A162436 - OEIS

%I #34 Dec 31 2022 13:18:59

%S 1,3,3,9,9,27,27,81,81,243,243,729,729,2187,2187,6561,6561,19683,

%T 19683,59049,59049,177147,177147,531441,531441,1594323,1594323,

%U 4782969,4782969,14348907,14348907,43046721,43046721,129140163,129140163,387420489,387420489,1162261467

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

%C Interleaving of A000244 and 3*A000244.

%C Unsigned version of A128019.

%C Partial sums are in A164123.

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

%C 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.

%H Vincenzo Librandi, <a href="/A162436/b162436.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,3).

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

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

%F a(n+3) = a(n+2)*a(n+1)/a(n). - _Reinhard Zumkeller_, Mar 04 2011

%F E.g.f.: cosh(sqrt(3)*x) - 1 + sinh(sqrt(3)*x)/sqrt(3). - _Stefano Spezia_, Dec 31 2022

%t CoefficientList[Series[(-3*x - 1)/(3*x^2 - 1), {x, 0, 200}], x] (* _Vladimir Joseph Stephan Orlovsky_, Jun 10 2011 *)

%t 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 *)

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

%o (PARI) a(n)=3^(n>>1) \\ _Charles R Greathouse IV_, Jul 15 2011

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

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

%K nonn,easy

%O 1,2

%A _Klaus Brockhaus_, Jul 03 2009, Jul 05 2009

%E G.f. corrected, formula simplified, comments added by _Klaus Brockhaus_, Sep 18 2009