A124647 - OEIS

%I #44 May 08 2023 09:35:25

%S 1,9,45,189,729,2673,9477,32805,111537,373977,1240029,4074381,

%T 13286025,43046721,138706101,444816117,1420541793,4519905705,

%U 14334558093,45328197213,142958160441,449795187729,1412147682405,4424729404869,13839047287569,43211719081593,134718888901437

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

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

%C 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

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

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

%H G. C. Greubel, <a href="/A124647/b124647.txt">Table of n, a(n) for n = 0..1000</a>

%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>

%H A. V. Kitaev and A. Vartanian, <a href="https://arxiv.org/abs/2304.05671">Algebroid Solutions of the Degenerate Third Painlevé Equation for Vanishing Formal Monodromy Parameter</a>, arXiv:2304.05671 [math.CA], 2023. See p. 14.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-9).

%F G.f.: (1+3*x)/(1-3*x)^2. - _Jaume Oliver Lafont_, Mar 07 2009

%F a(n) = 6*a(n-1) - 9*a(n-2) for n > 1; a(0) = 1, a(1) = 9. - _Klaus Brockhaus_, Sep 23 2009

%F a(n) = 9*A081038(n-1) for n > 0. - _Klaus Brockhaus_, Sep 23 2009

%F 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

%F E.g.f.: exp(3*x)*(1 + 6*x). - _Stefano Spezia_, May 07 2023

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

%t Table[3^n*(2*n+1), {n,0,30}] (* _G. C. Greubel_, May 01 2021 *)

%o (Magma) [ (2*n+1)*3^n: n in [0..23] ]; // _Klaus Brockhaus_, Sep 23 2009

%o (Sage) [3^n*(2*n+1) for n in (0..30)] # _G. C. Greubel_, May 01 2021

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

%K nonn,easy

%O 0,2

%A _Gary W. Adamson_, Dec 22 2006

%E More terms from _Klaus Brockhaus_, Sep 23 2009