A146965 a(n) = 10*a(n-1) - 18*a(n-2) with a(0)=1, a(1)=5.

1, 5, 32, 230, 1724, 13100, 99968, 763880, 5839376, 44643920, 341330432, 2609713760, 19953189824, 152557050560, 1166413088768, 8918103977600, 68185604178176, 521330170184960, 3985960826642432, 30475665203095040

Offset: 0

Author

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

Keywords

• nonn

Comments

The Mathematica program implements the formula provided by Deleham and Brockhaus. - Harvey P. Dale, Feb 17 2011

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..145
• Index entries for linear recurrences with constant coefficients, signature (10,-18).

Programs

GAP

a:=[1,5];; for n in [3..30] do a[n]:=10*a[n-1]-18*a[n-2]; od; a; # G. C. Greubel, Jan 08 2020

Magma

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-7); S:=[ ((5+r7)^n+(5-r7)^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+18*x^2), x, n+1), x, n), n = 0..30); # G. C. Greubel, Jan 08 2020

Mathematica

Transpose[NestList[{#[[2]],10#[[2]]-18#[[1]]}&,{1,5},20]][[1]]  (* Harvey P. Dale, Feb 17 2011 *)
LinearRecurrence[{10,-18},{1,5},30] (* Harvey P. Dale, Aug 27 2013 *)

PARI

my(x='x+O('x^30)); Vec((1-5*x)/(1-10*x+18*x^2)) \\ G. C. Greubel, Jan 08 2020

Sage

def A146965_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-5*x)/(1-10*x+18*x^2) ).list()
A146965_list(30) # G. C. Greubel, Jan 08 2020

Formula

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

G.f.: (1-5*x)/(1-10*x+18*x^2). - Philippe Deléham and Klaus Brockhaus, Nov 05 2008

a(n) = (Sum_{k=0..n} A098158(n,k)*5^(2*k)*7^(n-k))/5^n. - Philippe Deléham, Nov 06 2008

E.g.f.: exp(5*x)*cosh(sqrt(7)*x). - G. C. Greubel, Jan 08 2020

Extensions

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

Name from Philippe Deléham and Klaus Brockhaus, Nov 05 2008