A225972 -id:A225972 - cp's OEIS frontend

A226048 Irregular triangle read by rows: T(n,k) is the number of binary pattern classes in the (2,n)-rectangular grid with k '1's and (2n-k) '0's: two patterns are in same class if one of them can be obtained by a reflection or 180-degree rotation of the other.

1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 6, 6, 6, 2, 1, 1, 2, 10, 14, 22, 14, 10, 2, 1, 1, 3, 15, 32, 60, 66, 60, 32, 15, 3, 1, 1, 3, 21, 55, 135, 198, 246, 198, 135, 55, 21, 3, 1, 1, 4, 28, 94, 266, 508, 777, 868, 777, 508, 266, 94, 28, 4, 1, 1, 4, 36

Offset: 0

Yosu Yurramendi, May 24 2013

Sum of rows (see example) gives A225826.
This triangle is to A225826 as Losanitsch's triangle A034851 is to A005418.
By columns:
T(n,1) is A004526.
T(n,2) is A000217.
T(n,3) is A225972.
T(n,4) is A071239.
T(n,5) is A222715.
T(n,6) is A228581.
T(n,7) is A228582.
T(n,8) is A228583.
Also the number of equivalence classes of ways of placing k 1 X 1 tiles in an n X 2 rectangle under all symmetry operations of the rectangle. - Christopher Hunt Gribble, Feb 16 2014

			n\k 0 1  2   3   4   5   6   7   8   9  10 11 12 13 14
0   1
1   1 1  1
2   1 1  3   1   1
3   1 2  6   6   6   2   1
4   1 2 10  14  22  14  10   2   1
5   1 3 15  32  60  66  60  32  15   3   1
6   1 3 21  55 135 198 246 198 135  55  21  3  1
7   1 4 28  94 266 508 777 868 777 508 266 94 28  4  1
8   1 4 36 140...
...
The length of row n is 2*n+1, so n>= floor((k+1)/2).

Yosu Yurramendi and María Merino, Rows 0..40 of triangle, flattened

Cf. A225826, A005418, A034851.

A226048 := proc(n,k)
    if type(k,'even') then
        binomial(2*n,k) +3*binomial(n,k/2) ;
    else
        binomial(2*n,k) +(1-(-1)^n)*binomial(n-1,(k-1)/2) ;
    end if ;
    %/4 ;
end proc:
seq(seq(A226048(n,k),k=0..2*n),n=0..8) ; # R. J. Mathar, Jun 07 2020

T[n_, k_] := If[EvenQ[k],
   Binomial[2n, k] + 3 Binomial[n, k/2],
   Binomial[2n, k] + (1-(-1)^n) Binomial[n-1, (k-1)/2]]/4;
Table[T[n, k], {n, 0, 8}, { k, 0, 2n}] // Flatten (* Jean-François Alcover, May 05 2023 *)

If k even, 4*T(n,k) = binomial(2*n,k) +3*binomial(n,k/2). - Yosu Yurramendi, María Merino, Aug 25 2013

If k odd, 4*T(n,k) = 4*T(n,k) = binomial(2*n,k) +(1-(-1)^n)*binomial(n-1,(k-1)/2). - Yosu Yurramendi, María Merino, Aug 25 2013 [corrected by Christian Barrientos, Jun 14 2018]

Definition corrected by María Merino, May 19 2017

A248011 Table T(n,k), n>=1, k>=1, read by antidiagonals: T(n,k) = number of equivalence classes of ways of placing three 1 X 1 tiles in an n X k rectangle under all symmetry operations of the rectangle.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 6, 6, 2, 6, 14, 27, 14, 6, 10, 32, 60, 60, 32, 10, 19, 55, 129, 140, 129, 55, 19, 28, 94, 218, 294, 294, 218, 94, 28, 44, 140, 363, 506, 608, 506, 363, 140, 44, 60, 208, 536, 832, 1038, 1038, 832, 536, 208, 60, 85, 285, 785, 1240, 1695

Offset: 1

Christopher Hunt Gribble, Sep 29 2014

			T(n,k) for 1<=n<=9 and 1<=k<=9 is:
   k    1     2     3     4     5     6     7     8     9 ...
n
1       0     0     1     2     6    10    19    28    44
2       0     1     6    14    32    55    94   140   208
3       1     6    27    60   129   218   363   536   785
4       2    14    60   140   294   506   832  1240  1802
5       6    32   129   294   608  1038  1695  2516  3642
6      10    55   218   506  1038  1785  2902  4324  6242
7      19    94   363   832  1695  2902  4703  6992 10075
8      28   140   536  1240  2516  4324  6992 10416 14988
9      44   208   785  1802  3642  6242 10075 14988 21544

