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A122510 Array T(d,n) = number of integer lattice points inside the d-dimensional hypersphere of radius sqrt(n), read along diagonals. 15
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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).

LINKS

Table of n, a(n) for n=1..72.

Index entries for sequences related to sums of squares

FORMULA

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.

EXAMPLE

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

MAPLE

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;

MATHEMATICA

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-Fran├žois Alcover, Jun 13 2013 *)

CROSSREFS

Cf. A001650, A057655, A117609, A046895.

Sequence in context: A300437 A208607 A159291 * A102662 A142048 A117563

Adjacent sequences:  A122507 A122508 A122509 * A122511 A122512 A122513

KEYWORD

nonn,tabl

AUTHOR

R. J. Mathar, Oct 29 2006, Oct 31 2006

STATUS

approved

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Last modified February 22 18:31 EST 2019. Contains 320400 sequences. (Running on oeis4.)