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A131130 Binomial transform of [1,1,7,1,7,1,7,1,...].

%I A131130 #61 Apr 13 2024 07:58:14
%S A131130 1,2,10,26,58,122,250,506,1018,2042,4090,8186,16378,32762,65530,
%T A131130 131066,262138,524282,1048570,2097146,4194298,8388602,16777210,
%U A131130 33554426,67108858,134217722,268435450,536870906,1073741818,2147483642
%N A131130 Binomial transform of [1,1,7,1,7,1,7,1,...].
%C A131130 For n >= 3, number of vertices of 4,4'-bipyridinium dendrimers (see the Arjomanfar and Gholami reference, p. 71). - _Emeric Deutsch_, Apr 12 2015
%C A131130 Number of ways to color a (2n-1) X (2n-1) chess board in a "balanced" way. A coloring is called balanced if, within every square subgrid made up of k^2 cells for 1 <= k <= 2*n-1, the number of black cells differs from the number of white cells by at most one. It is problem 3 from the British Maths Olympiad 2020. - _Ruediger Jehn_, Jan 27 2021
%H A131130 Vincenzo Librandi, <a href="/A131130/b131130.txt">Table of n, a(n) for n = 0..1000</a>
%H A131130 A. Arjomanfar and N. Gholami, <a href="http://ijmc.kashanu.ac.ir/article_5219_853.html">Computing the Szeged index of 4,4'-bipyridinium dendrimer</a>, Iranian J. Math. Chem., 3, 2012, 67-72.
%H A131130 British Mathematical Olympiad, <a href="https://bmos.ukmt.org.uk/home/bmo2-2020.pdf">2020 - Round 2</a>, Problem 3.
%H A131130 <a href="/index/O#Olympiads">Index to sequences related to Olympiads</a>.
%H A131130 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2).
%F A131130 Row sums of triangle A131131.
%F A131130 a(n) = 4*2^n - 6 for n >= 1; a(0)=1.
%F A131130 From _Philippe Deléham_, Jan 04 2009: (Start)
%F A131130 a(n) = 3*a(n-1) - 2*a(n-2), n > 2; a(0)=1, a(1)=2, a(2)=10.
%F A131130 G.f.: (1-x+6*x^2) / (1-3*x+2*x^2). (End)
%F A131130 a(n) = 2*a(n-1) + 6 for n > 1, a(0)=1, a(1)=2. - _Philippe Deléham_, Sep 25 2009
%F A131130 E.g.f.: 3 - 6*exp(x) + 4*exp(2*x). - _Stefano Spezia_, Feb 05 2021
%p A131130 1, seq(4*2^n -6, n = 1..30);
%t A131130 Join[{1},LinearRecurrence[{3,-2},{2,10},30]] (* _Harvey P. Dale_, Mar 07 2014 *)
%t A131130 CoefficientList[Series[(1 -x +6x^2)/(1 -3x +2x^2), {x, 0, 40}], x] (* _Vincenzo Librandi_, Mar 08 2014 *)
%Y A131130 Cf. A095121 (bin transf 1,1,3,1,3,...), A131128 (bin transf 1,1,5,1,5,..), A131131.
%K A131130 nonn,easy
%O A131130 0,2
%A A131130 _Gary W. Adamson_, Jun 16 2007
%E A131130 Edited by _Emeric Deutsch_, Jul 12 2007