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A122510 Array T(d,n) = number of integer lattice points inside the d-dimensional hypersphere of radius sqrt(n), read by ascending antidiagonals.

%I A122510 #33 Aug 03 2025 08:36:21
%S A122510 1,1,3,1,5,3,1,7,9,3,1,9,19,9,5,1,11,33,27,13,5,1,13,51,65,33,21,5,1,
%T A122510 15,73,131,89,57,21,5,1,17,99,233,221,137,81,21,5,1,19,129,379,485,
%U A122510 333,233,81,25,7,1,21,163,577,953,797,573,297,93,29,7,1,23,201,835,1713,1793
%N A122510 Array T(d,n) = number of integer lattice points inside the d-dimensional hypersphere of radius sqrt(n), read by ascending antidiagonals.
%C A122510 Number of solutions to sum_(i=1,..,d) x[i]^2 <= n, x[i] in Z.
%H A122510 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%F A122510 Recurrence along rows: T(d,n)=T(d,n-1)+A122141(d,n) for n>=1; T(d,n)=sum_{i=0..n} A122141(d,i). Recurrence along columns: cf. A123937.
%e A122510 T(2,2)=9 counts 1 pair (0,0) with sum 0, 4 pairs (-1,0),(1,0),(0,-1),(0,1) with sum 1 and 4 pairs (-1,-1),(-1,1),(1,1),(1,-1) with sum 2.
%e A122510 Array T(d,n) with rows d=1,2,3... and columns n=0,1,2,3.. reads
%e A122510   1  3   3    3    5     5     5     5      5      7      7
%e A122510   1  5   9    9   13    21    21    21     25     29     37
%e A122510   1  7  19   27   33    57    81    81     93    123    147
%e A122510   1  9  33   65   89   137   233   297    321    425    569
%e A122510   1 11  51  131  221   333   573   893   1093   1343   1903
%e A122510   1 13  73  233  485   797  1341  2301   3321   4197   5757
%e A122510   1 15  99  379  953  1793  3081  5449   8893  12435  16859
%e A122510   1 17 129  577 1713  3729  6865 12369  21697  33809  47921
%e A122510   1 19 163  835 2869  7189 14581 27253  49861  84663 129303
%e A122510   1 21 201 1161 4541 12965 29285 58085 110105 198765 327829
%p A122510 T := proc(d,n) local i,cnts ; cnts := 0 ; for i from -trunc(sqrt(n)) to trunc(sqrt(n)) do if n-i^2 >= 0 then if d > 1 then cnts := cnts+T(d-1,n-i^2) ; else cnts := cnts+1 ; fi ; fi ; od ; RETURN(cnts) ; end: for diag from 1 to 14 do for n from 0 to diag-1 do d := diag-n ; printf("%d,",T(d,n)) ; od ; od;
%t A122510 t[d_, n_] := t[d, n] = t[d, n-1] + SquaresR[d, n]; t[d_, 0] = 1; Table[t[d-n, n], {d, 1, 12}, {n, 0, d-1}] // Flatten (* _Jean-François Alcover_, Jun 13 2013 *)
%Y A122510 Rows d=1..10 give A001650, A057655, A117609, A046895, A175360, A175361, A341396, A341397, A341398, A341399.
%Y A122510 Cf. A005408 (column 1), A058331 (column 2), A161712 (column 3), A055426 (column 4), A055427 (column 9)
%K A122510 nonn,tabl
%O A122510 1,3
%A A122510 _R. J. Mathar_, Oct 29 2006, Oct 31 2006