A059524 -id:A059524 - cp's OEIS frontend

A059020 Number of 2 X n checkerboards (with at least one red square) in which the set of red squares is edge connected.

0, 3, 13, 40, 108, 275, 681, 1664, 4040, 9779, 23637, 57096, 137876, 332899, 803729, 1940416, 4684624, 11309731, 27304157, 65918120, 159140476, 384199155, 927538873, 2239276992, 5406092952, 13051462995, 31509019045, 76069501192, 183648021540, 443365544387

Offset: 0

John W. Layman, Dec 14 2000

In other words, the number of connected (non-null) induced subgraphs in the n-ladder graph P_2 X P_n. - Eric W. Weisstein, May 02 2017

Also, the number of cycles in the grid graph P_3 X P_{n+1}. - Andrew Howroyd, Jun 12 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Connected Graph.
Eric Weisstein's World of Mathematics, Induced Subgraph.
Eric Weisstein's World of Mathematics, Ladder Graph.
Index entries for linear recurrences with constant coefficients, signature (4,-4,0,1).

Row 2 of A287151 and row 2 of A231829.
See also A059021, A059524.
Cf. A000129. - Jaume Oliver Lafont, Sep 28 2009
Other sequences counting connected induced subgraphs: A020873, A059525, A286139, A286182, A286183, A286184, A286185, A286186, A286187, A286188, A286189, A286191, A285765, A285934, A286304.

I:=[0, 3, 13, 40];[n le 4 select I[n] else 4*Self(n-1) - 4*Self(n-2) + Self(n-4):n in [1..30]]; // Marius A. Burtea, Aug 25 2019

Join[{0},LinearRecurrence[{4, -4, 0, 1}, {3, 13, 40, 108}, 20]] (* Eric W. Weisstein, May 02 2017 *) (* adapted by Vincenzo Librandi, May 09 2017 *)
Table[(LucasL[n + 3, 2] - 8 n - 14)/4, {n, 0, 20}] (* Eric W. Weisstein, May 02 2017 *)

a(n) = 2*a(n-1) + a(n-2) + 4*n - 1.
From Jaume Oliver Lafont, Nov 23 2008: (Start)
a(n) = 3*a(n-1) - a(n-2) - a(n-3) + 4;
a(n) = 4*a(n-1) - 4*a(n-2) + a(n-4). (End)
G.f.: x*(3+x)/((1-2*x-x^2)*(1-x)^2). - Jaume Oliver Lafont, Sep 28 2009
Empirical observations (from Superseeker):
(1) if b(n) = a(n) + n then {b(n)} is A048777;
(2) if b(n) = a(n+3) - 3*a(n+2) - 3*a(n+1) + a(n) then {b(n)} is A052542;
(3) if b(n) = a(n+2) - 2*a(n+1) + a(n) then {b(n)} is A001333.
4*a(n) = A002203(n+3) - 8*n - 14. - Eric W. Weisstein, May 02 2017
a(n) = 3*A048776(n-1) + A048776(n-2). - R. J. Mathar, May 12 2019
E.g.f.: (1/2)*exp(x)*(-7-4*x+7*cosh(sqrt(2)*x)+5*sqrt(2)*sinh(sqrt(2)*x)). - Stefano Spezia, Aug 25 2019

A287151 Array read by antidiagonals: T(m,n) = number of nonzero m X n binary arrays with all 1's connected.

Original entry on oeis.org

1, 3, 3, 6, 13, 6, 10, 40, 40, 10, 15, 108, 218, 108, 15, 21, 275, 1126, 1126, 275, 21, 28, 681, 5726, 11506, 5726, 681, 28, 36, 1664, 28992, 116166, 116166, 28992, 1664, 36, 45, 4040, 146642, 1168586, 2301877, 1168586, 146642, 4040, 45, 55, 9779, 741556, 11749134, 45280509, 45280509, 11749134, 741556, 9779, 55

Offset: 1

Andrew Howroyd, May 20 2017

Also the number of connected induced (non-null) subgraphs of the grid graph P_m X P_n.

All rows (or columns) are linear recurrences with constant coefficients and the order of the recurrence of row m is at most 1 + A378941(m+1). At least for columns up to 7, this bound gives the actual order of the recurrence. The second differences of any column give those arrays that touch the top and bottom boundaries and have a recurrence order of 2 less since a finite state machine to enumerate these does not require states for empty rows. The number of states required is also considered in A140662 but does not take symmetry into account. - Andrew Howroyd, Dec 18 2024

			Table starts:
====================================================================
m\n|  1    2      3        4         5           6             7
---|----------------------------------------------------------------
1  |  1    3      6       10        15          21            28 ...
2  |  3   13     40      108       275         681          1664 ...
3  |  6   40    218     1126      5726       28992        146642 ...
4  | 10  108   1126    11506    116166     1168586      11749134 ...
5  | 15  275   5726   116166   2301877    45280509     889477656 ...
6  | 21  681  28992  1168586  45280509  1732082741   66037462454 ...
7  | 28 1664 146642 11749134 889477656 66037462454 4872949974666 ...
...

Andrew Howroyd, Table of n, a(n) for n = 1..435
Eric Weisstein's World of Mathematics, Connected Graph
Eric Weisstein's World of Mathematics, Grid Graph
Eric Weisstein's World of Mathematics, Induced Subgraph

Rows 2..5 are A059020, A059021, A059524, A378940.
Main diagonal is A059525.
Cf. A116469, A140662, A286139, A286189, A292357, A378941.

