A238833 a(n) = n-1 for n <= 2; thereafter a(n) = A238824(n-2) + A238832(n-1).

0, 1, 0, 2, 2, 7, 16, 40, 101, 246, 615, 1504, 3724, 9147, 22567, 55541, 136884, 337128, 830628, 2046145, 5040932, 12418320, 30593281, 75367352, 185670647, 457405836, 1126836394, 2776001211, 6838779857, 16847579205, 41504619640, 102248123906, 251891939366, 620544865783, 1528734638988, 3766092860744

Offset: 1

Author

N. J. A. Sloane, Mar 08 2014

Keywords

• nonn
• easy

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 j(n).
• Index entries for linear recurrences with constant coefficients, signature (1,5,-1,-7,-1,6,6,1,-1).

Crossrefs

Cf. A238823-A238832.

Programs

Magma

m:=40; R:=LaurentSeriesRing(RationalField(), m); [0] cat Coefficients(R! -x^2*(x^8+2*x^7+x^6-2*x^5-2*x^4-x^3+3*x^2+x-1) / ((x+1)^2*(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
h:=n->p(n+3)-p(n+1); [seq(h(n),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
b:=n-> if n=1 then 0 else d(n-1)+p(n); fi; [seq(b(n),n=1..32)]; #A238828
a:=n->g(n)-h(n); [seq(a(n),n=1..32)]; #A238829
i:=proc(n) option remember; global b,r; local t1; t1:=[0,0];
if n <= 2 then t1[n] else
i(n-2)+b(n-1)+r(n); fi; end proc;
[seq(i(n),n=1..32)]; # A238830
q:=n-> if n<=2 then 0 else r(n)+i(n-2); fi;
[seq(q(n),n=1..45)]; # A238831
e:=n-> if n<=1 then 0 else d(n-1)+i(n-1); fi;
[seq(e(n),n=1..45)]; # A238832
j:=n-> if n<=2 then n-1 else d(n-2)+e(n-1); fi;
[seq(j(n),n=1..45)]; # A238833

Mathematica

CoefficientList[Series[- x (x^8 + 2 x^7 + x^6 - 2 x^5 - 2 x^4 - x^3 + 3 x^2 + x - 1)/((x + 1)^2 (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[{1,5,-1,-7,-1,6,6,1,-1},{0,1,0,2,2,7,16,40,101,246},40] (* Harvey P. Dale, Jul 23 2021 *)

PARI

concat(0, Vec(-x^2*(x^8+2*x^7+x^6-2*x^5-2*x^4-x^3+3*x^2+x-1)/((x+1)^2*(x^7-3*x^6-x^5-x^4+4*x^3-3*x+1)) + O(x^100))) \\ Colin Barker, Mar 20 2014

Formula

G.f.: -x^2*(x^8+2*x^7+x^6-2*x^5-2*x^4-x^3+3*x^2+x-1) / ((x+1)^2*(x^7-3*x^6-x^5-x^4+4*x^3-3*x+1)). - Colin Barker, Mar 20 2014