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

Author

Clark Kimberling

Keywords

• nonn
• easy

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
• Sean A. Irvine, Walks on Graphs.
• 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).

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.

Programs

Haskell

a026532 n = a026532_list !! (n-1)
a026532_list = scanl (*) 1 $ a176059_list
-- Reinhard Zumkeller, Apr 04 2012

Magma

[(1/4)*(3-(-1)^n)*6^Floor(n/2) : n in [1..30]]; // Vincenzo Librandi, Jun 08 2011

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 *)

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

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

Extensions

New definition from Ralf Stephan, Dec 01 2004

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