A155001 a(n) = 9*a(n-1) + 72*a(n-2), n > 2; a(0)=1, a(1)=1, a(2)=17.

1, 1, 17, 225, 3249, 45441, 642897, 9057825, 127809009, 1802444481, 25424248977, 358594243425, 5057894117169, 71339832581121, 1006226869666257, 14192509772837025, 200180922571503729, 2823489006787799361

Offset: 0

Author

Philippe Deléham, Jan 18 2009

Keywords

• nonn

Comments

The sequences A155001, A155000, A154999, A154997 and A154996 have a common form: a(0)=a(1)=1, a(2)= 2*b+1, a(n) = (b+1)*(a(n-1) + b*a(n-2)), with b some constant. The generating function of these is (1 - b*x - b^2*x^2)/(1 - (b+1)*x - b*(1+b)*x^2). - R. J. Mathar, Jan 20 2009

Links

• G. C. Greubel, Table of n, a(n) for n = 0..850
• Index entries for linear recurrences with constant coefficients, signature (9,72).

Crossrefs

Cf. A154929, A154996, A154997, A154999, A155000.

Programs

Magma

I:=[1,17]; [1] cat [n le 2 select I[n] else 9*(Self(n-1) +8*Self(n-2)): n in [1..30]]; // G. C. Greubel, Apr 20 2021

Maple

a[0] := 1: a[1] := 1: a[2] := 17: for n from 3 to 25 do a[n] := 9*a[n-1]+72*a[n-2] end do: seq(a[n], n = 0 .. 17); # Emeric Deutsch, Jan 21 2009

Mathematica

LinearRecurrence[{9,72},{1,1,17},20] (* Harvey P. Dale, Apr 26 2016 *)

Sage

def A155001_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-8*x-64*x^2)/(1-9*x-72*x^2) ).list()
A155001_list(30) # G. C. Greubel, Apr 20 2021

Formula

a(n+1) = Sum_{k=0..n} A154929(n,k)*8^(n-k).

G.f.: (1 - 8*x - 64*x^2)/(1 - 9*x - 72*x^2). - G. C. Greubel, Apr 20 2021

Extensions

Corrected by Philippe Deléham, Jan 21 2009

Corrected and extended by Emeric Deutsch and R. J. Mathar, Jan 21 2009