A238829 a(n) = A238823(n) - A238826(n).

1, 1, 2, 5, 12, 31, 77, 192, 474, 1170, 2881, 7097, 17477, 43050, 106043, 261235, 643552, 1585421, 3905750, 9621993, 23704161, 58396118, 143860974, 354406732, 873093707, 2150897733, 5298813853, 13053818630, 32158552201, 79223751853, 195170567014, 480809724213

Offset

1,3

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 a(n).

Index entries for linear recurrences with constant coefficients, signature (3,0,-4,1,1,3,-1).

Formula

G.f.: -x*(x-1)*(2*x^5+x^4+x^3-2*x^2-x+1) / ( 1-3*x+4*x^3-x^4-x^5-3*x^6+x^7 ). - R. J. Mathar, Mar 20 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
[0, seq(d(n-1)+p(n), n=2..32)]; #A238828
a:=n->g(n)-h(n);
[seq(a(n), n=1..32)]; #A238829

Mathematica

CoefficientList[Series[(1 - x) (2 x^5 + x^4 + x^3 - 2 x^2 - x + 1)/(1 - 3 x + 4 x^3 - x^4 - x^5 - 3 x^6 + x^7), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 21 2014 *)
LinearRecurrence[{3, 0, -4, 1, 1, 3, -1}, {1, 1, 2, 5, 12, 31, 77}, 40] (* Harvey P. Dale, Jun 08 2018 *)

Prog

(Magma) I:=[1, 1, 2, 5, 12, 31, 77]; [n le 7 select I[n] else 3*Self(n-1)-4*Self(n-3)+Self(n-4)+Self(n-5)+3*Self(n-6)-Self(n-7): n in [1..35]]

Crossrefs

Cf. A238823-A238828.

Sequence in context: A305311 A291239 A193284 * A125023 A129804 A317882

Adjacent sequences: A238826 A238827 A238828 * A238830 A238831 A238832

Keyword

nonn,easy

Author

N. J. A. Sloane, Mar 08 2014

Status

approved