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 A128135 Row sums of A128134. 6
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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 LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..1000 Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy] Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020. Index entries for linear recurrences with constant coefficients, signature (4,-4). FORMULA Row sums of A128134. Equals A134315 * [1, 2, 3, ...]. - Gary W. Adamson, Oct 19 2007 a(n) = 2*a(n-1) + 2^(n-1) for n >= 2. - Philipp Zumstein (zuphilip(AT)inf.ethz.ch), Oct 05 2009 From Colin Barker, May 29 2012: (Start) 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 From Amiram Eldar, Aug 05 2020: (Start) 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) EXAMPLE a(4) = 28 = sum of row 4 of A128134 = 3 + 10 + 11 + 4. MATHEMATICA 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 *) PROG (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 */ CROSSREFS Cf. A128132, A128133, A128134, A134315. Sequence in context: A034351 A182737 A320244 * A350551 A191797 A355356 Adjacent sequences: A128132 A128133 A128134 * A128136 A128137 A128138 KEYWORD nonn,easy AUTHOR Gary W. Adamson, Feb 16 2007 EXTENSIONS More terms from Philipp Zumstein (zuphilip(AT)inf.ethz.ch), Oct 05 2009 Incorrect formula deleted by Colin Barker, May 29 2012 STATUS approved

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Last modified April 19 08:45 EDT 2024. Contains 371782 sequences. (Running on oeis4.)