A238827 a(n) = 0 for n <= 3; thereafter a(n) = a(n-2)+A238825(n-3).
0, 0, 0, 0, 0, 0, 1, 2, 6, 13, 33, 77, 191, 464, 1147, 2819, 6956, 17132, 42228, 104026, 256303, 631394, 1555488, 3831945, 9440141, 23256017, 57292037, 141140858, 347705663, 856585345, 2110229136, 5198625560, 12807001916, 31550510748, 77725820617, 191480359254, 471718764310, 1162096170669
Offset: 1
Links
- Vincenzo Librandi, Table of n, a(n) for n = 1..1000
- V. M. Zhuravlev, Horizontally-convex polyiamonds and their generating functions, Mat. Pros. 17 (2013), 107-129 (in Russian). See the sequence r(n).
- Index entries for linear recurrences with constant coefficients, signature (2,3,-4,-3,2,4,2,-1).
Programs
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Magma
m:=40; R
:=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 -
Maple
g:=proc(n) option remember; local t1; t1:=[2,3,6,14,34,84,208,515]; if n <= 7 then t1[n] else 3*g(n-1)-4*g(n-3)+g(n-4)+g(n-5)+3*g(n-6)-g(n-7); fi; end proc; [seq(g(n),n=1..32)]; # A238823 d:=proc(n) option remember; global g; local t1; t1:=[0,1]; if n <= 2 then t1[n] else g(n-1)-2*d(n-1)-d(n-2); fi; end proc; [seq(d(n),n=1..32)]; # A238824 p:=proc(n) option remember; global d; local t1; t1:=[0,0,0,1]; if n <= 4 then t1[n] else p(n-2)+p(n-3)+2*(d(n-3)+d(n-4)); fi; end proc; [seq(p(n),n=1..32)]; # A238825 [seq(p(n+3)-p(n+1),n=1..32)]; #A238826 r:=proc(n) option remember; global p; local t1; t1:=[0,0,0,0]; if n <= 4 then t1[n] else r(n-2)+p(n-3); fi; end proc; [seq(r(n),n=1..32)]; # A238827
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Mathematica
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 *) 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 *)
Formula
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