Christopher Hunt Gribble, Table of n, a(n) for n = 1..9870

Cf. A034851, A226048, A226290, A225812, A228022, A228165, A228166, A243866, A006918, A244306, A244307, A248016, A248059, A248060, A248017, A248027.

b := proc (n::integer, k::integer)::integer;
(4*k^3*n^3 - 12*k^2*n^2 + 2*k^3 + 6*k^2*n + 6*k*n^2 + 2*n^3 - 12*k^2 + 11*k*n - 12*n^2 + 4*k + 4*n - 3 - (2*k^3 + 6*k^2*n - 12*k^2 + 3*k*n + 4*k - 3)*(-1)^n - (6*k*n^2 + 2*n^3 + 3*k*n - 12*n^2 + 4*n - 3)*(-1)^k + (3*k*n - 3)*(-1)^k*(-1)^n)*(1/96);
end proc;
f := seq(seq(b(n, k - n + 1), n = 1 .. k), k = 1 .. 140);

Empirically,
T(n,k) = (4*k^3*n^3 - 12*k^2*n^2 + 2*k^3 + 6*k^2*n + 6*k*n^2 + 2*n^3 - 12*k^2 + 11*k*n - 12*n^2 + 4*k + 4*n - 3 - (2*k^3 + 6*k^2*n - 12*k^2 + 3*k*n + 4*k - 3)*(-1)^n - (6*k*n^2 + 2*n^3 + 3*k*n - 12*n^2 + 4*n - 3)*(-1)^k + (3*k*n - 3)*(-1)^k*(-1)^n)/96;
T(1,k) = A005993(k-3) = (k-1)*(2*(k-2)*k + 3*(1-(-1)^k))/24;
T(2,k) = A225972(k) = (k-1)*(2*k*(2*k-1) + 3*(1-(-1)^k))/12;
T(2,k) - T(1,k) = A199771(k-1) and A212561(k) = (k-1)*(6*k^2 + 3*(1-(-1)^k))/24.

Terms corrected and extended by Christopher Hunt Gribble, Apr 01 2015

A289179 Edge count of the n X n white bishop graph.

Original entry on oeis.org

0, 1, 4, 14, 28, 55, 88, 140, 200, 285, 380, 506, 644, 819, 1008, 1240, 1488, 1785, 2100, 2470, 2860, 3311, 3784, 4324, 4888, 5525, 6188, 6930, 7700, 8555, 9440, 10416, 11424, 12529, 13668, 14910, 16188, 17575, 19000, 20540, 22120, 23821, 25564, 27434, 29348, 31395

Offset: 1

Eric W. Weisstein, Jun 27 2017

nonn

Sequence extended to a(1) using formula.

Eric Weisstein's World of Mathematics, Edge Count
Eric Weisstein's World of Mathematics, White Bishop Graph
Index entries for linear recurrences with constant coefficients, signature (2, 1, -4, 1, 2, -1).

Cf. A225972 (black bishop graph edge count).

Table[(n - 1) (4 n^2 - 2 n - 3 + 3 (-1)^n)/12, {n, 20}]
LinearRecurrence[{2, 1, -4, 1, 2, -1}, {0, 1, 4, 14, 28, 55}, 20]
CoefficientList[Series[x(x + 2 x^2+ 5 x^3)/((-1 + x)^4 (1 + x)^2), {x, 0, 20}], x] (* Corrected by Georg Fischer, May 19 2019 *)

a(n) = ((-1 + n)*(-3 + 3*(-1)^n - 2*n + 4*n^2))/12.
a(n) = 2*a(n-1)+a(n-2)-4*a(n-3)+a(n-4)+2*a(n-5)-a(n-6).
G.f. = x*(x + 2x^2 + 5x^3)/((-1 + x)^4*(1 + x)^2). [Corrected by Georg Fischer, May 19 2019]

cp's OEIS Frontend

A226048 Irregular triangle read by rows: T(n,k) is the number of binary pattern classes in the (2,n)-rectangular grid with k '1's and (2n-k) '0's: two patterns are in same class if one of them can be obtained by a reflection or 180-degree rotation of the other.

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A248011 Table T(n,k), n>=1, k>=1, read by antidiagonals: T(n,k) = number of equivalence classes of ways of placing three 1 X 1 tiles in an n X k rectangle under all symmetry operations of the rectangle.

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Author

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Examples

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Crossrefs

Programs

Maple

Formula

Extensions

A289179 Edge count of the n X n white bishop graph.

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Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula