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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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

First differs from A014529 at a(8).

REFERENCES

Mohammad K. Azarian, A Trigonometric Characterization of Equilateral Triangle, Problem 336, Mathematics and Computer Education, Vol. 31, No. 1, Winter 1997, p. 96. Solution published in Vol. 32, No. 1, Winter 1998, pp. 84-85.

Mohammad K. Azarian, Equating Distances and Altitude in an Equilateral Triangle, Problem 316, Mathematics and Computer Education, Vol. 28, No. 3, Fall 1994, p. 337. Solution published in Vol. 29, No. 3, Fall 1995, pp. 324-325.

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: A139630 A245738 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

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Last modified January 26 01:48 EST 2020. Contains 331270 sequences. (Running on oeis4.)