A059021 Number of 3 X n checkerboards (with at least one red square) in which the set of red squares is edge-connected.

Original entry on oeis.org

0, 6, 40, 218, 1126, 5726, 28992, 146642, 741556, 3749816, 18961450, 95880894, 484833212, 2451616864, 12396892316, 62686360476, 316981037374, 1602852315476, 8105013367472, 40983964057352, 207240288658392

Offset: 0

John W. Layman, Dec 14 2000

Number of nonzero 3 X n binary arrays with all 1's connected. Equivalently, the number of connected (non-null) induced subgraphs in the grid graph P_3 X P_n. - Andrew Howroyd, May 20 2017

Colin Barker, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Connected Graph
Eric Weisstein's World of Mathematics, Grid Graph
Eric Weisstein's World of Mathematics, Vertex-Induced Subgraph
Index entries for linear recurrences with constant coefficients, signature (9,-26,35,-22,-3,16,-9,1).

Row 3 of A287151.

See A059020 for the 2 X n case and A059524 for the 4 X n case.

Table[-7/4 - 3 n/2 - RootSum[-1 + 7 # - #^2 - 6 #^3 + 11 #^4 - 7 #^5 + #^6 &, -60219359 #^n + 44281168 #^(1 + n) + 293383797 #^(2 + n) - 152425571 #^(3 + n) - 51762232 #^(4 + n) + 12785939 #^(5 + n) &]/2083234808, {n, 20}] (* Eric W. Weisstein, Aug 09 2017 *)
LinearRecurrence[{9, -26, 35, -22, -3, 16, -9, 1}, {6, 40, 218, 1126, 5726, 28992, 146642, 741556}, 20] (* Eric W. Weisstein, Aug 09 2017 *)

concat(0, Vec(-2*x*(x^5-4*x^4-3*x^3+7*x^2-7*x+3)/((x-1)^2*(x^6-7*x^5+x^4+6*x^3-11*x^2+7*x-1)) + O(x^100))) \\ Colin Barker, Nov 06 2014

a(n) = 9a(n-1) - 26a(n-2) + 35a(n-3) - 22a(n-4) - 3a(n-5) + 16a(n-6) - 9a(n-7) + a(n-8). - David Radcliffe, Jan 19 2001

G.f.: -2*x*(x^5-4*x^4-3*x^3+7*x^2-7*x+3) / ((x-1)^2*(x^6-7*x^5+x^4+6*x^3-11*x^2+7*x-1)). - Colin Barker, Nov 06 2014

A378941 Number of Motzkin paths of length n up to reversal.

Original entry on oeis.org

1, 1, 2, 3, 7, 13, 32, 70, 179, 435, 1142, 2947, 7889, 21051, 57192, 155661, 427795, 1179451, 3271214, 9102665, 25434661, 71282431, 200406472, 564905068, 1596435581, 4521772933, 12835116530, 36504093693, 104012240063, 296871993373, 848694481664, 2429882584254, 6966789756243

Offset: 0

Andrew Howroyd, Dec 17 2024

nonn

A Motzkin path of length n is a path from (0,0) to (n,0) using only steps U = (1,1), H = (1,0) and D = (1,-1). This sequence considers a path and its reversal to be the same. The number of symmetric paths of length 2n (and also 2n+1) is given by A005773(n+1).

a(n) + 1 is an upper bound on the order of the linear recurrence of column n-1 of A287151. At least for columns up to 7, this bound gives the actual order of the recurrence. For example, a(5) = 13 and the order of the recurrence of column 4 (=A059524) is 14.

			The Motzkin paths for a(1)..a(5) are:
a(1) = 1: H;
a(2) = 2: HH, UD;
a(3) = 3: HHH, UHD, HUD=UDH;
a(4) = 7: HHHH, HUDH, UHHD, UUDD, UDUD, HHUD=UDHH, HUHD=UHDH.
a(5) = 13: HHHHH, HUHDH, UHHHD, UUHDD, UDHUD, HHHUD=UDHHH, HHUHD=UHDHH, HHUDH=HUDHH, HUHHD=UHHDH, HUUDD=UUDDH, HUDUD=UDUDH, UHUDD=UUHDD, UHDUD=UPUHD.

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Cf. A001006, A005773, A007123 (similar for Dyck paths), A175954, A185100, A287151, A292357.

Vec(-3/(4*x)-(1+sqrt(1-2*x-3*x^2+O(x^40)))/(4*x^2)+(1+x)/(-1+3*x^2+sqrt(1-2*x^2-3*x^4+O(x^40)))) \\ Thomas Scheuerle, Dec 18 2024

a(n) = (A001006(n) + A005773(floor(1 + n/2))) / 2.

cp's OEIS Frontend

A059020 Number of 2 X n checkerboards (with at least one red square) in which the set of red squares is edge connected.

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A287151 Array read by antidiagonals: T(m,n) = number of nonzero m X n binary arrays with all 1's connected.

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A059021 Number of 3 X n checkerboards (with at least one red square) in which the set of red squares is edge-connected.

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A378941 Number of Motzkin paths of length n up to reversal.

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