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A152261
a(n) = ((9 + sqrt(5))^n + (9 - sqrt(5))^n)/2.
2
1, 9, 86, 864, 9016, 96624, 1054016, 11628864, 129214336, 1442064384, 16136869376, 180866755584, 2029199527936, 22779718078464, 255815761289216, 2873425129242624, 32279654468386816, 362653470608523264
OFFSET
0,2
COMMENTS
Binomial transform of A152109. - Philippe Deléham, Dec 03 2008
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
MATHEMATICA
LinearRecurrence[{18, -76}, {1, 9}, 41] (* G. C. Greubel, May 23 2023 *)
PROG
(Magma) Z<x>:= PolynomialRing(Integers()); N<r5>:=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
(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
CROSSREFS
Cf. A152109.
Sequence in context: A204465 A276242 A015581 * A028339 A180751 A266828
KEYWORD
nonn
AUTHOR
Al Hakanson (hawkuu(AT)gmail.com), Dec 01 2008
EXTENSIONS
Extended beyond a(6) by Klaus Brockhaus, Dec 03 2008
STATUS
approved