A026532 Ratios of successive terms are 3, 2, 3, 2, 3, 2, 3, 2, ...

1, 3, 6, 18, 36, 108, 216, 648, 1296, 3888, 7776, 23328, 46656, 139968, 279936, 839808, 1679616, 5038848, 10077696, 30233088, 60466176, 181398528, 362797056, 1088391168, 2176782336, 6530347008, 13060694016, 39182082048, 78364164096, 235092492288, 470184984576

Offset

1,2

Comments

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

Partial products of A176059. - Reinhard Zumkeller, Apr 04 2012

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..700

José L. Ramírez, Bi-periodic incomplete Fibonacci sequences, Annales Mathematicae et Informaticae 42 (2013), 83-92. See the 1st column of Table 1 on p. 85.

Index entries for linear recurrences with constant coefficients, signature (0,6).

Formula

a(n) = T(n, 0) + T(n, 1) + ... + T(n, 2n-2), T given by A026519.

From Benoit Cloitre, Nov 14 2003: (Start)

a(n) = (1/2)*(5+(-1)^n)*a(n-1) for n>1, a(1) = 1.

a(n) = (1/4)*(3-(-1)^n)*6^floor(n/2). (End)

From Ralf Stephan, Feb 03 2004: (Start)

G.f.: x*(1+3*x)/(1-6*x^2).

a(n+2) = 6*a(n). (End)

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

a(n) = (1/2)*6^((n-2)/2)*(3*(1+(-1)^n) + sqrt(6)*(1-(-1)^n)). - G. C. Greubel, Dec 21 2021

Sum_{n>=1} 1/a(n) = 8/5. - Amiram Eldar, Feb 13 2023

Mathematica

FoldList[(2 + Boole[EvenQ@ #2]) #1 &, Range@ 28] (* or *)
CoefficientList[Series[x*(1+3x)/(1-6x^2), {x, 0, 31}], x] (* Michael De Vlieger, Aug 02 2017 *)
LinearRecurrence[{0, 6}, {1, 3}, 30] (* Harvey P. Dale, Jul 11 2018 *)

Prog

(Magma) [(1/4)*(3-(-1)^n)*6^Floor(n/2) : n in [1..30]]; // Vincenzo Librandi, Jun 08 2011
(Haskell)
a026532 n = a026532_list !! (n-1)
a026532_list = scanl (*) 1 $ a176059_list
-- Reinhard Zumkeller, Apr 04 2012
(PARI) a(n)=if(n%2, 3, 1)*6^(n\2) \\ Charles R Greathouse IV, Jul 02 2013
(Python)
def a(n): return (3 if n%2 else 1)*6**(n//2)
print([a(n) for n in range(31)]) # Indranil Ghosh, Aug 02 2017
(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

Crossrefs

Cf. A026519, A026520, A026521, A026522, A026523, A026524, A026525, A026526, A026527, A026528, A026529, A026530, A026531, A026533, A026534, A027262, A027263, A027264, A027265, A027266.

Cf. A026534, A026549, A176059, A208131.

Cf. A038730, A038792, and A134511 for incomplete Fibonacci sequences, and A324242 for incomplete Lucas sequences.

Sequence in context: A101726 A034457 A268529 * A160505 A081150 A362014

Adjacent sequences: A026529 A026530 A026531 * A026533 A026534 A026535

Keyword

nonn,easy

Author

Clark Kimberling

Extensions

New definition from Ralf Stephan, Dec 01 2004

Offset changed from 0 to 1 by Vincenzo Librandi, Jun 08 2011

Status

approved