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 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 n-th 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 (list; graph; refs; listen; history; text; internal format)
 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, a == 1, a == 3, a == 5, a == 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, a == 1, a == 3, a == 5, a == 11,   a == 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

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Last modified December 2 11:14 EST 2021. Contains 349440 sequences. (Running on oeis4.)