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%I #14 Feb 06 2024 16:22:55
%S 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 15,73,131,89,57,21,5,1,17,99,233,221,137,81,21,5,1,19,129,379,485,
%U 333,233,81,25,7,1,21,163,577,953,797,573,297,93,29,7,1,23,201,835,1713,1793
%N Array T(d,n) = number of integer lattice points inside the d-dimensional hypersphere of radius sqrt(n), read by ascending antidiagonals.
%C Number of solutions to sum_(i=1,..,d) x[i]^2 <= n, x[i] in Z. T(1,n)=A001650(n+1); T(2,n)=A057655(n); T(3,n)=A117609(n); T(4,n)=A046895(n); T(d,1)=A005408(d); T(d,2)=A058331(d).
%H <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%F 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 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 Array T(d,n) with rows d=1,2,3... and columns n=0,1,2,3.. reads
%e 1 3 3 3 5 5 5 5 5 7 7
%e 1 5 9 9 13 21 21 21 25 29 37
%e 1 7 19 27 33 57 81 81 93 123 147
%e 1 9 33 65 89 137 233 297 321 425 569
%e 1 11 51 131 221 333 573 893 1093 1343 1903
%e 1 13 73 233 485 797 1341 2301 3321 4197 5757
%e 1 15 99 379 953 1793 3081 5449 8893 12435 16859
%e 1 17 129 577 1713 3729 6865 12369 21697 33809 47921
%e 1 19 163 835 2869 7189 14581 27253 49861 84663 129303
%e 1 21 201 1161 4541 12965 29285 58085 110105 198765 327829
%p 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 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 Cf. A001650, A057655, A117609, A046895.
%K nonn,tabl
%O 1,3
%A _R. J. Mathar_, Oct 29 2006, Oct 31 2006
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