A154246 a(n) = ( (5 + sqrt(7))^n - (5 - sqrt(7))^n )/(2*sqrt(7)).

1, 10, 82, 640, 4924, 37720, 288568, 2206720, 16872976, 129008800, 986374432, 7541585920, 57661119424, 440862647680, 3370726327168, 25771735613440, 197044282245376, 1506551581411840, 11518718733701632, 88069258871603200

Offset

1,2

Comments

Second binomial transform of A086901 without initial term 1.

Lim_{n -> infinity} a(n)/a(n-1) = 5 + sqrt(7) = 7.6457513110....

Links

G. C. Greubel, Table of n, a(n) for n = 1..750

Index entries for linear recurrences with constant coefficients, signature (10,-18).

Formula

From Philippe Deléham, Jan 06 2009: (Start)

a(n) = 10*a(n-1) - 18*a(n-2) for n > 1, with a(0)=0, a(1)=1.

G.f.: x/(1 - 10*x + 18*x^2). (End)

E.g.f.: (1/sqrt(7))*exp(5*x)*sinh(sqrt(7)*x). - G. C. Greubel, Sep 07 2016

Mathematica

Table[Simplify[((5+Sqrt[7])^n -(5-Sqrt[7])^n)/(2*Sqrt[7])], {n, 1, 25}] (* or *) LinearRecurrence[{10, -18}, {1, 10}, 25] (* G. C. Greubel, Sep 07 2016 *)

Prog

(Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-7); S:=[((5+r)^n-(5-r)^n)/(2*r): n in [1..25]]; [Integers()!S[j]: j in [1..#S]]; // Klaus Brockhaus, Jan 07 2009
(Magma) I:=[1, 10]; [n le 2 select I[n] else 10*Self(n-1)-18*Self(n-2): n in [1..25]]; // Vincenzo Librandi, Sep 08 2016
(Sage) [lucas_number1(n, 10, 18) for n in range(1, 25)] # Zerinvary Lajos, Apr 26 2009
(PARI) my(x='x+O('x^25)); Vec(x/(1-10*x+18*x^2)) \\ G. C. Greubel, May 31 2019

Crossrefs

Cf. A010465 (decimal expansion of square root of 7), A086901.

Sequence in context: A076288 A362508 A232332 * A037549 A238276 A287825

Adjacent sequences: A154243 A154244 A154245 * A154247 A154248 A154249

Keyword

nonn,easy

Author

Al Hakanson (hawkuu(AT)gmail.com), Jan 05 2009

Extensions

Extended beyond a(7) by Klaus Brockhaus, Jan 07 2009

Edited by Klaus Brockhaus, Oct 06 2009

Status

approved