|
%I #27 Aug 29 2023 16:00:31
%S 0,0,1,2,5,12,28,70,169,420,1030,2546,6266,15452,38056,93774,230993,
%T 569084,1401913,3453690,8508214,20960336,51636447,127208350,313382262,
%U 772028708,1901920456,4685449914,11542774524,28436041324,70053211913,172578611878
%N a(0)=0; thereafter a(n) = A238824(n-1)+A238825(n).
%H Vincenzo Librandi, <a href="/A238828/b238828.txt">Table of n, a(n) for n = 0..999</a> [Offset shifted by _Georg Fischer_, Oct 18 2021]
%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 b(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^2*(1-2*x^2+2*x^5) / ( (1+x)*(x^7-3*x^6-x^5-x^4+4*x^3-3*x+1) ). - _R. J. Mathar_, Mar 20 2014, adapted to offset Jun 19 2021
%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
%p [0,seq(d(n-1)+p(n),n=2..32)]; #A238828
%t CoefficientList[Series[x^2 (1 - 2 x^2 + 2 x^5)/((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},{0,0,1,2,5,12,28,70},40] (* _Harvey P. Dale_, Aug 29 2023 *)
%o (Magma) m:=40; R<x>:=LaurentSeriesRing(RationalField(), m); [0,0] cat Coefficients(R! x^3*(1-2*x^2+2*x^5) / ( (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-A238827.
%K nonn,easy
%O 0,4
%A _N. J. A. Sloane_, Mar 08 2014
%E Offset corrected by _N. J. A. Sloane_, Jun 16 2021
|
