

A269964


Start with a square; at each stage add a square at each expandable vertex so that the ratio between 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, 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
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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(n2) + 2*a(n3) + 2*A269965(n) + 1.
a(n) = 2*a(n1) + a(n2)  2*a(n3) + 2*a(n4) + 2*a(n5)  2.
a(n) = 3*a(n1)  a(n2)  3*a(n3) + 4*a(n4)  2*a(n6).
G.f.: x*(12*x+x^22*x^4) / ((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]  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*(12*x+x^22*x^4)/((1x)*(1+x)*(13*x+2*x^22*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



