A000925 -id:A000925 - cp's OEIS frontend

A000161 Number of partitions of n into 2 squares.

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

Offset: 0

N. J. A. Sloane

Number of ways of writing n as a sum of 2 (possibly zero) squares when order does not matter.
Number of similar sublattices of square lattice with index n.
Let Pk = the number of partitions of n into k nonzero squares. Then we have A000161 = P0 + P1 + P2, A002635 = P0 + P1 + P2 + P3 + P4, A010052 = P1, A025426 = P2, A025427 = P3, A025428 = P4. - Charles R Greathouse IV, Mar 08 2010, amended by M. F. Hasler, Jan 25 2013
a(A022544(n))=0; a(A001481(n))>0; a(A125022(n))=1; a(A118882(n))>1. - Reinhard Zumkeller, Aug 16 2011

			25 = 3^2+4^2 = 5^2, so a(25) = 2.

J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 339

T. D. Noe, Table of n, a(n) for n = 0..10000
B. K. Agarwala and F. C. Auluck, Statistical mechanics and partitions into non-integral powers of integers, Proc. Camb. Phil. Soc., 47 (1951), 207-216. [Annotated scanned copy]
Henry Bottomley, Illustration of initial terms
R. T. Bumby, Sums of four squares, in Number theory (New York, 1991-1995), 1-8, Springer, New York, 1996.
J. H. Conway, E. M. Rains and N. J. A. Sloane, On the existence of similar sublattices, Canad. J. Math. 51 (1999), 1300-1306 (Abstract, pdf, ps).
Michael Gilleland, Some Self-Similar Integer Sequences
E. Grosswald, Representations of Integers as Sums of Squares, Springer-Verlag, NY, 1985, p. 84.
M. D. Hirschhorn, Some formulas for partitions into squares, Discrete Math, 211 (2000), pp. 225-228. [From _Ant King_, Oct 05 2010]
Index entries for sequences related to sublattices
Index entries for sequences related to sums of squares
Index entries for "core" sequences

Cf. A002654, A001481, A025427, A025428, A063725, A025426, A000290.
Cf. A000925, A247367.
Equivalent sequences for other numbers of squares: A010052 (1), A000164 (3), A002635 (4), A000174 (5).

a000161 n =
   sum $ map (a010052 . (n -)) $ takeWhile (<= n `div` 2) a000290_list
a000161_list = map a000161 [0..]
-- Reinhard Zumkeller, Aug 16 2011

A000161 := proc(n) local i,j,ans; ans := 0; for i from 0 to n do for j from i to n do if i^2+j^2=n then ans := ans+1 fi od od; RETURN(ans); end; [ seq(A000161(i), i=0..50) ];
A000161 := n -> nops( numtheory[sum2sqr](n) ); # M. F. Hasler, Nov 23 2007

Length[PowersRepresentations[ #,2,2]] &/@Range[0,150] (* Ant King, Oct 05 2010 *)

a(n)=sum(i=0,n,sum(j=0,i,if(i^2+j^2-n,0,1))) \\ for illustrative purpose

A000161(n)=sum(k=sqrtint((n-1)\2)+1,sqrtint(n),issquare(n-k^2)) \\ Charles R Greathouse IV, Mar 21 2014, improves earlier code by M. F. Hasler, Nov 23 2007

A000161(n)=#sum2sqr(n) \\ See A133388 for sum2sqr(). - M. F. Hasler, May 13 2018

from math import prod
from sympy import factorint
def A000161(n):
    f = factorint(n)
    return int(not any(e&1 for e in f.values())) + (((m:=prod(1 if p==2 else (e+1 if p&3==1 else (e+1)&1) for p, e in f.items()))+((((~n & n-1).bit_length()&1)<<1)-1 if m&1 else 0))>>1) if n else 1 # Chai Wah Wu, Sep 08 2022

a(n) = card { { a,b } c N | a^2+b^2 = n }. - M. F. Hasler, Nov 23 2007

Let f(n)= the number of divisors of n that are congruent to 1 modulo 4 minus the number of its divisors that are congruent to 3 modulo 4, and define delta(n) to be 1 if n is a perfect square and 0 otherwise. Then a(n)=1/2 (f(n)+delta(n)+delta(1/2 n)). - Ant King, Oct 05 2010

