A269965 Start with a square; at each stage add a square at each expandable vertex so that the ratio of the side of the squares at stage n+1 and at stage n is the golden ratio phi=0.618...; a(n) is the number of squares in a portion of the n-th stage (see below)

1, 3, 10, 26, 63, 145, 332, 760, 1745, 4007, 9198, 21102, 48403, 111021, 254656, 584132, 1339893, 3073459, 7049906, 16171066, 37093175, 85084313, 195166404, 447672720, 1026871705, 2355438303, 5402904310, 12393181766, 28427480091, 65206953349, 149571708488

Offset

5,2

Comments

This is an auxiliary sequence, the main one being A269962.

a(n) is the number of squares colored red in the illustration.

The ratio phi=0.618... is chosen so that from the fourth stage on some squares overlap perfectly. The figure displays some kind of fractal behavior. See illustration.

Links

Colin Barker, Table of n, a(n) for n = 5..1000

Paolo Franchi, Illustration of initial terms

Index entries for linear recurrences with constant coefficients, signature (3,-1,-3,4,0,-2).

Formula

a(1)=a(2)=a(3)=a(4)=0, for n>= 5, a(n) = A269963(n-4)+a(n-1).

a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) + 2*a(n-4) + 2*a(n-5) + 5.

a(n) = 3*a(n-1) - a(n-2) - 3*a(n-3) + 4*a(n-4) - 2*a(n-6).

G.f.: x^5*(1+2*x^2+2*x^3) / ((1-x)*(1+x)*(1-3*x+2*x^2-2*x^4)). - Colin Barker, Mar 09 2016

Mathematica

RecurrenceTable[{a[n + 1] ==
   2 a[n] + a[n - 1] - 2 a[n - 2] + 2 a[n - 3] + 2 a[n - 4] + 5,
  a[5] == 1, a[6] == 3, a[7] == 10, a[8] == 26, a[9] == 63}, a, {n, 5,
   30}]
RecurrenceTable[{a[n + 1] ==
   3 a[n] - a[n - 1] - 3 a[n - 2] + 4 a[n - 3] - 2 a[n - 5],
  a[5] == 1, a[6] == 3, a[7] == 10, a[8] == 26, a[9] == 63,
  a[10] == 145}, a, {n, 5, 30}]

Prog

(PARI) Vec(x^5*(1+2*x^2+2*x^3)/((1-x)*(1+x)*(1-3*x+2*x^2-2*x^4)) + O(x^50)) \\ Colin Barker, Mar 09 2016

Crossrefs

Main sequence: A269962.

Other auxiliary sequences: A269963, A269964.

Sequence in context: A301410 A333897 A316937 * A110158 A301308 A196232

Adjacent sequences: A269962 A269963 A269964 * A269966 A269967 A269968

Keyword

nonn,easy

Author

Paolo Franchi, Mar 09 2016

Status

approved