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

0,2

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

Cf. A108411 (powers of 3 repeated), A086405, A162558.

Cf. A162558. [R. J. Mathar, Jul 17 2009]

Sequence in context: A026292 A100193 A158869 * A134425 A332598 A305573

Adjacent sequences: A162554 A162555 A162556 * A162558 A162559 A162560

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