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] == 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*(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
KEYWORD
nonn,easy
AUTHOR
Paolo Franchi, Mar 09 2016
STATUS
approved