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A122510
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Array T(d,n) = number of integer lattice points inside the d-dimensional hypersphere of radius sqrt(n), read by ascending antidiagonals.
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19
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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, 15, 73, 131, 89, 57, 21, 5, 1, 17, 99, 233, 221, 137, 81, 21, 5, 1, 19, 129, 379, 485, 333, 233, 81, 25, 7, 1, 21, 163, 577, 953, 797, 573, 297, 93, 29, 7, 1, 23, 201, 835, 1713, 1793
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OFFSET
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1,3
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COMMENTS
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LINKS
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FORMULA
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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.
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EXAMPLE
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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.
Array T(d,n) with rows d=1,2,3... and columns n=0,1,2,3.. reads
1 3 3 3 5 5 5 5 5 7 7
1 5 9 9 13 21 21 21 25 29 37
1 7 19 27 33 57 81 81 93 123 147
1 9 33 65 89 137 233 297 321 425 569
1 11 51 131 221 333 573 893 1093 1343 1903
1 13 73 233 485 797 1341 2301 3321 4197 5757
1 15 99 379 953 1793 3081 5449 8893 12435 16859
1 17 129 577 1713 3729 6865 12369 21697 33809 47921
1 19 163 835 2869 7189 14581 27253 49861 84663 129303
1 21 201 1161 4541 12965 29285 58085 110105 198765 327829
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MAPLE
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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;
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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