A283028 Number of inequivalent 4 X 4 matrices with entries in {1,2,3,...,n} up to vertical and horizontal reflections.

0, 1, 16576, 10766601, 1073790976, 38147265625, 705278736576, 8308236966001, 70368756760576, 463255079498001, 2500000075000000, 11487432626662201, 46221065046245376, 166354152907593001, 544488335559184576, 1642102090850390625, 4611686021648613376

Offset: 0

Author

David Nacin, Feb 27 2017

Keywords

• nonn
• easy

Comments

Cycle index of dihedral group D2 acting on the 16 entries is (3s(2)^8 + s(1)^16)/4.

Examples

			For n=2 we get a(2)=16576 inequivalent 4 X 4 binary matrices up to vertical and horizontal reflections.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (17, -136, 680, -2380, 6188, -12376, 19448, -24310, 24310, -19448, 12376, -6188, 2380, -680, 136, -17, 1).

Crossrefs

Cf. A282612, A282613, A283026, A283027, A283029, A283030, A283031, A283033.

Cf. A283032 (5 X 5 version), A282614 (3 X 3 version), A039623 (2 X 2 version).

Programs

GAP

List([0..20], n -> n^8*(n^8+3)/4); # G. C. Greubel, Dec 07 2018

Magma

[n^8*(n^8+3)/4: n in [0..20]]; // G. C. Greubel, Dec 07 2018

Maple

[n^8*(n^8+3)/4$n=0..18]; # Muniru A Asiru, Dec 07 2018

Mathematica

Table[n^8(n^8 + 3)/4,{n,0,30}]

PARI

a(n) = n^8 * (n^8 + 3)/4; \\ Altug Alkan, Feb 27 2017

Python

def A283028(n): return n**8*(n**8 + 3)/4 # Indranil Ghosh, Feb 27 2017

Sage

[n^8*(n^8+3)/4 for n in range(20)] # G. C. Greubel, Dec 07 2018

Formula

a(n) = n^8 * (n^8 + 3)/4.

From Chai Wah Wu, Dec 07 2018: (Start)

a(n) = 17*a(n-1) - 136*a(n-2) + 680*a(n-3) - 2380*a(n-4) + 6188*a(n-5) - 12376*a(n-6) + 19448*a(n-7) - 24310*a(n-8) + 24310*a(n-9) - 19448*a(n-10) + 12376*a(n-11) - 6188*a(n-12) + 2380*a(n-13) - 680*a(n-14) + 136*a(n-15) - 17*a(n-16) + a(n-17) for n > 16.

G.f.: -x*(x + 1)*(x^14 + 16558*x^13 + 10468387*x^12 + 882544028*x^11 + 20463263441*x^10 + 175065686258*x^9 + 626804969739*x^8 + 968894839176*x^7 + 626804969739*x^6 + 175065686258*x^5 + 20463263441*x^4 + 882544028*x^3 + 10468387*x^2 + 16558*x + 1)/(x - 1)^17. (End)

E.g.f.: (1/4)*x*exp(x)*(x^15 + 120*x^14 + 6020*x^13 + 165620*x^12 + 2757118*x^11 + 28936908*x^10 + 193754990*x^9 + 820784250*x^8 + 2141764056*x^7 + 3281882688*x^6 + 2734927356*x^5 + 1096193700*x^4 + 171804004*x^3 + 7144584*x^2 + 33148*x + 4). - Stefano Spezia, Dec 07 2018