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%I #45 Feb 25 2024 10:48:20
%S 1,4,13,34,79,172,361,742,1507,3040,6109,12250,24535,49108,98257,
%T 196558,393163,786376,1572805,3145666,6291391,12582844,25165753,
%U 50331574,100663219,201326512,402653101,805306282,1610612647,3221225380,6442450849,12884901790
%N a(n) = 6*2^n - 3*n - 5.
%C Sequence generated from a 3 X 3 matrix, companion to A101945.
%C Characteristic polynomial of M = x^3 - 4x^2 + 5x - 2.
%C Sequence can also be generated by the same method as A061777 with slightly different rules. Refer to A061777, which is the "vertex to vertex" expansion version. For this case, the expandable vertices of the existing generation will contact the sides of the new ones, i.e., "vertex to side" expansion version. Let us assign the label "1" to the triangle at the origin; at n-th generation add a triangle at each expandable vertex, i.e., each vertex where the added generations will not overlap the existing ones, although overlaps among new generations are allowed. The non-overlapping triangles will have the same label value as a predecessor; for the overlapping ones, the label value will be sum of label values of predecessors. a(n) is the sum of all label values at n-th generation. The triangles count is A005448. See illustration. - _Kival Ngaokrajang_, Sep 26 2014
%C The number of ways to select 0 or more nodes of a 2 X n rectangular grid such that the selected nodes are connected, but do not fill any 2 X 2 square. This question arises in logic puzzles such as Nurikabe. - _Hugo van der Sanden_, Feb 22 2024
%H G. C. Greubel, <a href="/A101946/b101946.txt">Table of n, a(n) for n = 0..1000</a>
%H Kival Ngaokrajang, <a href="/A101946/a101946.pdf">Illustration of initial terms</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,2).
%F a(0)=1, a(1)=4, a(2)=13 and for n>2, a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3).
%F a(n) = right term in M^n * [1 1 1], where M = the 3X3 matrix [1 0 0 / 2 2 0 / 1 2 1]. M^n * [1 1 1] = [1 A033484(n) a(n)].
%F a(0) = 1, for n >= 1, a(n) = 3*A000225(n) + a(n-1). - _Kival Ngaokrajang_, Sep 26 2014
%F G.f.: (1+2*x^2)/((1-x)^2*(1-2*x)). - _Colin Barker_, Sep 26 2014
%F E.g.f.: 6*exp(2*x) - (5+3*x)*exp(x). - _G. C. Greubel_, Feb 06 2022
%e a(4) = 79 = 4*34 - 5*13 + 2*4 = 4*a(3) - 5*a(2) + 2*a(1).
%e a(4) = right term in M^4 * [1 1 1], since M^4 * [1 1 1] = [1 46 a(4)], where 46 = A033484(4).
%t a[0]=1; a[1]=4; a[2]=13; a[n_]:= a[n]= 4a[n-1] -5a[n-2] +2a[n-3]; Table[ a[n], {n, 0, 30}] (* Or *)
%t a[n_] := (MatrixPower[{{1, 0, 0}, {2, 2, 0}, {1, 2, 1}}, n].{{1}, {1}, {1}})[[3, 1]]; Table[ a[n], {n, 0, 30}] (* _Robert G. Wilson v_, Jan 12 2005 *)
%t Table[6*2^n-3n-5,{n,0,40}] (* or *) LinearRecurrence[{4,-5,2},{1,4,13},40] (* _Harvey P. Dale_, Jun 03 2017 *)
%o (PARI) a(n) = if (n<1, 1, 5*(2^n-1)+a(n-1))\\ _Kival Ngaokrajang_, Sep 26 2014
%o (PARI) Vec(-(2*x^2+1)/((x-1)^2*(2*x-1)) + O(x^100)) \\ _Colin Barker_, Sep 26 2014
%o (Magma) [6*2^n -3*n-5: n in [0..40]]; // _G. C. Greubel_, Feb 06 2022
%o (Sage) [3*(2^(n+1) -n-2) +1 for n in (0..40)] # _G. C. Greubel_, Feb 06 2022
%Y Cf. A033484, A101945.
%Y Cf. A000225, A005448, A061777.
%K nonn,easy
%O 0,2
%A _Gary W. Adamson_, Dec 22 2004
%E New definition from _Ralf Stephan_, May 17 2007
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