A283027 Number of inequivalent 4 X 4 matrices with entries in {1,2,3,...,n} up to rotations.

0, 1, 16456, 10763361, 1073758336, 38147070625, 705277897416, 8308234084801, 70368748374016, 463255057977921, 2500000025005000, 11487432519490081, 46221064831264896, 166354152499741921, 544488334821309256, 1642102089568970625, 4611686019501162496

Offset: 0

Author

David Nacin, Feb 27 2017

Keywords

• nonn
• easy

Comments

Cycle index of cyclic group C4 acting on the 16 entries is (2s(4)^4 + s(2)^8 + s(1)^16)/24.

Examples

			For n=2 we get a(2)=16456 inequivalent 4 X 4 binary matrices up to rotations.
		

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

Row n=4 of A343095.

Cf. A282612, A282614, A283026, A283028, A283029, A283030, A283032, A283033.

Cf. A283031 (5 X 5 version), A282613 (3 X 3 version), A006528 (2 X 2 version).

Programs

GAP

List([0..20], n -> n^4*(n^4+1)*(n^8-n^4+2)/4); # G. C. Greubel, Dec 07 2018

Magma

[n^4*(n^4+1)*(n^8-n^4+2)/4: n in [0..20]]; // G. C. Greubel, Dec 07 2018

Maple

[n^4*(n^4+1)*(n^8-n^4+2)/4$n=0..18]; # Muniru A Asiru, Dec 07 2018

Mathematica

Table[n^4*(n^4 + 1)*(n^8 - n^4 + 2)/4,{n,0,30}]

PARI

a(n) = n^4*(n^4 + 1)*(n^8 - n^4 + 2)/4; \\ Indranil Ghosh, Feb 27 2017

Python

def A283027(n) : return n**4*(n**4 + 1)*(n**8 - n**4 + 2)/4 # Indranil Ghosh, Feb 27 2017

Python

def A283027(n): return n**4*(n**4*(n**8+1)+2)>>2 # Chai Wah Wu, Jan 24 2023

Sage

[n^4*(n^4+1)*(n^8-n^4+2)/4 for n in range(20)] # G. C. Greubel, Dec 07 2018

Formula

a(n) = n^4*(n^4 + 1)*(n^8 - n^4 + 2)/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 + 16438*x^13 + 10467307*x^12 + 882551228*x^11 + 20463257081*x^10 + 175065647018*x^9 + 626805100059*x^8 + 968894657736*x^7 + 626805100059*x^6 + 175065647018*x^5 + 20463257081*x^4 + 882551228*x^3 + 10467307*x^2 + 16438*x + 1)/(x - 1)^17. (End)

E.g.f.: (1/4)*x*(x^15 + 120*x^14 + 6020*x^13 + 165620*x^12 + 2757118*x^11 + 28936908*x^10 + 193754990*x^9 + 820784250*x^8 + 2141764054*x^7 + 3281882632*x^6 + 2734926824*x^5 + 1096191600*x^4 + 171800604*x^3 + 7142664*x^2 + 32908*x + 4)*exp(x). - Stefano Spezia, Dec 07 2018

a(n) = n^4*(n^12 + n^4 + 2)/4. - Chai Wah Wu, Jan 24 2023