A128135 Row sums of A128134.

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

Offset: 1

Author

Gary W. Adamson, Feb 16 2007

Keywords

• nonn
• easy

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

For n>=2, a(n) is the total number of ones in runs of ones of length >=3 over all binary strings of length n+1. - Félix Balado, Aug 06 2025

Examples

			a(4) = 28 = sum of row 4 of A128134 = 3 + 10 + 11 + 4.
		

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).

Crossrefs

Cf. A128132, A128133, A128134, A134315.

Programs

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

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 *)

PARI

a(n)=if(n<=2,[1,3][n],2*a(n-1)+2^(n-1)); /* Joerg Arndt, Sep 29 2012 */

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)

Extensions

More terms from Philipp Zumstein (zuphilip(AT)inf.ethz.ch), Oct 05 2009

Incorrect formula deleted by Colin Barker, May 29 2012