A146962 a(n) = 10*a(n-1) - 19*a(n-2) with a(0)=1, a(1)=5.

1, 5, 31, 215, 1561, 11525, 85591, 636935, 4743121, 35329445, 263175151, 1960492055, 14604592681, 108796577765, 810478516711, 6037650189575, 44977410078241, 335058747180485, 2496016680318271, 18594050606753495

Offset: 0

Author

Al Hakanson (hawkuu(AT)gmail.com), Nov 03 2008

Keywords

• nonn
• easy

Comments

Binomial transform of A143648.

Inverse binomial transform of A145301.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (10,-19).

Crossrefs

Cf. A098158, A143648, A145301.

Programs

GAP

a:=[1,5];; for n in [3..30] do a[n]:=10*a[n-1]-19*a[n-2]; od; a; # G. C. Greubel, Jan 08 2020

Magma

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-6); S:=[ ((5+r6)^n+(5-r6)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Nov 05 2008

Maple

seq(coeff(series((1-5*x)/(1-10*x+19*x^2), x, n+1), x, n), n = 0..30); # G. C. Greubel, Jan 08 2020

Mathematica

LinearRecurrence[{10,-19},{1,5},30] (* Harvey P. Dale, Apr 27 2014 *)
CoefficientList[Series[(1-5x)/(1-10x+19x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Apr 28 2014 *)

PARI

my(x='x+O('x^30)); Vec((1-5*x)/(1-10*x+19*x^2)) \\ G. C. Greubel, Jan 08 2020

Sage

def A146962_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-5*x)/(1-10*x+19*x^2) ).list()
A146962_list(30) # G. C. Greubel, Jan 08 2020

Formula

a(n) = ((5 + sqrt(6))^n + (5 - sqrt(6))^n)/2.

G.f.: (1-5*x)/(1-10*x+19*x^2). - Philippe Deléham and Klaus Brockhaus, Nov 05 2008

a(n) = (Sum_{k=0..n} A098158(n,k)*5^(2*k)*6^(n-k))/5^n. - Philippe Deléham, Nov 06 2008

E.g.f.: exp(5*x)*cosh(sqrt(6)*x). - G. C. Greubel, Jan 08 2020

Extensions

Extended beyond a(7) by Klaus Brockhaus, Nov 05 2008

Edited by Klaus Brockhaus, Jul 15 2009

Name from Philippe Deléham and Klaus Brockhaus, Nov 05 2008