A269964 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, 1, 3, 5, 11, 23, 53, 121, 279, 639, 1465, 3357, 7699, 17659, 40509, 92921, 213143, 488903, 1121441, 2572357, 5900475, 13534515, 31045477, 71212113, 163346335, 374683807, 859449705, 1971405725, 4522010435, 10372587467, 23792640941, 54575559337

Offset

1,3

Comments

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

a(n) gives 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 = 1..1000

Paolo Franchi, Illustration of initial terms

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

Formula

a(n) = 2*a(n-2) + 2*a(n-3) + 2*A269965(n) + 1.

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

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

G.f.: x*(1-2*x+x^2-2*x^4) / ((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] - 2,
  a[1] == 1, a[2] == 1, a[3] == 3, a[4] == 5, a[5] == 11}, a, {n, 1,
  30}]
RecurrenceTable[{a[n + 1] ==
   3 a[n] - a[n - 1] - 3 a[n - 2] + 4 a[n - 3] - 2 a[n - 5],
  a[1] == 1, a[2] == 1, a[3] == 3, a[4] == 5, a[5] == 11,
  a[6] == 23}, a, {n, 1, 30}]

Prog

(PARI) Vec(x*(1-2*x+x^2-2*x^4)/((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, A269965.

Sequence in context: A113281 A037446 A113151 * A305412 A094810 A139376

Adjacent sequences: A269961 A269962 A269963 * A269965 A269966 A269967

Keyword

nonn,easy

Author

Paolo Franchi, Mar 09 2016

Status

approved