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

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).

Links

G. C. Greubel, Table of n, a(n) for n = 1..2000

Index entries for linear recurrences with constant coefficients, signature (1,1,1,-1,1,-1).

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

Cf. A000931, A014529, A133043.

Sequence in context: A245738 A368032 A265093 * A014529 A095015 A024367

Adjacent sequences: A133041 A133042 A133043 * A133045 A133046 A133047

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