A255870 a(n) is the total number of pentagrams in a pentagram fractal after n iterations.

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

Colin Barker, Table of n, a(n) for n = 0..1000

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).

Kival Ngaokrajang, Illustration of initial terms, Excel calculation sheet

Index entries for linear recurrences with constant coefficients, signature (3,-1,-4,3,1,-1).

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

Cf. A000071, A006327, A249852, A253688.

Sequence in context: A001701 A241452 A175898 * A286188 A335648 A094162

Adjacent sequences: A255867 A255868 A255869 * A255871 A255872 A255873

Keyword

nonn,easy

Author

Kival Ngaokrajang, Mar 08 2015

Status

approved