%I #71 Feb 13 2023 02:51:55
%S 1,3,6,18,36,108,216,648,1296,3888,7776,23328,46656,139968,279936,
%T 839808,1679616,5038848,10077696,30233088,60466176,181398528,
%U 362797056,1088391168,2176782336,6530347008,13060694016,39182082048,78364164096,235092492288,470184984576
%N Ratios of successive terms are 3, 2, 3, 2, 3, 2, 3, 2, ...
%C Preface the series with a 1: (1, 1, 3, 6, 18, 36, ...); then the next term in the series = (1, 1, 3, 6, ...) dot (1, 2, 1, 2, ...). Example: 36 = (1, 1, 3, 6, 18) dot (1, 2, 1, 2, 1) = (1 + 2 + 3 + 12 + 18). - _Gary W. Adamson_, Apr 18 2009
%C Partial products of A176059. - _Reinhard Zumkeller_, Apr 04 2012
%H Vincenzo Librandi, <a href="/A026532/b026532.txt">Table of n, a(n) for n = 1..700</a>
%H José L. Ramírez, <a href="http://ami.ektf.hu/uploads/papers/finalpdf/AMI_42_from83to92.pdf">Bi-periodic incomplete Fibonacci sequences</a>, Annales Mathematicae et Informaticae 42 (2013), 83-92. See the 1st column of Table 1 on p. 85.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,6).
%F a(n) = T(n, 0) + T(n, 1) + ... + T(n, 2n-2), T given by A026519.
%F From _Benoit Cloitre_, Nov 14 2003: (Start)
%F a(n) = (1/2)*(5+(-1)^n)*a(n-1) for n>1, a(1) = 1.
%F a(n) = (1/4)*(3-(-1)^n)*6^floor(n/2). (End)
%F From _Ralf Stephan_, Feb 03 2004: (Start)
%F G.f.: x*(1+3*x)/(1-6*x^2).
%F a(n+2) = 6*a(n). (End)
%F a(n+3) = a(n+2)*a(n+1)/a(n). - _Reinhard Zumkeller_, Mar 04 2011
%F a(n) = (1/2)*6^((n-2)/2)*(3*(1+(-1)^n) + sqrt(6)*(1-(-1)^n)). - _G. C. Greubel_, Dec 21 2021
%F Sum_{n>=1} 1/a(n) = 8/5. - _Amiram Eldar_, Feb 13 2023
%t FoldList[(2 + Boole[EvenQ@ #2]) #1 &, Range@ 28] (* or *)
%t CoefficientList[Series[x*(1+3x)/(1-6x^2), {x,0,31}], x] (* _Michael De Vlieger_, Aug 02 2017 *)
%t LinearRecurrence[{0,6},{1,3},30] (* _Harvey P. Dale_, Jul 11 2018 *)
%o (Magma) [(1/4)*(3-(-1)^n)*6^Floor(n/2) : n in [1..30]]; // _Vincenzo Librandi_, Jun 08 2011
%o (Haskell)
%o a026532 n = a026532_list !! (n-1)
%o a026532_list = scanl (*) 1 $ a176059_list
%o -- _Reinhard Zumkeller_, Apr 04 2012
%o (PARI) a(n)=if(n%2,3,1)*6^(n\2) \\ _Charles R Greathouse IV_, Jul 02 2013
%o (Python)
%o def a(n): return (3 if n%2 else 1)*6**(n//2)
%o print([a(n) for n in range(31)]) # _Indranil Ghosh_, Aug 02 2017
%o (Sage) [(1/2)*6^((n-2)/2)*(3*(1+(-1)^n) + sqrt(6)*(1-(-1)^n)) for n in (1..30)] # _G. C. Greubel_, Dec 21 2021
%Y Cf. A026519, A026520, A026521, A026522, A026523, A026524, A026525, A026526, A026527, A026528, A026529, A026530, A026531, A026533, A026534, A027262, A027263, A027264, A027265, A027266.
%Y Cf. A026534, A026549, A176059, A208131.
%Y Cf. A038730, A038792, and A134511 for incomplete Fibonacci sequences, and A324242 for incomplete Lucas sequences.
%K nonn,easy
%O 1,2
%A _Clark Kimberling_
%E New definition from _Ralf Stephan_, Dec 01 2004
%E Offset changed from 0 to 1 by _Vincenzo Librandi_, Jun 08 2011