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URL: https://oeis.org/A255870

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A255870
a(n) is the total number of pentagrams in a pentagram fractal after n iterations.
10
1, 6, 26, 76, 191, 411, 816, 1521, 2726, 4741, 8081, 13566, 22536, 37146, 60896, 99436, 161921, 263151, 427086, 692481, 1122056, 1817281, 2942351, 4762926, 7708866, 12475686, 20188766, 32668996, 52862651, 85536891, 138405156, 223948041, 362359586, 586314421
OFFSET
0,2
COMMENTS
Inspired by the "calice" (see detail in the CNRS link).
The pentagrams appearing in the calice are scaled down by a factor of 1/phi = 0.61803398... from the pentagrams whose vertex-to-vertex length = d. See illustration of the nested pentagrams in the links.
LINKS
Shalom Eliahou, Pavages, symétrie d'ordre 5 et Fibonacci: un amateur passionné, Images des Mathématiques, CNRS, 2015 (in French).
Frédéric Mansuy, Supersymétrie d'ordre cinq périodique (in French).
FORMULA
a(n) = 3*a(n-1)-a(n-2)-4*a(n-3)+3*a(n-4)+a(n-5)-a(n-6) for n>6. - Colin Barker, Mar 12 2015
G.f.: -(5*x^6+x^5-10*x^4-8*x^3-9*x^2-3*x-1) / ((x-1)^3*(x+1)*(x^2+x-1)). - Colin Barker, Mar 12 2015
a(n) = (-307 + 5*(-1)^n - 3*2^(2-n)*s*((11-5*s)*(1-s)^n - (1+s)^n*(11+5*s)) - 150*(1+n) - 50*(1+n)*(2+n)) / 8 for n>0 where s=sqrt(5). - Colin Barker, Mar 12 2017
E.g.f.: 3*exp(x/2)*(25*cosh(sqrt(5)*x/2) + 11*sqrt(5)*sinh(sqrt(5)*x/2)) - (12 + 5*x)*(23 + 5*x)*cosh(x)/4 - (281 + 25*x*(7 + x))*sinh(x)/4 - 5. - Stefano Spezia, Dec 07 2022
MATHEMATICA
CoefficientList[Series[(5 x^6 + x^5 - 10 x^4 - 8 x^3 - 9 x^2 - 3 x - 1)/((1-x)^3 (x+1) (x^2+x-1)), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 18 2015 *)
PROG
(Excel) See links.
(PARI) Vec(-(5*x^6+x^5-10*x^4-8*x^3-9*x^2-3*x-1)/((x-1)^3*(x+1)*(x^2+x-1)) + O(x^100)) \\ Colin Barker, Mar 12 2015
(Magma) I:=[1, 6, 26, 76, 191, 411, 816]; [n le 7 select I[n] else 3*Self(n-1)-Self(n-2)-4*Self(n-3)+3*Self(n-4)+Self(n-5)-Self(n-6): n in [1..40]]; // Vincenzo Librandi, Mar 18 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Kival Ngaokrajang, Mar 08 2015
STATUS
approved