A026549 Ratios of successive terms are 2, 3, 2, 3, 2, 3, 2, 3, ...

1, 2, 6, 12, 36, 72, 216, 432, 1296, 2592, 7776, 15552, 46656, 93312, 279936, 559872, 1679616, 3359232, 10077696, 20155392, 60466176, 120932352, 362797056, 725594112, 2176782336, 4353564672, 13060694016, 26121388032, 78364164096, 156728328192, 470184984576, 940369969152

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• easy

Comments

Appears to be the number of permutations p of {1,2,...,n} such that p(i)+p(i+1)>=n for every i=1,2,...,n-1 (if offset is 1). - Vladeta Jovovic, Dec 15 2003

Equals eigensequence of a triangle with 1's in even columns and (1,3,3,3,...) in odd columns. a(5) = 72 = (1, 3, 1, 3, 1, 1) dot (1, 1, 2, 6, 12, 36) = (1 + 3 + 2 + 18 + 12 + 36), where (1, 3, 1, 3, 1, 1) = row 5 of the generating triangle. - Gary W. Adamson, Aug 02 2010

Partial products of A010693. - Reinhard Zumkeller, Mar 29 2012

Satisfies Benford's law [Theodore P. Hill, Personal communication, Feb 06, 2017]. - N. J. A. Sloane, Feb 08 2017

For n >= 2, a(n) is the least k > a(n-1) such that both k and a(n-2) + a(n-1) + k have exactly n prime factors, counted with multiplicity. - Robert Israel, Aug 06 2024

Examples

			G.f. = 1 + 2*x + 6*x^2 + 12*x^3 + 36*x^4 + 72*x^5 + 216*x^6 + ... - _Michael Somos_, Apr 09 2022
		

References

• Arno Berger and Theodore P. Hill, An Introduction to Benford's Law, Princeton University Press, 2015.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..700
• Paul Barry, Embedding structures associated with Riordan arrays and moment matrices, International Journal of Combinatorics, Vol. 2014 (2014), Article ID 301394, 7 pages; arXiv preprint, arXiv:1312.0583 [math.CO], 2013.
• Sean A. Irvine, Walks on Graphs.
• Index entries for linear recurrences with constant coefficients, signature (0,6).
• Index entries for sequences related to Benford's law.

Crossrefs

Cf. A026532, A026536, A026551, A026567.

Cf. A010693, A208131, A109827.

Programs

Haskell

a026549 n = a026549_list !! n
a026549_list = scanl (*) 1 $ a010693_list
-- Reinhard Zumkeller, Mar 29 2012

Magma

[(1/2)*(3-(-1)^n)*6^Floor(n/2): n in [0..30]]; // Vincenzo Librandi, Jun 08 2011

Maple

seq(seq(2^i*3^j, i=j..j+1),j=0..30); # Robert Israel, Aug 06 2024

Mathematica

LinearRecurrence[{0,6},{1,2},30] (* Harvey P. Dale, May 29 2016 *)

PARI

{a(n) = 6^(n\2) * (n%2+1)}; /* Michael Somos, Apr 09 2022 */

SageMath

[(1+(n%2))*6^(n//2) for n in (0..30)] # G. C. Greubel, Apr 09 2022

Formula

Equals T(n, 0) + T(n, 1) + ... + T(n, 2n), T given by A026536.

a(n) = 2*A026532(n), for n > 0.

G.f.: (1+2*x)/(1-6*x^2) - Paul Barry, Aug 25 2003

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

a(n) = (1/2)*(3 - (-1)^n)*6^floor(n/2), or a(n) = 6*a(n-2). - Vincenzo Librandi, Jun 08 2011

a(n) = 1/a(-n) if n is even and (2/3)/a(-n) if n is odd for all n in Z. - Michael Somos, Apr 09 2022

Sum_{n>=0} 1/a(n) = 9/5. - Amiram Eldar, Feb 13 2023

Extensions

New definition from Ralf Stephan, Dec 01 2004