim_{n -> infinity} a(n)/a(n-1) = 9 + sqrt(2) = 10.4142135623....

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..100
• 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 (18,-79).

Crossrefs

Cf. A000129 (Pell numbers), A007070, A081185, A081184, A081183, A081182, A081180, A081179. - Sergio Falcon, Sep 21 2009

Cf. A002193 (decimal expansion of sqrt(2)), A048697, A164600.

Programs

Magma

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((9+r)^n-(9-r)^n)/(2*r): n in [1..18] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Dec 31 2008

Mathematica

Join[{a=1,b=18},Table[c=18*b-79*a;a=b;b=c,{n,40}]] (* Vladimir Joseph Stephan Orlovsky, Feb 09 2011 *)
LinearRecurrence[{18,-79},{1,18},25] (* G. C. Greubel, Aug 22 2016 *)

Formula

a(n) = 18*a(n-1) - 79*a(n-2) for n>1; a(0)=0, a(1)=1. - Philippe Deléham, Jan 01 2009

G.f.: x/(1 - 18*x + 79*x^2). - Klaus Brockhaus, Dec 31 2008, corrected Oct 11 2009

a(n) = Sum[Binomial[n - 1 - i, i] (-1)^i * 18^(n - 1 - 2 i) * 79^i, {i, 0, Floor[(n - 1)/2]}]. - Sergio Falcon, Sep 21 2009

E.g.f.: exp(9*x)*sinh(sqrt(2)*x)/sqrt(2). - Ilya Gutkovskiy, Aug 12 2017

Extensions

Extended beyond a(7) by Klaus Brockhaus, Dec 31 2008

Edited by Klaus Brockhaus, Oct 11 2009