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A006499
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Number of restricted circular combinations.
(Formerly M2768)
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2
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1, 3, 9, 12, 16, 28, 49, 77, 121, 198, 324, 522, 841, 1363, 2209, 3572, 5776, 9348, 15129, 24477, 39601, 64078, 103684, 167762, 271441, 439203, 710649, 1149852, 1860496, 3010348, 4870849, 7881197, 12752041, 20633238, 33385284, 54018522, 87403801, 141422323, 228826129
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OFFSET
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0,2
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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G.f.: (1+2x+6x^2+2x^3)/((1+x^2)*(1-x-x^2)). - Ralf Stephan, Apr 23 2004
a(n) = Lucas(n+2) - i^n - (-i)^n - (1/2)*i^(n-1) - (1/2)*(-i)^(n-1) where i = sqrt(-1).
a(n) = (1/2)*(Lucas(n+2) - 3*(-1)^floor(n/2) + (-1)^floor((n-1)/2)). (End)
a(n) = Lucas(floor(n/2+1))*Lucas(ceiling(n/2+1));
a(2*n) = Lucas(n+1)^2;
a(2*n+1) = Lucas(n+1)*Lucas(n+2). (End)
E.g.f.: exp(x/2)*(3*cosh(sqrt(5)*x/2) + sqrt(5)*sinh(sqrt(5)*x/2)) - 2*cos(x) - sin(x). - Stefano Spezia, Mar 12 2024
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MAPLE
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A006499:=-(1+2*z+6*z**2+2*z**3)/((z**2+z-1)*(1+z**2)); # [conjectured (correctly) by Simon Plouffe in his 1992 dissertation]
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MATHEMATICA
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CoefficientList[ Series[(1 + 2x + 6x^2 + 2x^3)/((1 + x^2)(1 - x - x^2)), {x, 0, 35}], x] (* Robert G. Wilson v, Feb 25 2005 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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