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A164545
a(n) = 4*a(n-1) + 4*a(n-2) for n > 1; a(0) = 1, a(1) = 8.
4
1, 8, 36, 176, 848, 4096, 19776, 95488, 461056, 2226176, 10748928, 51900416, 250597376, 1209991168, 5842354176, 28209381376, 136206942208, 657665294336, 3175488946176, 15332616962048, 74032423632896, 357460162379776
OFFSET
0,2
COMMENTS
Binomial transform of A164544. Second binomial transform of A164640. Inverse binomial transform of A038761.
LINKS
Harvey P. Dale, Table of n, a(n) for n = 0..1000 [extending from n(164) by Vincenzo Librandi]
Martin Burtscher, Igor Szczyrba and RafaƂ Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
FORMULA
a(n) = 4*a(n-1) + 4*a(n-2) for n > 1; a(0) = 1, a(1) = 8.
a(n) = ((2+3*sqrt(2))*(2+2*sqrt(2))^n + (2-3*sqrt(2))*(2-2*sqrt(2))^n)/4.
G.f.: (1 + 4*x)/(1 - 4*x - 4*x^2).
a(n) = (2*i)^n*( ChebyshevU(n, -i) - 2*i*ChebyshevU(n-1, -i) ). - G. C. Greubel, Jul 17 2021
MATHEMATICA
LinearRecurrence[{4, 4}, {1, 8}, 30] (* Harvey P. Dale, Dec 25 2011 *)
PROG
(Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((2+3*r)*(2+2*r)^n+(2-3*r)*(2-2*r)^n)/4: n in [0..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 19 2009
(Sage) [(2*i)^n*(chebyshev_U(n, -i) - 2*i*chebyshev_U(n-1, -i)) for n in (0..30)] # G. C. Greubel, Jul 17 2021
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Al Hakanson (hawkuu(AT)gmail.com), Aug 15 2009
EXTENSIONS
Edited and extended beyond a(5) by Klaus Brockhaus, Aug 19 2009
STATUS
approved