OFFSET
0,2
COMMENTS
Binomial transform of A086405.
Inverse binomial transform of A162558.
4th binomial transform of A108411.
2nd binomial transform of A079935. [R. J. Mathar, Jul 17 2009]
From J. Conrad, Aug 29 2016: (Start)
Partial sum of A136777.
Backward difference of Sum_{k=0..n} A027907(n+1,2k+2)*3^k.
(End)
String length in substitution system {0 -> 1001001, 1 -> 11011} at step n from initial string "1" (1 -> 11011 -> 110111101110010011101111011 -> ...). - Ilya Gutkovskiy, Aug 30 2016
LINKS
Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-13).
FORMULA
a(n) = 8*a(n-1)-13*a(n-2) for n > 1; a(0) = 1, a(1) = 5.
G.f.: (1-3*x)/(1-8*x+13*x^2).
MAPLE
seq(simplify(((3+sqrt(3))*(4+sqrt(3))^n+(3-sqrt(3))*(4-sqrt(3))^n)*1/6), n = 0..20); # Emeric Deutsch, Jul 14 2009
MATHEMATICA
Table[FullSimplify[((3 + #) (4 + #)^n + (3 - #) (4 - #)^n)/6 &@ Sqrt@ 3], {n, 0, 23}] (* Michael De Vlieger, Aug 30 2016 *)
LinearRecurrence[{8, -13}, {1, 5}, 30] (* Harvey P. Dale, Oct 23 2020 *)
PROG
(Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-3); S:=[ ((3+r)*(4+r)^n+(3-r)*(4-r)^n)/6: n in [0..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 13 2009
(Magma) I:=[1, 5]; [n le 2 select I[n] else 8*Self(n-1)-13*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Aug 30 2016
CROSSREFS
KEYWORD
nonn
AUTHOR
Al Hakanson (hawkuu(AT)gmail.com), Jul 06 2009
EXTENSIONS
Edited, corrected and extended beyond a(5) by Klaus Brockhaus, Emeric Deutsch and R. J. Mathar, Jul 07 2009
More terms from Vincenzo Librandi, Aug 30 2016
STATUS
approved