A238827 - OEIS

%I #18 Sep 08 2022 08:46:07

%S 0,0,0,0,0,0,1,2,6,13,33,77,191,464,1147,2819,6956,17132,42228,104026,

%T 256303,631394,1555488,3831945,9440141,23256017,57292037,141140858,

%U 347705663,856585345,2110229136,5198625560,12807001916,31550510748,77725820617,191480359254,471718764310,1162096170669

%N a(n) = 0 for n <= 3; thereafter a(n) = a(n-2)+A238825(n-3).

%H Vincenzo Librandi, <a href="/A238827/b238827.txt">Table of n, a(n) for n = 1..1000</a>

%H V. M. Zhuravlev, <a href="http://www.mccme.ru/free-books/matpros/mph.pdf">Horizontally-convex polyiamonds and their generating functions</a>, Mat. Pros. 17 (2013), 107-129 (in Russian). See the sequence r(n).

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (2,3,-4,-3,2,4,2,-1).

%F G.f.: -x^7*(-1+x^2+x^3) / ( (1+x)*(x^7-3*x^6-x^5-x^4+4*x^3-3*x+1) ). - _R. J. Mathar_, Mar 20 2014

%p g:=proc(n) option remember; local t1;

%p t1:=[2,3,6,14,34,84,208,515];

%p if n <= 7 then t1[n] else

%p 3*g(n-1)-4*g(n-3)+g(n-4)+g(n-5)+3*g(n-6)-g(n-7); fi; end proc;

%p [seq(g(n),n=1..32)]; # A238823

%p d:=proc(n) option remember; global g; local t1;

%p t1:=[0,1];

%p if n <= 2 then t1[n] else

%p g(n-1)-2*d(n-1)-d(n-2); fi; end proc;

%p [seq(d(n),n=1..32)]; # A238824

%p p:=proc(n) option remember; global d; local t1;

%p t1:=[0,0,0,1];

%p if n <= 4 then t1[n] else

%p p(n-2)+p(n-3)+2*(d(n-3)+d(n-4)); fi; end proc;

%p [seq(p(n),n=1..32)]; # A238825

%p [seq(p(n+3)-p(n+1),n=1..32)]; #A238826

%p r:=proc(n) option remember; global p; local t1;

%p t1:=[0,0,0,0];

%p if n <= 4 then t1[n] else

%p r(n-2)+p(n-3); fi; end proc;

%p [seq(r(n),n=1..32)]; # A238827

%t CoefficientList[Series[- x^6 (- 1 + x^2 + x^3)/((1 + x) (x^7 - 3 x^6 - x^5 - x^4 + 4 x^3 - 3 x + 1)), {x, 0, 40}], x] (* _Vincenzo Librandi_, Mar 21 2014 *)

%t LinearRecurrence[{2,3,-4,-3,2,4,2,-1},{0,0,0,0,0,0,1,2,6,13},40] (* _Harvey P. Dale_, Jun 26 2020 *)

%o (Magma) m:=40; R<x>:=LaurentSeriesRing(RationalField(), m); [0,0,0,0,0,0] cat Coefficients(R! -x^7*(-1+x^2+x^3) / ( (1+x)*(x^7-3*x^6-x^5-x^4+4*x^3-3*x+1))); // _Vincenzo Librandi_, Mar 21 2014

%Y Cf. A238823-A238826.

%K nonn,easy

%O 1,8

%A _N. J. A. Sloane_, Mar 08 2014