login
Number of horizontally convex polyoctagons containing n regular polygons (squares or octagons).
9

%I #17 Jan 11 2019 06:22:43

%S 0,2,6,24,96,390,1582,6422,26062,105768,429228,1741898,7068978,

%T 28687370,116419254,472453328,1917312976,7780851238,31576298030,

%U 128143125598,520031215214,2110393854816,8564413235420,34756154117738,141047636989250,572400382184434,2322918728158854,9426882974883464

%N Number of horizontally convex polyoctagons containing n regular polygons (squares or octagons).

%H Colin Barker, <a href="/A323260/b323260.txt">Table of n, a(n) for n = 0..1000</a>

%H K. A. Van'kov, V. M. Zhuravlyov, <a href="https://www.mccme.ru/free-books/matpros/pdf/mp-22.pdf#page=127">Regular tilings and generating functions</a>, Mat. Pros. Ser. 3, issue 22, 2018 (127-157) [in Russian]. See Table 1, g_n.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-3,-5,7,-1).

%F G.f. = 2*x*(1-x)^3*(1+x)/(1-5*x+3*x^2+5*x^3-7*x^4+x^5).

%F a(n) = 2*A323261(n).

%F a(n) = 5*a(n-1) - 3*a(n-2) - 5*a(n-3) + 7*a(n-4) - a(n-5) for n>5. - _Colin Barker_, Jan 10 2019

%t CoefficientList[Series[2*x*(1-x)^3*(1+x)/(1-5*x+3*x^2+5*x^3-7*x^4+x^5), {x, 0, 27}], x] (* _Amiram Eldar_, Jan 10 2019 *)

%o (PARI) concat(0, Vec(2*x*(1 - x)^3*(1 + x) / (1 - 5*x + 3*x^2 + 5*x^3 - 7*x^4 + x^5) + O(x^30))) \\ _Colin Barker_, Jan 10 2019

%Y Cf. A323261-A323269.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, Jan 09 2019