A282613 Number of inequivalent 3 X 3 matrices with entries in {1,2,3,..,n} up to rotations.

0, 1, 140, 4995, 65824, 489125, 2521476, 10092775, 33562880, 96870249, 250025500, 589527851, 1290008160, 2651218765, 5165397524, 9611031375, 17180133376, 29647326545, 49590297900, 80672546899, 128000804000, 198571037301, 301818598180, 450289780535

Offset

0,3

Comments

Cycle index of symmetry group (cyclic rotation group of order 4 acting on the 9 cells of the square) is (2s(4)^2*s(1) + s(2)^4*s(1) + s(1)^9)/4.

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Formula

a(n) = n^3*(n^2+1)*(n^4-n^2+2)/4.

G.f.: x*(1 + 130*x + 3640*x^2 + 22054*x^3 + 39070*x^4 + 22054*x^5 + 3640*x^6 + 130*x^7 + x^8) / (1 - x)^10. - Colin Barker, Feb 23 2017

Example

The number of 3 X 3 binary matrices up to rotations is 140.

Mathematica

Table[(2n^3+n^5+n^9)/4, {n, 0, 24}]

Prog

(PARI) concat(0, Vec(x*(1 + 130*x + 3640*x^2 + 22054*x^3 + 39070*x^4 + 22054*x^5 + 3640*x^6 + 130*x^7 + x^8) / (1 - x)^10 + O(x^30))) \\ Colin Barker, Feb 23 2017

Crossrefs

Row n=3 of A343095.

Cf. A282612, A282614, A217331, A168555.

Cf. A006528 (2 x 2 version), A283027 (4 X 4 version).

Sequence in context: A128193 A061607 A223349 * A202786 A035820 A075915

Adjacent sequences: A282610 A282611 A282612 * A282614 A282615 A282616

Keyword

nonn,easy

Author

David Nacin, Feb 19 2017

Status

approved