A283027 - OEIS

%I #28 Jan 25 2023 02:21:14

%S 0,1,16456,10763361,1073758336,38147070625,705277897416,8308234084801,

%T 70368748374016,463255057977921,2500000025005000,11487432519490081,

%U 46221064831264896,166354152499741921,544488334821309256,1642102089568970625,4611686019501162496

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

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

%H G. C. Greubel, <a href="/A283027/b283027.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_17">Index entries for linear recurrences with constant coefficients</a>, signature (17,-136,680,-2380,6188,-12376,19448,-24310,24310,-19448,12376,-6188,2380,-680,136,-17,1).

%F a(n) = n^4*(n^4 + 1)*(n^8 - n^4 + 2)/4.

%F From _Chai Wah Wu_, Dec 07 2018: (Start)

%F 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.

%F 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)

%F 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

%F a(n) = n^4*(n^12 + n^4 + 2)/4. - _Chai Wah Wu_, Jan 24 2023

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

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

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

%o (PARI) a(n) = n^4*(n^4 + 1)*(n^8 - n^4 + 2)/4; \\ _Indranil Ghosh_, Feb 27 2017

%o (Python) def A283027(n) : return n**4*(n**4 + 1)*(n**8 - n**4 + 2)/4 # _Indranil Ghosh_, Feb 27 2017

%o (Python)

%o def A283027(n): return n**4*(n**4*(n**8+1)+2)>>2 # _Chai Wah Wu_, Jan 24 2023

%o (Magma) [n^4*(n^4+1)*(n^8-n^4+2)/4: n in [0..20]]; // _G. C. Greubel_, Dec 07 2018

%o (Sage) [n^4*(n^4+1)*(n^8-n^4+2)/4 for n in range(20)] # _G. C. Greubel_, Dec 07 2018

%o (GAP) List([0..20], n -> n^4*(n^4+1)*(n^8-n^4+2)/4); # _G. C. Greubel_, Dec 07 2018

%Y Row n=4 of A343095.

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

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

%K nonn,easy

%O 0,3

%A _David Nacin_, Feb 27 2017