login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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 wrong. 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 *)

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 * A191797 A027252 A104574

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified June 21 04:10 EDT 2021. Contains 345354 sequences. (Running on oeis4.)