A164545 a(n) = 4*a(n-1) + 4*a(n-2) for n > 1; a(0) = 1, a(1) = 8.

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 Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.

Index entries for linear recurrences with constant coefficients, signature (4,4).

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

Cf. A038761, A164544, A164640.

Sequence in context: A055064 A014347 A200053 * A199310 A316107 A054627

Adjacent sequences: A164542 A164543 A164544 * A164546 A164547 A164548

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