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%I A128135 #62 Feb 18 2026 11:54:51
%S A128135 1,3,10,28,72,176,416,960,2176,4864,10752,23552,51200,110592,237568,
%T A128135 507904,1081344,2293760,4849664,10223616,21495808,45088768,94371840,
%U A128135 197132288,411041792,855638016,1778384896,3690987520,7650410496,15837691904,32749125632,67645734912,139586437120,287762808832
%N A128135 Row sums of A128134.
%C A128135 Conjecture: a(n)/a(n-1) tends to sqrt(5). (E.g., a(10)/a(9) = 2.235294....)
%C A128135 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
%C A128135 This sequence is a subsequence of a greedily and recursively defined sequence (see links). - _Sela Fried_, Aug 30 2024
%C A128135 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
%H A128135 Vincenzo Librandi, <a href="/A128135/b128135.txt">Table of n, a(n) for n = 1..1000</a>
%H A128135 Félix Balado and Guénolé C. M. Silvestre, <a href="https://arxiv.org/abs/2602.10005">Systematic Enumeration of Fundamental Quantities Involving Runs in Binary Strings</a>, arXiv:2602.10005 [math.CO], 2026. See p. 111.
%H A128135 Sela Fried, <a href="/A128135/a128135.pdf">On integer sequence A128135</a>, 2024.
%H A128135 Sela Fried, <a href="https://arxiv.org/abs/2410.07237">Proofs of some Conjectures from the OEIS</a>, arXiv:2410.07237 [math.NT], 2024. See p. 11.
%H A128135 Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a>, 2011. [Cached copy]
%H A128135 Guo-Niu Han, <a href="https://arxiv.org/abs/2006.14070">Enumeration of Standard Puzzles</a>, arXiv:2006.14070 [math.CO], 2020.
%H A128135 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4).
%F A128135 Row sums of A128134.
%F A128135 Equals A134315 * [1, 2, 3, ...]. - _Gary W. Adamson_, Oct 19 2007
%F A128135 a(n) = 2*a(n-1) + 2^(n-1) for n >= 2. - Philipp Zumstein (zuphilip(AT)inf.ethz.ch), Oct 05 2009
%F A128135 From _Colin Barker_, May 29 2012: (Start)
%F A128135 a(n) = 2^(n - 2)*(2*n - 1) for n > 1.
%F A128135 a(n) = 4*a(n-1) - 4*a(n-2) for n > 3.
%F A128135 G.f.: x*(1 - x + 2*x^2)/(1 - 2*x)^2. (End)
%F A128135 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
%F A128135 From _Amiram Eldar_, Aug 05 2020: (Start)
%F A128135 Sum_{n>=1} 1/a(n) = 2*sqrt(2)*arcsinh(1) - 1.
%F A128135 Sum_{n>=1} (-1)^(n+1)/a(n) = 2*sqrt(2)*arccot(sqrt(2)) - 1. (End)
%e A128135 a(4) = 28 = sum of row 4 of A128134 = 3 + 10 + 11 + 4.
%t A128135 CoefficientList[Series[(1-x+2*x^2)/(1-2*x)^2,{x,0,40}],x] (* _Vincenzo Librandi_, Jun 28 2012 *)
%t A128135 LinearRecurrence[{4,-4},{1,3,10},40] (* _Harvey P. Dale_, May 26 2023 *)
%o A128135 (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
%o A128135 (PARI) a(n)=if(n<=2,[1,3][n],2*a(n-1)+2^(n-1)); /* _Joerg Arndt_, Sep 29 2012 */
%Y A128135 Cf. A128132, A128133, A128134, A134315.
%K A128135 nonn,easy
%O A128135 1,2
%A A128135 _Gary W. Adamson_, Feb 16 2007
%E A128135 More terms from Philipp Zumstein (zuphilip(AT)inf.ethz.ch), Oct 05 2009
%E A128135 Incorrect formula deleted by _Colin Barker_, May 29 2012

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