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

%I A255870 #46 Oct 19 2024 15:57:32
%S A255870 1,6,26,76,191,411,816,1521,2726,4741,8081,13566,22536,37146,60896,
%T A255870 99436,161921,263151,427086,692481,1122056,1817281,2942351,4762926,
%U A255870 7708866,12475686,20188766,32668996,52862651,85536891,138405156,223948041,362359586,586314421
%N A255870 a(n) is the total number of pentagrams in a pentagram fractal after n iterations.
%C A255870 Inspired by the "calice" (see detail in the CNRS link).
%C A255870 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.
%H A255870 Colin Barker, <a href="/A255870/b255870.txt">Table of n, a(n) for n = 0..1000</a>
%H A255870 Shalom Eliahou, <a href="http://images-archive.math.cnrs.fr/Pavages-symetrie-d-ordre-5-et.html">Pavages, symétrie d'ordre 5 et Fibonacci: un amateur passionné</a>, Images des Mathématiques, CNRS, 2015 (in French).
%H A255870 Frédéric Mansuy, <a href="http://supersymetrie.fr/">Supersymétrie d'ordre cinq périodique</a> (in French).
%H A255870 Kival Ngaokrajang, <a href="/A255870/a255870.pdf">Illustration of initial terms</a>, <a href="/A255870/a255870_2.pdf">Excel calculation sheet</a>
%H A255870 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-4,3,1,-1).
%F A255870 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
%F A255870 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
%F A255870 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
%F A255870 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
%t A255870 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 *)
%o A255870 (Excel) See links.
%o A255870 (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
%o A255870 (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
%Y A255870 Cf. A000071, A006327, A249852, A253688.
%K A255870 nonn,easy
%O A255870 0,2
%A A255870 _Kival Ngaokrajang_, Mar 08 2015