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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.
2
1, 2, 3, 7, 11, 20, 36, 61, 110, 191, 335, 591, 1032, 1816, 3185, 5586, 9811, 17207, 30203, 53004, 93004, 163229, 286430, 502655, 882111, 1547967, 2716528, 4767152, 8365761, 14680930, 25763171, 45211271, 79340235, 139232356, 244335860, 428779421, 752455502, 1320467391
OFFSET
1,2
COMMENTS
First differs from A014529 at a(8).
FORMULA
From Colin Barker, Sep 18 2013: (Start)
Conjecture: a(n) = a(n-1) + a(n-2) + a(n-3) - a(n-4) + a(n-5) - a(n-6).
G.f.: x*(x^3+x+1) / ((x^3-x^2+2*x-1)*(x^3-x-1)).
(End)
From Félix Breton, Dec 17 2015: (Start)
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
a(n) = a(n-1) + p(n+4)^2
= a(n-1) + (p(n+1) + p(n+2))^2
= 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
= a(n-1) + (a(n-3)-a(n-4)) + (a(n-2)-a(n-3)) + a(n-3) + (a(n-5)-a(n-6))
= a(n-1) + a(n-2) + a(n-3) - a(n-4) + a(n-5) - a(n-6). (End)
MATHEMATICA
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 *)
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Omar E. Pol, Nov 04 2007
EXTENSIONS
a(27) and beyond taken from G. C. Greubel's table. - Omar E. Pol, Dec 18 2015
a(589) in b-file corrected by Andrew Howroyd, Feb 21 2018
STATUS
approved