A238829 -id:A238829 - cp's OEIS frontend

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.

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]

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

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

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Maple

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

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Maple

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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.