A124647 a(n) = (2n + 1)*3^n.

1, 9, 45, 189, 729, 2673, 9477, 32805, 111537, 373977, 1240029, 4074381, 13286025, 43046721, 138706101, 444816117, 1420541793, 4519905705, 14334558093, 45328197213, 142958160441, 449795187729, 1412147682405, 4424729404869, 13839047287569, 43211719081593, 134718888901437

Offset: 0

Author

Gary W. Adamson, Dec 22 2006

Keywords

• nonn
• easy

Comments

1 - 1/9 + 1/45 - 1/189 + ... = Pi/(2*sqrt(3)) = A093766. [Jolley eq 271].

If X_1,X_2,...,X_n are 3-blocks of a (4n+1)-set X then, for n>=1, a(n) is the number of (n+1)-subsets of X intersecting each X_i, (i=1,2,...,n). - Milan Janjic, Nov 23 2007

Sum_{k>=0} 1/a(k) = log(2+sqrt(3))*sqrt(3)/2 = 1.1405189944... - Jaume Oliver Lafont, Nov 30 2009

Examples

			a(3) = 189 = 7*(3^3).
		

References

• L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, p. 50

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Milan Janjic, Two Enumerative Functions
• A. V. Kitaev and A. Vartanian, Algebroid Solutions of the Degenerate Third Painlevé Equation for Vanishing Formal Monodromy Parameter, arXiv:2304.05671 [math.CA], 2023. See p. 14.
• Index entries for linear recurrences with constant coefficients, signature (6,-9).

Crossrefs

Cf. A000010, A000244, A018804, A036290, A081038, A093766.

Programs

Magma

[ (2*n+1)*3^n: n in [0..23] ]; // Klaus Brockhaus, Sep 23 2009

Mathematica

Table[3^n*(2*n+1), {n,0,30}] (* G. C. Greubel, May 01 2021 *)

Sage

[3^n*(2*n+1) for n in (0..30)] # G. C. Greubel, May 01 2021

Formula

G.f.: (1+3*x)/(1-3*x)^2. - Jaume Oliver Lafont, Mar 07 2009

a(n) = 6*a(n-1) - 9*a(n-2) for n > 1; a(0) = 1, a(1) = 9. - Klaus Brockhaus, Sep 23 2009

a(n) = 9*A081038(n-1) for n > 0. - Klaus Brockhaus, Sep 23 2009

a(n) = Sum_{i=1..2*3^n-1} gcd(i,2*3^n) = A018804(2*3^n) -2*3^n. This is an application of the multiplicative property of the gcd sum-function A018804. So we get: 2*3^0 * phi(3^n) + ... + 2*3^(n-1) * phi(3^1) + 2*3^n * phi(3^0)+3^0 * phi(2*3^n) + ... + 3^n * phi(2*3^0) - gcd(2*3^n,2*3^n) = a(n), where phi=A000010 is Euler's totient. A general formula is Sum_{i=1..2*p^n-1} gcd(i,2*p^n) = n*3*p^n * n - 3*n*p^(n-1) + p^n, for p an odd prime. This sequence correspondes to p=3. - Jeffrey R. Goodwin, Nov 10 2011

E.g.f.: exp(3*x)*(1 + 6*x). - Stefano Spezia, May 07 2023

Extensions

More terms from Klaus Brockhaus, Sep 23 2009