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#42 by Joerg Arndt at Mon Jan 01 11:50:43 EST 2024
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#41 by Paolo P. Lava at Mon Jan 01 11:46:41 EST 2024
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| FORMULA
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a(n) = -(1/6)*sqrt(3)*(7 - 4*sqrt(3))^n + (1/6)*sqrt(3)*(7 + 4*sqrt(3))^n + (1/2)*(7 + 4*sqrt(3))^n + (1/2)*(7 - 4*sqrt(3))^n. - Paolo P. Lava, Aug 06 2008
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| STATUS
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approved
editing
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#40 by Joerg Arndt at Mon Jan 27 01:28:36 EST 2020
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#39 by Jon E. Schoenfield at Mon Jan 27 01:17:26 EST 2020
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#38 by Jon E. Schoenfield at Mon Jan 27 01:17:23 EST 2020
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| FORMULA
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a(n) = -(1/6)*sqrt(3)*(7 - 4*sqrt(3))^n + (1/6)*sqrt(3)*(7 + 4*sqrt(3))^n + (1/2)*(7 + 4*sqrt(3))^n + (1/2)*(7 - 4*sqrt(3))^n. [_. - _Paolo P. Lava_, Aug 06 2008]
G.f.: x*(1 - 3*x)/(1 - 14*x + x^2). [_). - _Philippe Deléham_, Nov 17 2008]
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| STATUS
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approved
editing
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#37 by Susanna Cuyler at Mon Jan 07 11:16:22 EST 2019
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#36 by Franck Maminirina Ramaharo at Mon Jan 07 11:01:39 EST 2019
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#35 by Franck Maminirina Ramaharo at Mon Jan 07 11:01:32 EST 2019
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E.g.f.: (1/3)*((9*cosh(4*sqrt(3)*x) - 5*sqrt(3)*sinh(4*sqrt(3)*x))*exp(7*x) - 9). - Franck Maminirina Ramaharo, Jan 07 2019
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#34 by Franck Maminirina Ramaharo at Mon Jan 07 10:58:09 EST 2019
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| COMMENTS
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The Diophantine equation A000290(x) = A000326(y) + A000326(y-1) has the solutions x = a(n) and y = (4^n+( + (1+ + sqrt(3))^(4*n- - 3)+() + (1- - sqrt(3))^(4*n- - 3))/(3*2^(2*n- - 1)). - Bruno Berselli, Mar 04 2013
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| FORMULA
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a(n) = ((3 - b)*(7 - 4*b)^n + (3 + b)*(7 + 4*b)^n)/6, b= = sqrt(3).
a(n) = -(1/6)*sqrt(3)*(7- - 4*sqrt(3))^n+( + (1/6)*sqrt(3)*(7+ + 4*sqrt(3))^n+( + (1/2)*(7+ + 4*sqrt(3))^n+( + (1/2)*(7- - 4*sqrt(3))^n. [Paolo P. Lava, Aug 06 2008]
G.f.: x*(1- - 3*x)/(1- - 14*x+ + x^2). [Philippe Deléham, Nov 17 2008]
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| STATUS
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proposed
editing
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#33 by Jean-François Alcover at Mon Jan 07 10:22:20 EST 2019
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