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1, 3, 10, 28, 72, 176, 416, 960, 2176, 4864, 10752, 23552, 51200, 110592, 237568, 507904, 1081344, 2293760, 4849664, 10223616, 21495808, 45088768, 94371840, 197132288, 411041792, 855638016, 1778384896, 3690987520, 7650410496, 15837691904, 32749125632, 67645734912, 139586437120, 287762808832
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OFFSET
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1,2
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COMMENTS
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Conjecture: a(n)/a(n-1) tends to sqrt(5). (E.g., a(10)/a(9) = 2.235294....)
The conjecture is false. The fraction a(n)/a(n-1) tends to 2 as n grows. - Philipp Zumstein (zuphilip(AT)inf.ethz.ch), Oct 05 2009
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LINKS
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FORMULA
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a(n) = 2*a(n-1) + 2^(n-1) for n >= 2. - Philipp Zumstein (zuphilip(AT)inf.ethz.ch), Oct 05 2009
a(n) = 2^(n - 2)*(2*n - 1) for n > 1.
a(n) = 4*a(n-1) - 4*a(n-2) for n > 3.
G.f.: x*(1 - x + 2*x^2)/(1 - 2*x)^2. (End)
G.f.: (1 - G(0))/2 where G(k) = 1 - (2*k + 2)/(1 - x/(x - (k + 1)/G(k+1))) (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 06 2012
Sum_{n>=1} 1/a(n) = 2*sqrt(2)*arcsinh(1) - 1.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*sqrt(2)*arccot(sqrt(2)) - 1. (End)
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EXAMPLE
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a(4) = 28 = sum of row 4 of A128134 = 3 + 10 + 11 + 4.
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MATHEMATICA
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CoefficientList[Series[(1-x+2*x^2)/(1-2*x)^2, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 28 2012 *)
LinearRecurrence[{4, -4}, {1, 3, 10}, 40] (* Harvey P. Dale, May 26 2023 *)
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PROG
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(Magma) I:=[1, 3, 10]; [n le 3 select I[n] else 4*Self(n-1)-4*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Jun 28 2012
(PARI) a(n)=if(n<=2, [1, 3][n], 2*a(n-1)+2^(n-1)); /* Joerg Arndt, Sep 29 2012 */
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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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More terms from Philipp Zumstein (zuphilip(AT)inf.ethz.ch), Oct 05 2009
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STATUS
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approved
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