A162436 a(n) = 3*a(n-2) for n > 2; a(1) = 1, a(2) = 3.

1, 3, 3, 9, 9, 27, 27, 81, 81, 243, 243, 729, 729, 2187, 2187, 6561, 6561, 19683, 19683, 59049, 59049, 177147, 177147, 531441, 531441, 1594323, 1594323, 4782969, 4782969, 14348907, 14348907, 43046721, 43046721, 129140163, 129140163, 387420489, 387420489, 1162261467

Offset: 1

Author

Klaus Brockhaus, Jul 03 2009, Jul 05 2009

Keywords

• nonn
• easy

Comments

Interleaving of A000244 and 3*A000244.

Unsigned version of A128019.

Partial sums are in A164123.

Apparently a(n) = A056449(n-1) for n > 1. a(n) = A108411(n) for n >= 1.

Binomial transform is A026150 without initial 1, second binomial transform is A001834, third binomial transform is A030192, fourth binomial transform is A161728, fifth binomial transform is A162272.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (0,3).

Crossrefs

Cf. A000244 (powers of 3), A128019 (expansion of (1-3x)/(1+3x^2)), A164123, A026150, A001834, A030192, A161728, A162272.

Essentially the same as A056449 (3^floor((n+1)/2)) and A108411 (powers of 3 repeated).

Programs

Magma

[ n le 2 select 2*n-1 else 3*Self(n-2): n in [1..35] ];

Mathematica

CoefficientList[Series[(-3*x - 1)/(3*x^2 - 1), {x, 0, 200}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 10 2011 *)
Transpose[NestList[{Last[#],3*First[#]}&,{1,3},40]][[1]] (* or *) With[{c= 3^Range[20]},Join[{1},Riffle[c,c]]](* Harvey P. Dale, Feb 17 2012 *)

PARI

a(n)=3^(n>>1) \\ Charles R Greathouse IV, Jul 15 2011

Formula

a(n) = 3^((1/4)*(2*n - 1 + (-1)^n)).

G.f.: x*(1 + 3*x)/(1 - 3*x^2).

a(n+3) = a(n+2)*a(n+1)/a(n). - Reinhard Zumkeller, Mar 04 2011

E.g.f.: cosh(sqrt(3)*x) - 1 + sinh(sqrt(3)*x)/sqrt(3). - Stefano Spezia, Dec 31 2022

Extensions

G.f. corrected, formula simplified, comments added by Klaus Brockhaus, Sep 18 2009