A004652 -id:A004652 - cp's OEIS frontend

A277226 Number of inequivalent (modulo C_4 rotations) square n X n grids with squares coming in two colors and four squares have one of the colors.

1, 34, 464, 3182, 14769, 53044, 158976, 416140, 980625, 2124310, 4295376, 8199674, 14907809, 25992232, 43700224, 71167704, 112680801, 173990730, 262690000, 388656070, 564571601, 806527964, 1134722304, 1574255332, 2156041329, 2917838014, 3905408976, 5173826770, 6788930625

Offset: 2

Wolfdieter Lang, Oct 06 2016

See the k=4 column of table A054772(n, k), with more explanations there.

Colin Barker, Table of n, a(n) for n = 2..1000
Index entries for linear recurrences with constant coefficients, signature (6,-12,2,27,-36,0,36,-27,-2,12,-6,1).

Cf. A054772, A000012 (k=0), A004652 (k=1), A212714 (k=2), A275799 (k=3).

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(x^2*(1+28*x+272*x^2+804*x^3+1150*x^4+804*x^5 +272*x^6+28*x^7+x^8)/((1-x)^9*(1+x)^3))); // G. C. Greubel, Oct 22 2018

CoefficientList[Series[x^2*(1+28*x+272*x^2+804*x^3+1150*x^4+804*x^5 +272*x^6+28*x^7+x^8)/((1-x)^9*(1+x)^3), {x, 0, 50}], x] (* G. C. Greubel, Oct 22 2018 *)

Vec(x^2*(1+28*x+272*x^2+804*x^3+1150*x^4+804*x^5+272*x^6+28*x^7 +x^8)/((1-x)^9*(1+x)^3) + O(x^40)) \\ Colin Barker, Oct 16 2016

a(n) = A054772(n, 4) = A054772(n, n^2-4), n >= 2.
From Colin Barker, Oct 09 2016: (Start)
G.f.: x^2*(1+28*x+272*x^2+804*x^3+1150*x^4+804*x^5+272*x^6+28*x^7+x^8) / ((1-x)^9*(1+x)^3).
a(n) = (n^8-6*n^6+14*n^4)/96 for n even.
a(n) = (n^8-6*n^6+14*n^4-6*n^2-3)/96 for n odd. (End)
From Stefan Hollos, Oct 16 2016: (Start)
a(n) = (C(n^2,4) + C(n^2/2,2) + n^2/2)/4 for n even,
a(n) = (C(n^2,4) + C((n^2-1)/2,2) + (n^2-1)/2)/4 for n odd. (End)

A318958 A(n, k) is a square array read in the decreasing antidiagonals, for n >= 0 and k >= 0.

Original entry on oeis.org

0, 0, 0, 0, -1, -1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 2, 3, 2, 2, 0, 1, 3, 3, 4, 3, 3, 0, 3, 4, 6, 6, 7, 6, 6, 0, 2, 5, 6, 8, 8, 9, 8, 8, 0, 4, 6, 9, 10, 12, 12, 13, 12, 12, 0, 3, 7, 9, 12, 13, 15, 15, 16, 15, 15, 0, 5, 8, 12, 14, 17, 18, 20, 20, 21, 20, 20

Offset: 0

Paul Curtz, Sep 06 2018

			The array starts:
[n\k][0,   1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, ...]
[0]   0,   0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0, ... = A000004
[1]   0,  -1,  1,  0,  2,  1,  3,  2,  4,  3,  5,  4, ... = A028242(n-2)
[2]  -1,   0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, ... = A023443(n)
[3]   0,   0,  3,  3,  6,  6,  9,  9, 12, 12, 15, 15, ... = 3*A004526(n)
[4]   0,   2,  4,  6,  8, 10, 12, 14, 16, 18, 20, 22, ... = A005843(n)
[5]   2,   3,  7,  8, 12, 13, 17, 18, 22, 23, 27, 28, ... = A047221(n+1)
[6]   3,   6,  9, 12, 15, 18, 21, 24, 27, 30, 33, 36, ... = A008585(n+1)
[7]   6,   8, 13, 15, 20, 22, 27, 29, 34, 36, 41, 43, ... = A047336(n+2)
[8]   8,  12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, ... = A008586(n+2)
Successive columns: A198442(n-2), A198442(n-1), A004652(n), A198442(n+1), A198442(n+2), A079524(n), ... .
First subdiagonal: 0, 0, 3, 6, ... = A242477(n).
First upperdiagonal: 0, 1, 2, 6, 10, ... = A238377(n-1).
Array written as a triangle:
0;
0,  0;
0, -1, -1;
0,  1,  0, 0;
0,  0,  1, 0, 0;
0,  2,  2, 3, 2, 2;
etc.

Cf. A000004, A004526, A004652, A005843, A008585, A008586, A023443, A028242, A047221, A047336, A079524, A198442, A238377, A242477.

A := proc(n, k) option remember; local h;
h := n -> `if`(n<3, [0, 0, -1][n+1], iquo(n^2-4*n+3, 4));
if k = 0 then h(n) elif k = 1 then h(n+1) else A(n, k-2) + n fi end: # Peter Luschny, Sep 08 2018

h[n_] := If[n < 3, {0, 0, -1}[[n + 1]], Quotient[n^2 - 4 n + 3, 4]];
A[n_, k_] := A[n, k] = If[k == 0, h[n], If[k == 1, h[n+1], A[n, k-2] + n]];
Table[A[n - k, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Jul 22 2019, after Peter Luschny *)

Let h(n) = 0, 0, -1, A198442(1), A198442(2), A198442(3), ... Then A(n, 0) = h(n), A(n, 1) = h(n+1) and A(n, k) = A(n, k-2) + n otherwise.

cp's OEIS Frontend

A277226 Number of inequivalent (modulo C_4 rotations) square n X n grids with squares coming in two colors and four squares have one of the colors.

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