Sum_{k>=0}(A(n,2*k)) = (n^6 - 19*n^4 + 27*n^3 + 93*n^2 - 242*n + 112 + (-n^3+7*n^2-18*n+16)*(-1)^n)/8. (losing/checkmated positions)
Sum_{k>=0}(A(n,2*k+1)) = (n-2)*(n-1)*(3*n^4 + 3*n^3 - 22*n^2 + 19*n - 15 + (3-n)*(-1)^n)/24. (winning positions)
Sum_{k>=0}(A(n,k)) = (6*n^6 - 6*n^5 - 82*n^4 + 172*n^3 + 163*n^2 - 643*n + 306 + (2-n)*(4*n^2 - 19*n + 27)*(-1)^n)/24.
Note that for n>=3, there are n+1 nonequivalent positions that are stalemates, each one with the king to move at a1 (under symmetry), 2 with the rook on b2 and its king on c2 or c3, and n-1 with the other at c1 and the rook in one of the squares on the second rank and file >= b.
There are also (n-2)*(4*n^3 + 2*n^2 - 43*n + 31 + (3-n)*(-1)^n)/4 positions in which the rook can be immediately captured (making it unwinnable) that are not included in the sequence, and
A357723(n) in the resulting KvK endgame.
The winning positions are those in which the side with the rook is to move, but to get the number in which the side without the rook is to, adding the losing and stalemated positions yields (n-2)*(n^5 + 2*n^4 - 15*n^3 - 3*n^2 + 87*n - 60 - (n^2 - 5*n + 8)*(-1)^n)/8, then adding yields (n-2)*(n-1)*(n^4 + 3*n^3 - 4*n^2 - 3*n - 2 + (2-n)*(-1)^n)/8, so the equations for each side to move both have roots at n=1 and n=2.
In total, there are (n-2)*(n-1)*(6*n^4 + 12*n^3 - 34*n^2 + 10*n - 21 + (-4*n+9)*(-1)^n)/24 nonequivalent positions in the KRvK endgame.
Sum_{k>=0}(A(n,k)-A(n-1,k)) = (18*n^5 - 60*n^4 - 74*n^3 + 429*n^2 - 226*n - 282 + (-4*n^3 + 33*n^2 - 98*n + 102)*(-1)^n)/12.
.
The values of n at which each value of k begins to abide by a polynomial are ((2,1,2,4,6,5,5,6,8,7)[k] if k<10 else (floor(k/2)*2 =
A052928(k))). This value can be changed (reduced for k>=10) to ((2,1,1,4,6,5,6,5,7,7,7,8,9,9,11,11,12,13,14,15)[k] if k<20 else k-5) by adding the output of the following function to each polynomial at each value (offsetting only 5-k%2 values):
define o(n,k):
.if k is odd:
..if n=k-2:
...add (n-1)/2-3
..else if n=k-3:
...add n-4
..else if n=k-4:
...add n^2-n-15
..else if n=k-5:
...add 3*n^2-(9*n+23)/2
.else:
..if k-n=1:
...add (n-1)/2-1
..else if n=k-2:
...add n-1
..else if n=k-3:
...add 4*n-15
..else if n=k-4:
...add 8*n-35
..else if n=k-5:
...add (25*n-137)/2
.
A(n, k) = 0 if n<3 or k>2*
A225552(n).
The sequence measures distance to mate, as is convention in chess (so A(n,0) counts positions in checkmate), however if kings were to be allowed to be captured, the number of positions after a move into check from a position in checkmate (usually considered illegal) would be
A(n,-1) = (n-1)*(6*n^4 + 22*n^3 - 60*n^2 + 59*n - 24 + (n-2*n^2)*(-1)^n)/24
(note that (n-1)*(4*n + 1 - (-1)^n)/4 of these cases have the rook captured)
A(n, 0) = 0 if n<2 else (n-2)*(n+1)/2 = n^2/2 - n/2 - 1 =
A000096(n-2).
A(n, 1) = 0 if n<1 else (n-2)*(n-1)*(n+1)/2 = n^3/2 - n^2 - n/2 + 1 =
A077414(n-1).
A(n, 2) = 0 if n<2 else (n-2)*(2*n-3) = 2*n^2 - 7*n + 6 =
A014105(n-2).
A(n, 3) = (0,0,0,5)[n] if n<4 else n^3 + 4*n^2 - 26*n + 27 = m^3 - 94*m/3 + 1793/27, where m = n + 4/3.
