A025443 -id:A025443

A341040

Number T(n,k) of partitions of n into k distinct nonzero squares; triangle T(n,k), n>=0, 0<=k<=A248509(n), read by rows.

+10
20

1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

0

COMMENTS

T(n,k) is defined for n, k >= 0. The triangle contains only the terms with 0 <= k <= A248509(n). T(n,k) = 0 for k > A248509(n).

LINKS

Alois P. Heinz, Rows n = 0..4000, flattened

FORMULA

T(n,k) = [x^n*y^k] Product_{j>=1} (1 + y*x^(j^2)).

T(A000330(n),n) = 1.

Row n = [0] <=> n in { A001422 }.

Sum_{k>=0} 2^k * T(n,k) = A279360(n).

Sum_{k>=0} k * T(n,k) = A281542(n).

Sum_{k>=0} (-1)^k * T(n,k) = A276516(n).

EXAMPLE

T(62,3) = 2 is the first term > 1 and counts partitions [49,9,4] and [36,25,1].

Triangle T(n,k) begins:

1;

0, 1;

0;

0, 1;

0, 0, 1;

0;

0, 1;

0, 0, 1;

0;

0, 0, 1;

0, 0, 0, 1;

...

MAPLE

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

b(n, i-1)+`if`(i^2>n, 0, expand(b(n-i^2, i-1)*x))))

end:

T:= n->(p->seq(coeff(p, x, i), i=0..max(0, degree(p))))(b(n, isqrt(n))):

seq(T(n), n=0..45);

MATHEMATICA

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0,

b[n, i - 1] + If[i^2 > n, 0, Expand[b[n - i^2, i - 1]*x]]]];

T[n_] := CoefficientList[b[n, Floor@Sqrt[n]], x] /. {} -> {0};

T /@ Range[0, 45] // Flatten (* Jean-François Alcover, Feb 15 2021, after Alois P. Heinz *)

CROSSREFS

Columns k=0-10 give: A000007, A010052 (for n>0), A025441, A025442, A025443, A025444, A340988, A340998, A340999, A341000, A341001.

Row sums give A033461.

Cf. A000290, A000330, A001422, A243148, A248509, A276516, A279360, A281542, A337165.

KEYWORD

nonn,look,tabf

AUTHOR

Alois P. Heinz, Feb 03 2021

STATUS

approved

A004433

Numbers that are the sum of 4 distinct nonzero squares: of form w^2+x^2+y^2+z^2 with 0<w<x<y<z.

+10
17

30, 39, 46, 50, 51, 54, 57, 62, 63, 65, 66, 70, 71, 74, 75, 78, 79, 81, 84, 85, 86, 87, 90, 91, 93, 94, 95, 98, 99, 102, 105, 106, 107, 109, 110, 111, 113, 114, 116, 117, 118, 119, 120, 121, 122, 123, 125, 126, 127, 129, 130, 131, 133, 134, 135, 137

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

Index entries for sequences related to sums of squares

FORMULA

{n: A025443(n) >=1}. Union of A025386 and A025376. - R. J. Mathar, Jun 15 2018

EXAMPLE

30 = 1^2+2^2+3^2+4^2.

MATHEMATICA

data = Flatten[ DeleteCases[ FindInstance[ w^2 + x^2 + y^2 + z^2 == # && 0 < w < x < y < z < #, {w, x, y, z}, Integers] & /@ Range[137], {}], 1]; w^2 + x^2 + y^2 + z^2 /. data (* Ant King, Oct 17 2010 *)

Select[Union[Total[#^2]&/@Subsets[Range[10], {4}]], #<=137&] (* Harvey P. Dale, Jul 03 2011 *)

PROG

(Haskell)

a004433 n = a004433_list !! (n-1)

a004433_list = filter (p 4 $ tail a000290_list) [1..] where

p k (q:qs) m = k == 0 && m == 0 ||

q <= m && k >= 0 && (p (k - 1) qs (m - q) || p k qs m)

-- Reinhard Zumkeller, Apr 22 2013

(PARI) list(lim)=my(v=List()); lim\=1; for(z=4, sqrtint(lim\4), for(y=3, min(sqrtint((lim-z^2)\3), z-1), for(x=2, min(sqrtint((lim-y^2-z^2)\2), y-1), for(w=1, min(sqrtint(lim-x^2-y^2-z^2), x-1), listput(v, w^2+x^2+y^2+z^2))))); Set(v) \\ Charles R Greathouse IV, Feb 07 2017

CROSSREFS

Cf. A001944, A001995, A003995, A004431, A004432, A004434, A224981, A224982, A224983, A000290.

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

A025417

Least sum of 4 distinct nonzero squares in exactly n ways.

+10
10

30, 90, 78, 142, 126, 174, 190, 210, 286, 351, 294, 318, 270, 330, 430, 390, 414, 550, 450, 582, 594, 546, 462, 606, 510, 862, 786, 726, 770, 714, 690, 630, 702, 846, 810, 750, 1074, 798, 930, 870, 1026, 966, 1246, 1150, 1310, 1190, 1122, 1314, 1338, 1365, 1050

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Donovan Johnson, Table of n, a(n) for n = 1..1000

Index entries for sequences related to sums of squares

FORMULA

{min k: A025443(k) = n}. - R. J. Mathar, Jun 15 2018

KEYWORD

nonn

AUTHOR

David W. Wilson

STATUS

approved

A350405

a(n) is the smallest number which can be represented as the sum of n distinct nonzero n-gonal numbers in exactly n ways, or 0 if no such number exists.

