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A164540
a(n) = 4*a(n-1) + 4*a(n-2) for n > 1; a(0) = 1, a(1) = 14.
3
1, 14, 60, 296, 1424, 6880, 33216, 160384, 774400, 3739136, 18054144, 87173120, 420909056, 2032328704, 9812951040, 47381118976, 228776280064, 1104629596160, 5333623504896, 25753012404224, 124346543636480, 600398224162816
OFFSET
0,2
COMMENTS
Binomial transform of A164539. Second binomial transform of A164675. Inverse binomial transform of A164541.
LINKS
Martin Burtscher, Igor Szczyrba, 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) = 14.
G.f.: (1+10*x)/(1-4*x-4*x^2).
a(n) = ((1+3*sqrt(2))*(2+2*sqrt(2))^n + (1-3*sqrt(2))*(2-2*sqrt(2))^n)/2.
MATHEMATICA
LinearRecurrence[{4, 4}, {1, 14}, 30] (* Harvey P. Dale, Jul 18 2024 *)
PROG
(Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((1+3*r)*(2+2*r)^n+(1-3*r)*(2-2*r)^n)/2: n in [0..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 20 2009
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 20 2009
STATUS
approved