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.
The top left corner array is: 0, 1, 1, 2, 2, 3, 1, 2, 2, 3, 3, ... -1, 0, 0, 1, 1, 2, 0, 1, 1, 2, 2, ... -1, 0, 0, 1, 1, 2, 0, 1, 1, 2, 2, ... -2, -1, -1, 0, 0, 1, -1, 0, 0, 1, 1, ... -2, -1, -1, 0, 0, 1, -1, 0, 0, 1, 1, ... -3, -2, -2, -1, -1, 0, -2, -1, -1, 0, 0, ... -1, 0, 0, 1, 1, 2, 0, 1, 1, 2, 2, ... -2, -1, -1, 0, 0, 1, -1, 0, 0, 1, 1, ... -2, -1, -1, 0, 0, 1, -1, 0, 0, 1, 1, ... -3, -2, -2, -1, -1, 0, -2, -1, -1, 0, 0, ... -3, -2, -2, -1, -1, 0, -2, -1, -1, 0, 0, ... ...
a(25) = n^3*(n+1)*(n+2)/6 = 25^3*(25+1)*(25+2)/6 = 15625*26*27/6 = 15625*13*9 = 1828125.
a[n_] := n^3*(n+1)*(n+2)/6; Array[a, 35, 0] (* Amiram Eldar, Feb 13 2023 *)
median{1, 1/2, 1/3, 1/4} = (1/2 + 1/3)/2 = 7/12, so that a(4) = 12.
A226725:=n->n^(1/2 + (-1)^n/2)*(n + 2^(1/2 + (-1)^n/2))/2: seq(A226725(n), n=1..100); # Wesley Ivan Hurt, Feb 27 2015
Denominator[Table[Median[Table[1/k, {k, n}]], {n, 120}]]
f[n_] := If[ OddQ@ n, Floor[(n + 1)/2], n(n/2 + 1)]; Array[f, 59] (* Robert G. Wilson v, Feb 27 2015 *)
With[{nn=30},Riffle[Range[nn],Table[2n+2n^2,{n,nn}]]] (* Harvey P. Dale, May 26 2019 *)
Riffle[Range[60],LinearRecurrence[{3,-3,1},{4,12,24},60]] (* Harvey P. Dale, Oct 03 2023 *)
Vec(x*(x^2-4*x-1)/((x-1)^3*(x+1)^3) + O(x^100)) \\ Colin Barker, Feb 27 2015
After the three moves a(1..3) = (1, 4, 2), the upper left becomes 1 4 3 ... 0 2 7 ... ... One further move '1' would place the tile = value 1 in its final position (row 2, column 1); then moves 4 and 2 would lead to 4 2 3 ... 1 0 ... ... where 1 and 2 have their final position. (This is called goal (2) in COMMENTS.) However, to achieve goal [2] with also 0 in its initial position (row 1, column 1), one has to proceed differently, see a(4..12). Read as a table whose n-th row completes transposition of the (1+n)-th antidiagonal, the sequence reads: row 1: [1, 4, 2, 5, 8, 12, 7, 3, 4, 2, 5, 1] \\ here antidiagonals 1 & 2 are transposed row 2: [2, 5, 3, 4, 5, 2, 1, 3, 4, 7, 12, 8, 3, 1] \\ here, antidiagonals 1-3 are transposed.
(init(n=4)={M=X=Y=vector(2*(n-1)*n); NM=!X0=Y0=1; B=matrix(n, n, y, x, if( n=(x+y)*(x+y-1)\2-y, X[n]=x; Y[n]=y; n))})(); M349244=Map()/*for memoization*/
move(m/*0 = undo*/)={if(m>0, #M>NM||M=Vec(M,NM+9); M[NM++]=m, m=M[NM]; NM--); B[Y0, X0]=m; [X0, Y0, X[m], Y[m]]= [X[m], Y[m], X0, Y0]; B[Y0, X0]=0}
movelist(L=#B)={setminus( Set([ B[Y0+imag(I^k), X0+real(I^k)] | k<-[0..3], if(k>2, Y0>1, k>1, X0>1, k, Y00, check goal if d=0, if d<0 do DFS with d=1, 2, 3...*/
find(goal, d=-1, L=#B)={if( !d, !for(y=1, #goal, for(x=1, #goal[y], B[y, x]==goal[y][x]||return)), d<0, d=0; until(find(goal, d++, L),); M[NM-d+1..NM], foreach(movelist(), m, move(m); find(goal, d-1, L)&& return(1); move()))}
goal(k, g=[[0]])={for(j=1, k, if(#g[#g]>1, g=concat(g, [[j]]), g[k=j-g[#g][1]]=concat(g[k], j))); g}
A349244_row(r)=iferr(mapget(M349244, r), E, init(r+2); for(k=1, r-1, apply(move, A349244_row(k))); mapput(M349244, r, r=find(goal(r*(r+3)\2))); r)
Starting from the initial configuration (cf. comments), the first possible move "1" means to slide the 1 from row 1, column 2 to the "empty square" 0 at (1,1); then move "4" slides the 4 one up, and move "2" slides the 2 to the right:
0 1 3 ... (1) 1 0 3 ... (4) 1 4 3 ... (2) 1 4 3 ...
2 4 7 ... ==> 2 4 7 ... ==> 2 0 7 ... ==> 0 2 7 ...
... ... ... ... ... ... ... ... ... ... ... ...
The next move, 1, will place that tile in its final position (row 2, column 1):
(1) 0 4 3 ...
==> 1 2 7 ...
... ... ...
Given that the 0 is also in its final position (1,1), this achieves what we call goal [1]. Now further moves 4 and 2 would move the 2 in its final position (1,2), so {1, 2} are in their final position, but 0 isn't. (This is called goal (2) in comments.)
However, to achieve goal [2] with also 0 in its initial position (row 1, column 1), in a minimum number of moves, one has to proceed differently: see a(4..18).
Formatted as an irregular table with rows ending with achieved goals [1], [2], [3], ... the sequence reads:
row 1: [1, 4, 2, 1] \\ here {0, 1} are at their final position
row 2: [4, 2, 1, 4, 2, 3, 7, 12, 8, 5, 4, 1, 3, 2] \\ here {0, 1, 2} are "done"
row 3: [1, 3, 5, 4, 3, 1] \\ here {0, 1, 2, 3} are in their final position
row 4: [2, 5, 4, 8, 12, 7, 5, 2] \\ now {0, ..., 4} are in their final position
row 5: [1, 3, 8, 13, 9, 8, 3, 1] \\ now {0, ..., 5} are in their final position
row 6: [2, 5, 6, 11, 7, 6, 5, 2, 1, 4, 6, 12, 13, 6, 4, 3, 6, 9, 8, 6, 3, 1]
row 7: [2, 4, 12, 7, 17, 13, 7, 12, 9, 7, 12, 9, 4, 2]
row 8: [1, 3, 7, 8, 18, 12, 8, 18, 12, 24, 13, 17, 9, 8, 18, 7, 3, 1]
row 9: [2, 4, 8, 18, 17, 23, 16, 10, 11, 9, 18, 8, 4, 2]
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