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A265676 a(n) is the total number of petals of the Flower of Life at the n-th iteration.

%I A265676 #16 Aug 26 2024 19:40:08
%S A265676 0,1,7,19,43,67,97,139,181,229,289,349,415,493,571,655,751,847,949,
%T A265676 1063,1177,1297,1429,1561,1699,1849,1999,2155,2323,2491,2665,2851,
%U A265676 3037,3229,3433,3637,3847,4069,4291,4519,4759,4999,5245,5503,5761,6025,6301,6577
%N A265676 a(n) is the total number of petals of the Flower of Life at the n-th iteration.
%C A265676 Inspired by A264788, but counting on petals of the Flower of Life instead of circles. For n >= 3, the second differences seem to be cyclic of 6, 12, 0.
%H A265676 Colin Barker, <a href="/A265676/b265676.txt">Table of n, a(n) for n = 0..1000</a>
%H A265676 Crystalinks, <a href="http://www.crystalinks.com/floweroflife.html">Flower of Life</a>
%H A265676 Kival Ngaokrajang, <a href="/A265676/a265676.pdf">Illustration of initial terms</a>, <a href="/A265676/a265676_1.pdf">For n = 11</a>
%H A265676 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,1,-2,1).
%F A265676 From _Colin Barker_, Dec 13 2015: (Start)
%F A265676 a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5) for n>5.
%F A265676 G.f.: x*(1+5*x+6*x^2+11*x^3-5*x^4) / ((1-x)^3*(1+x+x^2)).
%F A265676 (End)
%t A265676 CoefficientList[Series[x (1 + 5 x + 6 x^2 + 11 x^3 - 5 x^4)/((1 - x)^3 (1 + x + x^2)), {x, 0, 50}], x] (* _Vincenzo Librandi_, Dec 14 2015 *)
%t A265676 LinearRecurrence[{2,-1,1,-2,1},{0,1,7,19,43,67},50] (* _Harvey P. Dale_, Aug 26 2024 *)
%o A265676 (PARI) { a = 7; d1 = 6; print1("0, 1, ", a, ", "); for(n = 3, 100, if (Mod(n,3) == 0, d2 = 6); if (Mod(n,3) == 1, d2 = 12); if (Mod(n,3) == 2, d2 = 0); d1 = d1 + d2; a = a + d1; print1(a, ", "))}
%o A265676 (PARI) concat(0, Vec(x*(1+5*x+6*x^2+11*x^3-5*x^4) / ((1-x)^3*(1+x+x^2)) + O(x^100))) \\ _Colin Barker_, Dec 13 2015
%o A265676 (Magma) I:=[0,1,7,19,43,67]; [n le 6 select I[n] else 2*Self(n-1)-Self(n-2)+Self(n-3)-2*Self(n-4)+Self(n-5): n in [1..60]]; // _Vincenzo Librandi_, Dec 14 2015
%Y A265676 Cf. A264788.
%K A265676 nonn,easy
%O A265676 0,3
%A A265676 _Kival Ngaokrajang_, Dec 13 2015