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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
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
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
Sequence in context: A034351 A182737 A320244 * A350551 A191797 A355356
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.)