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
KEYWORD
nonn,easy
AUTHOR
Kival Ngaokrajang, Mar 08 2015
STATUS
approved