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 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 n-th 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 (list; graph; refs; listen; history; text; internal format)
 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 Colin Barker, Table of n, a(n) for n = 5..1000 Paolo Franchi, Illustration of initial terms Index entries for linear recurrences with constant coefficients, signature (3,-1,-3,4,0,-2). FORMULA a(1)=a(2)=a(3)=a(4)=0, for n>= 5, a(n) = A269963(n-4)+a(n-1). a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) + 2*a(n-4) + 2*a(n-5) + 5. a(n) = 3*a(n-1) - a(n-2) - 3*a(n-3) + 4*a(n-4) - 2*a(n-6). G.f.: x^5*(1+2*x^2+2*x^3) / ((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] + 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)/((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, A269964. Sequence in context: A301410 A333897 A316937 * A375140 A110158 A301308 Adjacent sequences: A269962 A269963 A269964 * A269966 A269967 A269968 KEYWORD nonn,easy AUTHOR Paolo Franchi, Mar 09 2016 STATUS approved

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Last modified September 8 21:25 EDT 2024. Contains 375759 sequences. (Running on oeis4.)