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
This sequence is a subsequence of a greedily and recursively defined sequence (see links). - Sela Fried, Aug 30 2024
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Sela Fried, On integer sequence A128135, 2024.
Sela Fried, Proofs of some Conjectures from the OEIS, arXiv:2410.07237 [math.NT], 2024. See p. 11.
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
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