A238825 -id:A238825 - cp's OEIS frontend

A238826 a(n) = p(n+3)-p(n+1), where p(n) = A238825(n).

1, 2, 4, 9, 22, 53, 131, 323, 798, 1968, 4853, 11958, 29463, 72581, 178803, 440474, 1085110, 2673183, 6585468, 16223521, 39967243, 98460769, 242561730, 597559646, 1472109847, 3626595728, 8934249307, 22009844973, 54222045921, 133577963318, 329074124992, 810685962909

Offset: 1

N. J. A. Sloane, Mar 08 2014

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

Cf. A238823-A238825.

m:=40; R:=LaurentSeriesRing(RationalField(), m); Coefficients(R! -x*(1+x)*(x^3+x^2-1)*(x-1)^2 / ( 1-3*x+4*x^3-x^4-x^5-3*x^6+x^7)); // Vincenzo Librandi, Mar 21 2014

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

CoefficientList[Series[-(1 + x) (x^3 + x^2 - 1) (x - 1)^2/(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},{1,2,4,9,22,53,131},40] (* Harvey P. Dale, Aug 15 2025 *)

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

A238827 a(n) = 0 for n <= 3; thereafter a(n) = a(n-2)+A238825(n-3).

Original entry on oeis.org

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

N. J. A. Sloane, Mar 08 2014

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).

Cf. A238823-A238826.

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

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

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 *)

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

A238828 a(0)=0; thereafter a(n) = A238824(n-1)+A238825(n).

Original entry on oeis.org

0, 0, 1, 2, 5, 12, 28, 70, 169, 420, 1030, 2546, 6266, 15452, 38056, 93774, 230993, 569084, 1401913, 3453690, 8508214, 20960336, 51636447, 127208350, 313382262, 772028708, 1901920456, 4685449914, 11542774524, 28436041324, 70053211913, 172578611878

Offset: 0

N. J. A. Sloane, Mar 08 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..999 [Offset shifted by _Georg Fischer_, Oct 18 2021]
V. M. Zhuravlev, Horizontally-convex polyiamonds and their generating functions, Mat. Pros. 17 (2013), 107-129 (in Russian). See the sequence b(n).
Index entries for linear recurrences with constant coefficients, signature (2,3,-4,-3,2,4,2,-1).

Cf. A238823-A238827.

m:=40; R:=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

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
[0,seq(d(n-1)+p(n),n=2..32)]; #A238828

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 *)
LinearRecurrence[{2,3,-4,-3,2,4,2,-1},{0,0,1,2,5,12,28,70},40] (* Harvey P. Dale, Aug 29 2023 *)

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

Offset corrected by N. J. A. Sloane, Jun 16 2021

A238823 a(n) = 3a(n-1)-4a(n-3)+a(n-4)+a(n-5)+3*a(n-6)-a(n-7) for n >= 8, with initial values as shown.

Original entry on oeis.org

2, 3, 6, 14, 34, 84, 208, 515, 1272, 3138, 7734, 19055, 46940, 115631, 284846, 701709, 1728662, 4258604, 10491218, 25845514, 63671404, 156856887, 386422704, 951966378, 2345203554, 5777493461, 14233063160, 35063663603, 86380598122, 212801715171, 524244692006, 1291495687122

Offset: 1

N. J. A. Sloane, Mar 08 2014

Number of horizontally convex polyiamonds with n triangles.

			The initial values of Zhuravlev's sequences are as follows.
(The columns give n, A238829, A238828, A238824 (twice), A238830, A238833, A238832, A238825, A238831, A238827, A238826, A238823, respectively):
n a b c d i j e p q r h g
1 1 0 1 0 0 0 0 0 0 0 1 2
2 1 0 0 1 0 1 0 0 0 0 2 3
3 2 1 1 1 0 0 1 0 0 0 4 6
4 5 2 1 3 1 2 1 1 0 0 9 14
5 12 5 3 7 2 2 4 2 0 0 22 34
6 31 12 7 17 6 7 9 5 1 0 53 84
7 77 28 17 43 15 16 23 11 3 1 131 208
8 192 70 43 105 36 40 58 27 8 2 323 515
9 474 169 105 262 91 101 141 64 21 6 798 1272

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
K. A. Van'kov, V. M. Zhuravlyov, Regular tilings and generating functions, Mat. Pros. Ser. 3, issue 22, 2018 (127-157) [in Russian]. See page 128. - _N. J. A. Sloane_, Jan 09 2019
Kirill Vankov, Valerii Zhuravlev, Regular and semiregular (uniform) tilings and generating functions, hal-02535947, [math.CO], 2020.
Wikipedia, Polyiamond
V. M. Zhuravlev, Horizontally-convex polyiamonds and their generating functions, Mat. Pros. 17 (2013), 107-129 (in Russian). See the sequence g(n). [Note the recurrence for g(n) in Theorem 1 contains a typo]
Index entries for linear recurrences with constant coefficients, signature (3,0,-4,1,1,3,-1).

Cf. A238824-A238833.

