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1, 9, 45, 189, 729, 2673, 9477, 32805, 111537, 373977, 1240029, 4074381, 13286025, 43046721, 138706101, 444816117, 1420541793, 4519905705, 14334558093, 45328197213, 142958160441, 449795187729, 1412147682405, 4424729404869
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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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 R. Janjic (agnus(AT)blic.net), Nov 23 2007
Sum_{k>=0} 1/a(k) = log(2+sqrt(3))*sqrt(3)/2 = 1.1405189944... [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Nov 30 2009]
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REFERENCES
| L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, p. 50
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LINKS
| Milan Janjic, Two Enumerative Functions
Index to sequences with linear recurrences with constant coefficients, signature (6,-9).
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FORMULA
| G.f.: (1+3*x)/(1-3*x)^2 [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), 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,...,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, ..., 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. [From Jeffrey R. Goodwin, Nov 10 2011]
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EXAMPLE
| a(3) = 189 = 7*(3^3)
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PROG
| (MAGMA) [ (2*n+1)*3^n: n in [0..23] ]; [From Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Sep 23 2009]
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CROSSREFS
| Sequence in context: A036826 A022574 A050574 * A111640 A024209 A179855
Adjacent sequences: A124644 A124645 A124646 * A124648 A124649 A124650
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KEYWORD
| nonn,easy
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AUTHOR
| Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 22 2006
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EXTENSIONS
| More terms from Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Sep 23 2009
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