

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 nth stage (see below)


4



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
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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



FORMULA

a(1)=a(2)=a(3)=a(4)=0, for n>= 5, a(n) = A269963(n4)+a(n1).
a(n) = 2*a(n1) + a(n2)  2*a(n3) + 2*a(n4) + 2*a(n5) + 5.
a(n) = 3*a(n1)  a(n2)  3*a(n3) + 4*a(n4)  2*a(n6).
G.f.: x^5*(1+2*x^2+2*x^3) / ((1x)*(1+x)*(13*x+2*x^22*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)/((1x)*(1+x)*(13*x+2*x^22*x^4)) + O(x^50)) \\ Colin Barker, Mar 09 2016


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



