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A269963
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).
4
1, 2, 7, 16, 37, 82, 187, 428, 985, 2262, 5191, 11904, 27301, 62618, 143635, 329476, 755761, 1733566, 3976447, 9121160, 20922109, 47991138, 110082091, 252506316, 579198985, 1328566598, 3047466007, 6990277456, 16034298325, 36779473258, 84364755139
OFFSET
1,2
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.
FORMULA
a(n) = 2*A269964(n) + a(n-1) - 1.
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) + 2*a(n-4) + 2*a(n-5).
G.f.: x*(1+2*x^2+2*x^3) / ((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],
a[1] == 1, a[2] == 2, a[3] == 7, a[4] == 16, a[5] == 37}, a, {n, 1,
30}]
PROG
(PARI) Vec(x*(1+2*x^2+2*x^3)/((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: A269964, A269965.
Sequence in context: A345025 A329324 A131405 * A176805 A224227 A260505
KEYWORD
nonn,easy
AUTHOR
Paolo Franchi, Mar 09 2016
STATUS
approved