A152261 a(n) = ((9 + sqrt(5))^n + (9 - sqrt(5))^n)/2.

1, 9, 86, 864, 9016, 96624, 1054016, 11628864, 129214336, 1442064384, 16136869376, 180866755584, 2029199527936, 22779718078464, 255815761289216, 2873425129242624, 32279654468386816, 362653470608523264

Offset: 0

Author

Al Hakanson (hawkuu(AT)gmail.com), Dec 01 2008

Keywords

• nonn

Comments

Binomial transform of A152109. - Philippe Deléham, Dec 03 2008

Links

• G. C. Greubel, Table of n, a(n) for n = 0..950
• Index entries for linear recurrences with constant coefficients, signature (18,-76).

Crossrefs

Cf. A152109.

Programs

Magma

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-5); S:=[ ((9+r5)^n+(9-r5)^n)/2: n in [0..17] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Dec 03 2008

Magma

[n le 2 select 9^(n-1) else 18*Self(n-1) -76*Self(n-2): n in [1..30]]; // G. C. Greubel, May 23 2023

Mathematica

LinearRecurrence[{18,-76}, {1,9}, 41] (* G. C. Greubel, May 23 2023 *)

SageMath

@CachedFunction
def a(n): # a = A152261
    if (n<2): return 9^n
    else: return 18*a(n-1) -76*a(n-2)
[a(n) for n in range(41)] # G. C. Greubel, May 23 2023

Formula

From Philippe Deléham, Dec 03 2008: (Start)

a(n) = 18*a(n-1) - 76*a(n-2), n > 1; a(0)=1, a(1)=9.

G.f.: (1-9*x)/(1-18*x+76*x^2).

a(n) = Sum_{k=0..n} A098158(n,k)*9^(2k-n)*5^(n-k). (End)

a(n) = m^n*(ChebyshevU(n, 9/m) - (9/m)*ChebyshevU(n-1, 9/m)), where m = 2*sqrt(19). - G. C. Greubel, May 23 2023

Extensions

Extended beyond a(6) by Klaus Brockhaus, Dec 03 2008