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A133044 Area of the spiral of equilateral triangles with side lengths which follow the Padovan sequence, divided by the area of the initial triangle.

%I A133044 #50 Jul 07 2023 14:48:57
%S A133044 1,2,3,7,11,20,36,61,110,191,335,591,1032,1816,3185,5586,9811,17207,
%T A133044 30203,53004,93004,163229,286430,502655,882111,1547967,2716528,
%U A133044 4767152,8365761,14680930,25763171,45211271,79340235,139232356,244335860,428779421,752455502,1320467391
%N A133044 Area of the spiral of equilateral triangles with side lengths which follow the Padovan sequence, divided by the area of the initial triangle.
%C A133044 First differs from A014529 at a(8).
%H A133044 G. C. Greubel, <a href="/A133044/b133044.txt">Table of n, a(n) for n = 1..2000</a>
%H A133044 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,1,-1,1,-1).
%F A133044 From _Colin Barker_, Sep 18 2013: (Start)
%F A133044 Conjecture: a(n) = a(n-1) + a(n-2) + a(n-3) - a(n-4) + a(n-5) - a(n-6).
%F A133044 G.f.: x*(x^3+x+1) / ((x^3-x^2+2*x-1)*(x^3-x-1)).
%F A133044 (End)
%F A133044 From _Félix Breton_, Dec 17 2015: (Start)
%F A133044 a(n) = 2*p(n+4)*p(n+5) - p(n+2)^2 where p is the Padovan sequence (A000931). This establishes Colin Barker's conjecture, because
%F A133044 a(n) = a(n-1) + p(n+4)^2
%F A133044 = a(n-1) + (p(n+1) + p(n+2))^2
%F A133044 = a(n-1) + p(n+1)^2 + p(n+2)^2 + 2*p(n+1)*p(n+2) - p(n-1)^2 + p(n-1)^2
%F A133044 = a(n-1) + (a(n-3)-a(n-4)) + (a(n-2)-a(n-3)) + a(n-3) + (a(n-5)-a(n-6))
%F A133044 = a(n-1) + a(n-2) + a(n-3) - a(n-4) + a(n-5) - a(n-6). (End)
%t A133044 RecurrenceTable[{a[n + 6] == a[n + 5] + a[n + 4] + a[n + 3] - a[n + 2] + a[n + 1] - a[n], a[1] == 1, a[2] == 2, a[3] == 3, a[4] == 7, a[5] == 11, a[6] == 20}, a, {n, 1, 2000}] (* _G. C. Greubel_, Dec 17 2015 *)
%t A133044 Rest@ CoefficientList[Series[x (x^3 + x + 1)/((x^3 - x^2 + 2 x - 1) (x^3 - x - 1)), {x, 0, 38}], x] (* _Michael De Vlieger_, Feb 21 2018 *)
%o A133044 (PARI) Vec((x^3+x+1)/((x^3-x^2+2*x-1)*(x^3-x-1)) + O(x^40)) \\ _Andrew Howroyd_, Feb 21 2018
%Y A133044 Cf. A000931, A014529, A133043.
%K A133044 nonn
%O A133044 1,2
%A A133044 _Omar E. Pol_, Nov 04 2007
%E A133044 a(27) and beyond taken from _G. C. Greubel_'s table. - _Omar E. Pol_, Dec 18 2015
%E A133044 a(589) in b-file corrected by _Andrew Howroyd_, Feb 21 2018