I:=[2,3,6,14,34,84,208,515]; [n le 8 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..40]]; // Vincenzo Librandi, Mar 10 2014

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)];

LinearRecurrence[{3, 0, -4, 1, 1, 3, -1}, {2, 3, 6, 14, 34, 84, 208}, 40] (* Vincenzo Librandi, Mar 10 2014 *)

G.f.: x*(2 - 3*x - 3*x^2 + 4*x^3 + 2*x^4 + x^5 - 3*x^6) / (1 - 3*x + 4*x^3 - x^4 - x^5 - 3*x^6 + x^7). [Bruno Berselli, Mar 10 2014]

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

Original entry on oeis.org

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

N. J. A. Sloane, Mar 08 2014

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).

Cf. A238823-A238828.

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]]

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

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 *)

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

A238830 a(1)=a(2)=0; thereafter a(n) = a(n-2)+A238828(n-1)+A238827(n).

Original entry on oeis.org

0, 0, 0, 1, 2, 6, 15, 36, 91, 218, 544, 1325, 3281, 8055, 19880, 48930, 120610, 297055, 731922, 1802994, 4441915, 10942602, 26957739, 66410994, 163606230, 403049273, 992926975, 2446110587, 6026082552, 14845470456, 36572353012, 90097307929

Offset: 1

N. J. A. Sloane, Mar 08 2014

V. M. Zhuravlev, Horizontally-convex polyiamonds and their generating functions, Mat. Pros. 17 (2013), 107-129 (in Russian). See the sequence i(n).
Index entries for linear recurrences with constant coefficients, signature (1,5,-1,-7,-1,6,6,1,-1).

Cf. A238823-A238829.

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

LinearRecurrence[{1,5,-1,-7,-1,6,6,1,-1},{0,0,0,1,2,6,15,36,91},40] (* Harvey P. Dale, Dec 29 2021 *)

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

A238831 a(n) = 0 if n <= 2; thereafter a(n) = A238827(n) + A238830(n-2).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 3, 8, 21, 49, 124, 295, 735, 1789, 4428, 10874, 26836, 66062, 162838, 401081, 988225, 2434388, 5997403, 14774547, 36397880, 89667011, 220898267, 544190131, 1340632638, 3302695932, 8136311688, 20044096016, 49379354928, 121647818677, 299683787423, 738281805364, 1818783831517

Offset: 1

N. J. A. Sloane, Mar 08 2014

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

Cf. A238823-A238830.

m:=40; R:=LaurentSeriesRing(RationalField(), m); [0,0,0,0,0] cat Coefficients(R! -x^6*(x-1)*(2*x+1)*(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

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

CoefficientList[Series[- x^5 (x - 1) (2 x + 1) (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 *)

concat([0,0,0,0,0], Vec(-x^6*(x-1)*(2*x+1)*(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

G.f.: -x^6*(x-1)*(2*x+1)*(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

A238832 a(1)=0; thereafter a(n) = A238824(n-1)+A238830(n-1).

Original entry on oeis.org

0, 0, 1, 1, 4, 9, 23, 58, 141, 353, 861, 2134, 5236, 12924, 31798, 78382, 193029, 475619, 1171600, 2886427, 7110657, 17517598, 43154977, 106314193, 261908415, 645221312, 1589525242, 3915853416, 9646844896, 23765351096, 58546797181, 144232146189, 355321086856, 875346302897, 2156447153427, 5312485264678

Offset: 1

N. J. A. Sloane, Mar 08 2014

V. M. Zhuravlev, Horizontally-convex polyiamonds and their generating functions, Mat. Pros. 17 (2013), 107-129 (in Russian). See the sequence e(n).
Index entries for linear recurrences with constant coefficients, signature (1,5,-1,-7,-1,6,6,1,-1).

Cf. A238823-A238831.

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

concat([0,0], Vec(x^3*(2*x^5+2*x^4+x^3-2*x^2+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

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

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

Original entry on oeis.org

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

N. J. A. Sloane, Mar 08 2014

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).

Cf. A238823-A238832.

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

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

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 *)

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

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

cp's OEIS Frontend

A238826 a(n) = p(n+3)-p(n+1), where p(n) = A238825(n).

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A238827 a(n) = 0 for n <= 3; thereafter a(n) = a(n-2)+A238825(n-3).

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A238828 a(0)=0; thereafter a(n) = A238824(n-1)+A238825(n).

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A238823 a(n) = 3*a(n-1)-4*a(n-3)+a(n-4)+a(n-5)+3*a(n-6)-a(n-7) for n >= 8, with initial values as shown.

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

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A238830 a(1)=a(2)=0; thereafter a(n) = a(n-2)+A238828(n-1)+A238827(n).

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A238831 a(n) = 0 if n <= 2; thereafter a(n) = A238827(n) + A238830(n-2).

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A238823 a(n) = 3a(n-1)-4a(n-3)+a(n-4)+a(n-5)+3*a(n-6)-a(n-7) for n >= 8, with initial values as shown.