The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A122141 Array: T(d,n) = number of ways of writing n as a sum of d squares, read by ascending antidiagonals. 29
 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 Wikipedia, Sum of squares function (transposed) 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 *) PROG (Python) from sympy.core.power import isqrt from functools import cache @cache def T(d, n): if n == 0: return 1 if n < 0 or d < 1: return 0 return T(d-1, n) + sum(T(d-1, n-(j**2)) for j in range(1, isqrt(n)+1)) * 2 # Darío Clavijo, Feb 06 2024 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 divides 64). Sequence in context: A309054 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified September 10 17:29 EDT 2024. Contains 375792 sequences. (Running on oeis4.)