A(n, 4) = (0,0,0,9,30,59)[n] if n<6 else (2*n^3 - 5*n + 5 + (3-n)*(-1)^(n))/4
A(n, 5) = (0,0,0,5,40)[n] if n<5 else (6*n^3 - 26*n^2 + 84*n - 135 + (7-2*n)*(-1)^n)/4 = 3*m^3/2 + 209*m/18 - 12109/972 + (m/2-37/36)*(-1)^n, where m = n - 13/9.
A(n, 6) = (0,0,0,4,26)[n] if n<5 else 14*n - 31.
A(n, 7) = (0,0,0,0,30, 87)[n] if n<6 else 2*n^2 + 24*n - 82.
A(n, 8) = (0,0,0,0,31, 79,116)[n] if n<7 else 2*n^2 + 14*n - 42 + o().
A(n, 9) = (0,0,0,0,22,174,310)[n] if n<7 else 11*n^2 - 7*n - 41.
A(n,10) = (0,0,0,0,65,178,262)[n] if n<7 else 7*n^2 + 7*n - 40 + o(n,10).
A(n,11) = (0,0,0,0, 7,180,492, 795)[n] if n<8 else 45*n^2/2 - 73*n/2 - 61 + o(n,11).
A(n,12) = (0,0,0,0,25,206,291, 449, 592)[n] if n<9 else 9*n^2 + 7*n - 64 + o(n,12).
A(n,13) = (0,0,0,0, 2,125,461,1002,1418)[n] if n<9 else (53*n^2 - 75*n)/2 - 58 + o(n,13).
A(n,14) = (0,0,0,0, 6,243,397, 552, 683, 805, 939)[n] if n<11 else (7*n^2 + 141*n)/2 - 160 + o(n,14).
A(n,15) = (0,0,0,0, 0, 49,447,1147,2149,2950,3709)[n] if n<11 else (91*n^2 - 205*n)/2 - 44 + o(n,15).
A(n,16) = (0,0,0,0, 0,136,619, 986,1433,1836,2254,2559)[n] if n<12 else (39*n^2 + 47*n)/2 - 134 + o(n,16).
A(n,17) = (0,0,0,0, 0, 19,473,1303,2514,4166,5703,6973,8306)[n] if n<13 else (139*n^2 - 341*n)/2 - 16 + o(n,17).
A(n,18) = (0,0,0,0, 0, 70,698,1207,1712,2376,3075, 3578, 4197, 4690)[n] if n<14 else (49*n^2 + 97*n)/2 - 179 + o(n,18).
A(n,19) = (0,0,0,0, 0, 2,292,1154,2382,4296,6722, 8563,10691,12530,14445)[n] if n<15 else (171*n^2 - 389*n)/2 - 94 + o(n,19).
A(n,20) = (0,0,0,0, 0, 7,782,1515,1985,2624,3532, 4518, 5466, 6308, 7261)[n] if n<15 else 34*n^2 + 42*n - 302 + o(n,20).
A(n,21) = (0,0,0,0, 0, 0, 96,1124,2565,4341,7182,10310,13031,15307,18270,20921)[n] if n<16 else 106*n^2 - 262*n + 6 + o(n,21).
A(n,22) = (0,0,0,0, 0, 0,313,2116,2854,3322,4186, 5212, 6278, 7313, 8479, 9494,10681)[n] if n<17 else 35*n^2 + 106*n - 358 + o(n,22).
A(n,23) = (0,0,0,0, 0, 0, 13, 832,2691,5011,7578,11884,16461,19759,23062,26187,30474,34401)[n] if n<18 else 133*n^2 - 306*n - 194 + o(n,23).
A(n,24) = (0,0,0,0, 0, 0, 45,2002,3597,5172,6558, 7114, 8891,10534,11965,13623,15618,17470,19443)[n] if n<19 else (109*n^2 + 223*n)/2 - 687 + o(n,24).
A(n,25) = (0,0,0,0, 0, 0, 0, 258,2234,5482,9412,13305,20056,25698,30163,33982,39027,43570,49523,55256) if n<20 else (345*n^2 - 931*n)/2 + 60 + o(n,25).
A(n,26) = (0,0,0,0, 0, 0, 0, 773,4194,6209,8937,10557,12341,13901,15940,18224,20672,23119,25666,28471,31426) if n<21 else 75*n^2 + 73*n - 589 + o(n,26).
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