This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A361870 #123 Jun 02 2025 16:49:53
%S A361870 2,2,1,2,2,1,2,3,2,1,2,6,6,2,1,2,10,102,22,2,1,2,20,8548,2852288,402,
%T A361870 2,1,2,36,4211744,384307306807269376,6296489398464125698304,1228158,2,
%U A361870 1
%N A361870 Array read by downward antidiagonals: A(n,k) is the number of nonequivalent 2-colorings of the cells of an n-dimensional hypercube with edges k cells long under action of symmetry.
%C A361870 A(0,0) = 2 by the convention that 0^0 = 1, in spirit with A003992.
%C A361870 A(n,2) = A000616(n), because the Boolean functions' truth tables are n-dimensional edge-length-2 hypercubes, and are considered equivalent under action of input permutation (transposition of dimensions) and applications of NOT to inputs (reflection in dimensions) that compose for the hypercube's symmetry group.
%C A361870 A(n,3) is the number of transitions in an isotropic non-totalistic Life-like cellular automaton with an n-dimensional Moore neighborhood.
%C A361870 A(n,k) ~ 2^(k^n)/(n!*2^n) for large k where n >= 1, and large n where k >= 2, and converges from above. Proof: When computing it with the Pólya enumeration theorem (for each action, count colorings of cycles under it instead of cells, average result over actions), the asymptotic form describes the number of states contributed by the identity action. Over k, the values contributed by the other actions are at most O(2^(k^n/2)), so the proportion that they contribute may be made arbitrarily small by choosing a large enough n. Over n, there are no more than n!*2^n such non-identity actions, assuming that they are all of order 2 (as an upper bound). Each one may have no more than a hyperplane (2^k^(n-1)) of fixed points, which (if it is of order 2) will multiply its colorings by 2^(k^(n-1)/2). The ratio of the identity term to the others is at least O(2^k^n/(n!*2^n*2^(k^(n-1)/2))), and its base-2 logarithm, by Stirling's approximation, is O(k^n-n*log(n)-n-k^(n-1)/2), so the 2^(k^n) term will dominate.
%C A361870 A(1,k) is the only row >= 1 that is linear-recurrent over k (and has a rational generating function), all other nontrivial rows and columns grow faster than any linear-recurrent function.
%C A361870 A000120(A(n,k)) is eventually periodic over k if and only if n <= 2.
%H A361870 Natalia L. Skirrow, <a href="/A361870/b361870.txt">Table of n, a(n) for n = 0..59</a>
%H A361870 Adam P. Goucher, <a href="https://cp4space.files.wordpress.com/2012/09/moda-ch01.pdf">Combinatorics I</a>, Mathematical Olympiad Dark Arts (2005), 8-11.
%H A361870 LifeWiki, <a href="https://conwaylife.com/wiki/Isotropic_non-totalistic_rule">Isotropic non-totalistic rule</a>, <a href="https://conwaylife.com/wiki/P%C3%B3lya_enumeration_theorem">Pólya enumeration theorem</a>.
%H A361870 Natalia L. Skirrow, <a href="https://gist.github.com/DroneBetter/90318584a0ad6a0f4d1c418d3da132cd">dimensional_INT_enumerator.py</a> (unreduced version of Python program below, that also finds equations for fixed n).
%F A361870 Where an expression can be simplified by dividing a power of 2's coefficient by 2 and incrementing its exponent by 1, it is left as-is, so that the 2^ can be changed to c^ for general c-colorings.
%F A361870 A(n,2) = A000616(n).
%F A361870 A(0,k) = 2.
%F A361870 Herein, c(x) denotes the ceiling function.
%F A361870 A(1,k) = A005418(k+1) = (2^k + 2^c(k/2))/2.
%F A361870 A(2,k) = A054247(k) = (2^k^2 + 2*2^(k*(k+1)/2) + 2*2^(c(k/2)*k) + 2^c(k^2/2) + 2*2^c(k^2/4))/8.
%F A361870 A(3,k) = (2^k^3 + 3*2^(c(k/2)*k^2) + 9*2^(c(k^2/2)*k) + 2^c(k^3/2) + 6*2^(k^2*(k+1)/2) + 6*2^(c(k^2/4)*k) + 6*2^c(c(k^2/2)*k/2) + 8*2^(k*(k^2+2)/3) + 8*2^c(k*(k^2+2)/6))/48.
%F A361870 A(4,k) = (2^k^4 + 4*2^(c(k/2)*k^3) + 30*2^(c(k^2/2)*k^2) + 16*2^(c(k^3/2)*k) + 2^c(k^4/2) + 12*2^(k^3*(k+1)/2) + 12*2^(c(k^2/4)*k^2) + 48*2^(c(c(k^2/2)*k/2)*k) + 12*2^c((c(k^2/2)*k^2)/2) + 32*2^(k^2*(k^2+2)/3) + 32*2^(c(k/2)*k*(k^2+2)/3) + 32*2^(c((k*(k^2-1)/3+k)/2)*k) + 32*2^c((k^2*(k^2+2)/3)/2) + 12*2^(k^2*(k^2+1)/2) + 12*2^c(k^4/4) + 48*2^(k*(k^3+k+2)/4) + 48*2^c(k^4/8))/384.
%e A361870 n\k|0, 1, 2, 3, 4, 5, 6, 7, ...
%e A361870 ---+------------------------------------------------------------------------------
%e A361870 0 |2, 2, 2, 2, 2, 2, 2, 2, ...
%e A361870 1 |1, 2, 3, 6, 10, 20, 36, ...
%e A361870 2 |1, 2, 6, 102, 8548, 4211744, ...
