A090139 a(n) = 10*a(n-1) - 20*a(n-2), a(0)=1,a(1)=5.

1, 5, 30, 200, 1400, 10000, 72000, 520000, 3760000, 27200000, 196800000, 1424000000, 10304000000, 74560000000, 539520000000, 3904000000000, 28249600000000, 204416000000000, 1479168000000000, 10703360000000000

Offset: 0

Author

Paul Barry, Nov 22 2003

Keywords

• easy
• nonn

Comments

Fifth binomial transform of (1, 0, 5, 0, 25, 0, ...).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (10,-20).

Programs

GAP

a:=[1,5];; for n in [3..30] do a[n]:=10*a[n-1]-20*a[n-2]; od; a; # G. C. Greubel, Aug 02 2019

Magma

I:=[1,5]; [n le 2 select I[n] else 10*Self(n-1) -20*Self(n-2): n in [1..30]]; // G. C. Greubel, Aug 02 2019

Mathematica

LinearRecurrence[{10, -20}, {1,5}, 30] (* G. C. Greubel, Aug 02 2019 *)

PARI

my(x='x+O('x^30)); Vec((1-5*x)/(1-10*x+20*x^2)) \\ G. C. Greubel, Aug 02 2019

Sage

((1-5*x)/(1-10*x+20*x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Aug 02 2019

Formula

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

a(n) = Sum_{k=0..floor(n/2)} C(n, 2k) * 5^(n-k).

a(n) = Sum_{k=0..n} C(n, k) * 5^(n-k/2) * (1+(-1)^k)/2.

a(n) = Sum_{k=0..n} 5^k*A098158(n,k). - Philippe Deléham, Dec 04 2006

G.f.: (1-5*x)/(1-10*x+20*x^2). - G. C. Greubel, Aug 02 2019