A162557 a(n) = ((3+sqrt(3))*(4+sqrt(3))^n+(3-sqrt(3))*(4-sqrt(3))^n)/6.
1, 5, 27, 151, 857, 4893, 28003, 160415, 919281, 5268853, 30200171, 173106279, 992248009, 5687602445, 32601595443, 186873931759, 1071170713313, 6140004593637, 35194817476027, 201738480090935, 1156375213539129, 6628401467130877, 37994333961038339, 217785452615605311
Offset: 0
Keywords
Links
- Harvey P. Dale, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (8,-13).
Crossrefs
Cf. A162558. [R. J. Mathar, Jul 17 2009]
Programs
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Magma
Z
:=PolynomialRing(Integers()); N :=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
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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
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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 *)
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).
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
Comments