+10
9

37, 142, 285, 536, 911, 1268, 1909, 2713, 3876, 5179, 6891, 8901, 11190, 14384, 18087, 21697, 27055, 32166, 39111, 46560, 53892, 64412, 73949, 86778, 98202, 113635, 130088, 148051, 167505, 190968, 214955, 240143, 269775, 297615, 331201, 367429, 409179, 451340, 497830

(list; graph; refs; listen; history; text; internal format)

OFFSET

3,1

LINKS

David A. Corneth, Table of n, a(n) for n = 3..102

Eric Weisstein's World of Mathematics, Polygonal Number

FORMULA

a(n) >= A006484(n). - David A. Corneth, Dec 30 2021

EXAMPLE

For n = 3: 37 = 1 + 15 + 21 = 3 + 6 + 28 = 6 + 10 + 21.

MATHEMATICA

Do[i=1; While[b=PolygonalNumber[n, Range@i++]; !IntegerQ[t=Min[First/@Select[Tally[Select[Total/@Subsets[b, {n}], #<=Max@b&]], Last@#==n&]]]]; Print@t, {n, 3, 10}] (* Giorgos Kalogeropoulos, Dec 30 2021 *)

CROSSREFS

Cf. A006484, A025443, A057145, A307598, A350207, A350241, A350288.

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Dec 29 2021

EXTENSIONS

a(10)-a(31) from Michael S. Branicky, Dec 29 2021

More terms from David A. Corneth, Dec 30 2021

STATUS

approved

A025444

Number of partitions of n into 5 distinct nonzero squares.

+10
8

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, 1, 0

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,104

LINKS

David A. Corneth, Table of n, a(n) for n = 0..9999

Index entries for sequences related to sums of squares

FORMULA

a(n) = [x^n y^5] Product_{k>=1} (1 + y*x^(k^2)). - Ilya Gutkovskiy, Apr 22 2019

EXAMPLE

a(111) = 2 via 1 + 4 + 9 + 16 + 81 = 1 + 9 + 16 + 36 + 49. - David A. Corneth, Feb 02 2021

MAPLE

From R. J. Mathar, Oct 18 2010: (Start)

A025444aux := proc(n, m, nmax) local a, m, upn, lv ; if m = 1 then if issqr(n) and nmax^2 >= n and n >= 1 then return 1; else return 0; end if; else a := 0 ; for upn from 1 to nmax do lv := n-upn^2 ; if lv <0 then break; end if; a := a + procname(lv, m-1, upn-1) ; end do: return a; end if; end proc:

A025444 := proc(n) A025444aux(n, 5, n) ; end proc: (End)

CROSSREFS

Cf. A000290, A008452, A010052, A025433, A025441, A025442, A025443, A025444, A045851, A340946, A340988, A340998, A340999, A341000, A341001.

Column k=5 of A341040.

KEYWORD

nonn,look

AUTHOR

David W. Wilson

STATUS

approved

A341000

Number of partitions of n into 9 distinct nonzero squares.

+10
7

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 2, 0, 0, 0, 1, 2, 0, 0, 2

(list; graph; refs; listen; history; text; internal format)

OFFSET

285,92

LINKS

David A. Corneth, Table of n, a(n) for n = 285..5000

Index entries for sequences related to sums of squares

EXAMPLE

a(381) = 2 via 1 + 4 + 9 + 16 + 36 + 49 + 64 + 81 + 121 = 1 + 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100. - David A. Corneth, Feb 02 2021

CROSSREFS

Cf. A000290, A008452, A010052, A025433, A025441, A025442, A025443, A025444, A045851, A340946, A340988, A340998, A340999, A341001.

Column k=9 of A341040.

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Feb 02 2021

STATUS

approved

A341001

Number of partitions of n into 10 distinct nonzero squares.

+10
7

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

385,97

LINKS

Alois P. Heinz, Table of n, a(n) for n = 385..20000

Index entries for sequences related to sums of squares

CROSSREFS

Cf. A000144, A000290, A010052, A025434, A025441, A025442, A025443, A025444, A045852, A340947, A340988, A340998, A340999, A341000.

Column k=10 of A341040.

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Feb 02 2021

STATUS

approved

A340988

Number of partitions of n into 6 distinct nonzero squares.

+10
6

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 2, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 3, 2, 0, 1, 2, 1, 1, 1, 0, 3, 0, 0, 4, 1, 0, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

91,46

LINKS

Table of n, a(n) for n=91..190.

Index entries for sequences related to sums of squares

CROSSREFS

Cf. A000141, A000290, A010052, A025430, A025441, A025442, A025443, A025444, A045848, A224981, A340905, A340998, A340999, A341000, A341001.

Column k=6 of A341040.

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Feb 02 2021

STATUS

approved

A340998

Number of partitions of n into 7 distinct nonzero squares.

+10
6

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 2, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 2, 0, 0, 1, 0, 1, 0, 1, 2, 0, 0, 0, 2, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 2, 3, 0, 0, 2, 2

(list; graph; refs; listen; history; text; internal format)

OFFSET

140,61

LINKS

Table of n, a(n) for n=140..240.

Index entries for sequences related to sums of squares

CROSSREFS

Cf. A000290, A008451, A010052, A025431, A025441, A025442, A025443, A025444, A045849, A224982, A340906, A340988, A340999, A341000, A341001.

Column k=7 of A341040.

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Feb 02 2021

STATUS

approved

A340999

Number of partitions of n into 8 distinct nonzero squares.

+10
6

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 0, 1, 0, 0, 2, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 3, 0, 0, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

204,73

LINKS

Table of n, a(n) for n=204..303.

Index entries for sequences related to sums of squares

CROSSREFS

Cf. A000143, A000290, A010052, A025432, A025441, A025442, A025443, A025444, A045850, A224983, A340915, A340988, A340998, A341000, A341001.

Column k=8 of A341040.

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Feb 02 2021

STATUS

approved