%e A361870 3 |1, 2, 22, 2852288, 384307306807269376, ...
%e A361870 4 |1, 2, 402, 6296489398464125698304, ...
%e A361870 5 |1, 2, 1228158, ...
%e A361870 6 |1, 2, ...
%e A361870 7 |1, ...
%e A361870 ...
%o A361870 (Python)
%o A361870 from functools import reduce
%o A361870 from itertools import accumulate
%o A361870 from math import isqrt,lcm,factorial as fact
%o A361870 tap=lambda f,*i:tuple(map(f,*i))
%o A361870 redumulate=lambda f,l,i=None: accumulate(l,f,initial=i)
%o A361870 expumulate=lambda f,l: lambda i: accumulate(range(l),lambda x,i: f(x),initial=i)
%o A361870 factorise=lambda m: tuple(filter(lambda n: not m%n,range(1,m//2+1)))+(m,)
%o A361870 def cycleLengths(dims,size):
%o A361870 convert=(lambda m,i,a: (lambda d,n,i: ('('*bool(i)+str(size)+'+~(')*n+a+("//"+str(size**d))*bool(d)+('%'+str(size))*(d<dims-1)+')'*n+(')'*n+'*'+str(size**i))*bool(i))(*m[i],i))
%o A361870 boardCells=size**dims
%o A361870 matrindex=(lambda p: sum((e-sum(e>d for d,m in p[:i]))*fact(len(p)+~i)*2**dims+2**i*n for i,(e,n) in enumerate(p)))
%o A361870 matrices=sorted((tuple((j,n>>i&1) for i,j in enumerate((lambda t: tuple(reduce(lambda e,n: e+(e>=n),t[i-1::-1],e)%dims for i,e in enumerate(t)))(tuple(a[0]%(dims-i) for i,a in enumerate(redumulate(lambda m,i: divmod(m[0],i),range(dims,1,-1),(m,0))))))) for m in range(fact(dims)) for n in range(2**dims)),key=matrindex) if dims else []
%o A361870 exps=tap(lambda m: tap(matrindex,expumulate(lambda i: tap(lambda j: (lambda k,l: (k,l^j[1]))(*i[j[0]]),m),lcm(4,fact(dims)))(matrices[0])),matrices)
%o A361870 lambdas=tap(lambda m: eval("lambda s: ["+','.join('s['+str(eval('+'.join(map(lambda i: convert(m,i,str(j)),range(dims)))))+']' for j in range(boardCells))+']'),matrices)
%o A361870 test=list(range(1,boardCells+1))
%o A361870 factors=tap(lambda e: factorise(e[1:].index(0)+1),exps)
%o A361870 subperiods=tuple(tuple(sum(map(int.__eq__,test,lambdas[exps[i][a]](test))) for a in f) for i,f in enumerate(factors))
%o A361870 return((lambda t: tap(lambda t: reduce(lambda r,t: r+((t[0],t[1]-sum(i[1] for i in r if not t[0]%i[0])),),t,()),t))(tap(lambda a,b: tuple(zip(a,b)),factors,subperiods)))
%o A361870 specific=(lambda cycles: int(bool(cycles) and sum(2**sum(i[1]//i[0] for i in c) for c in cycles)//len(cycles)))
%o A361870 line=lambda k: (1,2)[k] if k<2 else 1<<k-1|1<<(k-1>>1)
%o A361870 A054247=lambda n: (1,2,6)[n] if n<3 else 1<<n**2-3|2**(n+3>>1)+3**(~n&1)<<(n**2-5>>1)|1<<(n**2-5>>2)
%o A361870 cube=lambda k: (2**k**3+3*2**((k+1>>1)*k**2)+9*2**((k**2+1>>1)*k)+2**(k**3+1>>1)+6*2**(k**2*(k+1)>>1)+6*2**((k**2+3)//4*k)+6*2**((k**2+1>>1)*k+1>>1)+8*2**(k*(k**2+2)//3)+8*2**(k*(k**2+2)//3+1>>1))//48
%o A361870 tesseract=lambda k: (2**k**4+4*2**((k+1>>1)*k**3)+30*2**((k**2+1>>1)*k**2)+16*2**((k**3+1>>1)*k)+2**(k**4+1>>1)+12*2**(k**3*(k+1)>>1)+12*2**((k**2+3>>2)*k**2)+48*2**(((k**2+1>>1)*k+1>>1)*k)+12*2**((k**2+1>>1)*k**2+1>>1)+32*2**(k**2*(k**2+2)//3)+32*2**((k+1>>1)*k*(k**2+2)//3)+32*2**((k*(k**2-1)//3+k+1>>1)*k)+32*2**(k**2*(k**2+2)//3+1>>1)+12*2**(k**2*(k**2+1)>>1)+12*2**(k**4+3>>2)+48*2**(k*(k**3+k+2)>>2)+48*2**(k**4+7>>3))//384
%o A361870 nonequivalents=lambda n,k: (lambda k: 2,line,A054247,cube,tesseract)[n](k) if n<5 else 2**k if k<2 else specific(cycleLengths(n,k))
%o A361870 A002262=(lambda n: (lambda s: (lambda o: (o,s-o))(n-s*(s+1)//2))(isqrt((n<<3)+1)-1>>1))
%o A361870 print(tuple(map(lambda n: nonequivalents(*A002262(n)),range(28)))) # _Natalia L. Skirrow_, May 29 2023
%Y A361870 Cf. A000616, A005418, A054247, A003992.
%K A361870 tabl,nonn
%O A361870 0,1
%A A361870 _Natalia L. Skirrow_, May 28 2023