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A122141 Array: T(d,n) = number of ways of writing n as a sum of d squares, read along diagonals. 26
1, 1, 2, 1, 4, 0, 1, 6, 4, 0, 1, 8, 12, 0, 2, 1, 10, 24, 8, 4, 0, 1, 12, 40, 32, 6, 8, 0, 1, 14, 60, 80, 24, 24, 0, 0, 1, 16, 84, 160, 90, 48, 24, 0, 0, 1, 18, 112, 280, 252, 112, 96, 0, 4, 2, 1, 20, 144, 448, 574, 312, 240, 64, 12, 4, 0, 1, 22, 180, 672, 1136, 840, 544, 320, 24, 30, 8, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

This is the transpose of the array in A286815.

T(d,n) is divisible by 2d for any n != 0 iff d is a power of 2. - Jianing Song, Sep 05 2018

LINKS

Alois P. Heinz, Antidiagonals d = 1..141, flattened

Index entries for sequences related to sums of squares

FORMULA

T(n,n) = A066535(n). - Alois P. Heinz, Jul 16 2014

EXAMPLE

Array T(d,n) with rows d = 1,2,3... and columns n = 0,1,2,3.. reads

1 2 0 0 2 0 0 0 0 2 0 ...

1 4 4 0 4 8 0 0 4 4 8 ...

1 6 12 8 6 24 24 0 12 30 24 ...

1 8 24 32 24 48 96 64 24 104 144 ...

1 10 40 80 90 112 240 320 200 250 560 ...

1 12 60 160 252 312 544 960 1020 876 1560 ...

1 14 84 280 574 840 1288 2368 3444 3542 4424 ...

1 16 112 448 1136 2016 3136 5504 9328 12112 14112 ...

1 18 144 672 2034 4320 7392 12672 22608 34802 44640 ...

1 20 180 960 3380 8424 16320 28800 52020 88660 129064

MAPLE

A122141 := 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+procname(d-1, n-i^2) ; elif n-i^2 = 0 then cnts := cnts+1 ; fi ; fi ; od ; cnts ;

end:

for diag from 1 to 14 do for n from 0 to diag-1 do d := diag-n ; printf("%d, ", A122141(d, n)) ; od ; od;

# second Maple program:

A:= proc(d, n) option remember; `if`(n=0, 1, `if`(n<0 or d<1, 0,

      A(d-1, n) +2*add(A(d-1, n-j^2), j=1..isqrt(n))))

    end:

seq(seq(A(h-n, n), n=0..h-1), h=1..14); # Alois P. Heinz, Jul 16 2014

MATHEMATICA

Table[ SquaresR[d - n, n], {d, 1, 12}, {n, 0, d - 1}] // Flatten (* Jean-François Alcover, Jun 13 2013 *)

A[d_, n_] := A[d, n] = If[n==0, 1, If[n<0 || d<1, 0, A[d-1, n] + 2*Sum[A[d-1, n-j^2], {j, 1, Sqrt[n]}]]]; Table[A[h-n, n], {h, 1, 14}, {n, 0, h-1}] // Flatten (* Jean-François Alcover, Feb 28 2018, after Alois P. Heinz *)

CROSSREFS

Cf. A066535, A286815.

Cf. A000122 (1st row), A004018 (2nd row), A005875 (3rd row), A000118 (4th row), A000132 (5th row), A000141 (6th row), A008451 (7th row), A000143 (8th row), A008452 (9th row), A000144 (10th row), A008453 (11th row), A000145 (12th row), A276285 (13th row), A276286 (14th row), A276287 (15th row), A000152 (16th row).

Cf. A005843 (2nd column), A046092 (3rd column), A130809 (4th column).

Cf. A010052 (1st row divides 2), A002654 (2nd row divides 4), A046897 (4th row divides 8) , A008457 (8th row divides 16), A302855 (16th row divides 32), A302857 (32nd row divdes 64).

Sequence in context: A016585 A143316 A127192 * A091604 A200192 A137629

Adjacent sequences:  A122138 A122139 A122140 * A122142 A122143 A122144

KEYWORD

nonn,tabl

AUTHOR

R. J. Mathar, Oct 29 2006

STATUS

approved

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Last modified December 17 12:05 EST 2018. Contains 318201 sequences. (Running on oeis4.)