A045847 Matrix whose (i,j)-th entry is number of representations of j as a sum of i squares of nonnegative integers; read by diagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 1, 0, 0, 1, 4, 3, 0, 1, 0, 1, 5, 6, 1, 2, 0, 0, 1, 6, 10, 4, 3, 2, 0, 0, 1, 7, 15, 10, 5, 6, 0, 0, 0, 1, 8, 21, 20, 10, 12, 3, 0, 0, 0, 1, 9, 28, 35, 21, 21, 12, 0, 1, 1, 0, 1, 10, 36, 56, 42, 36, 30, 4, 3, 2, 0, 0, 1, 11, 45, 84, 78, 63, 61, 20, 6, 6, 2, 0, 0

Offset: 0

N. J. A. Sloane

			Rows are
1,0,0,..;
1,1,0,0,1,0..;
1,2,1,0,2,2,..;
1,3,3,1,...

Seiichi Manyama, Ascending antidiagonals n = 0..139, flattened
H. Wilf, A combinatorial determinant, arXiv:math/9809120 [math.CO], 1998.

Rows k=0-10 give: A000007, A010052, A000925, A002102, A014110, A038671, A045848, A045849, A045850, A045851, A045852.
Diagonal gives A287617.
Antidiagonal sums give A302018.

i-th row is expansion of (1+x+x^4+x^9+...)^i.

More terms from Erich Friedman

A014110 Number of ordered ways of writing n as a sum of 4 squares of nonnegative integers.

Original entry on oeis.org

1, 4, 6, 4, 5, 12, 12, 4, 6, 16, 18, 12, 8, 16, 24, 12, 5, 24, 30, 16, 18, 28, 24, 12, 12, 28, 42, 28, 12, 36, 48, 16, 6, 36, 42, 36, 29, 28, 48, 28, 18, 48, 60, 28, 24, 60, 48, 24, 8, 44, 72, 48, 30, 48, 84, 36, 24, 52, 54, 48, 36, 52, 72, 52, 5, 72, 96, 40, 42, 72

Offset: 0

Joe Keane (jgk(AT)jgk.org)

This counts ordered sums of squares of nonnegative integers, whereas A000118 counts ordered sums of squares of integers of any sign. - R. J. Mathar, May 16 2023

			From  _R. J. Mathar_, May 16 2023: (Start)
a(1)=4 counts 0^2+0^2+0^2+1^2 = 0^2+0^2+1^2+0^2 = 0^2+1^2+0^2+0^2 = 1^2+0^2+0^2+0^2.
a(2)=6 counts 0^2+0^2+1^2+1^2 = 0^2+1^2+0^2+1^2 = 0^2+1^2+1^2+0^2 = 1^2+0^2+0^2+1^2 = 1^2+0^2+1^2+0^2 = 1^2+1^2+0^2+0^2. (End)

Convolution square of A000925.

a = Compile[{{n, Integer}}, Block[{c = 0, k, j, i = Floor[Sqrt[n]]}, While[i > -1, j = Floor[Sqrt[n - i^2]]; While[j > -1, k = Floor[Sqrt[n - i^2 - j^2]]; While[k > -1, c += Boole[ Mod[ Sqrt[ n - i^2 - j^2 - k^2], 1] == 0]; k--]; j--]; i--]; c]]; Array[a, 70, 0] - (* _Robert G. Wilson v, Aug 13 2025 *)

Coefficient of q^n in (1/16)*(1 + theta_3(0, q))^4; or coeff. of q^n in (Sum q^(i^2), i=0..inf)^4.

A224212 Number of nonnegative solutions to x^2 + y^2 <= n.

Original entry on oeis.org

1, 3, 4, 4, 6, 8, 8, 8, 9, 11, 13, 13, 13, 15, 15, 15, 17, 19, 20, 20, 22, 22, 22, 22, 22, 26, 28, 28, 28, 30, 30, 30, 31, 31, 33, 33, 35, 37, 37, 37, 39, 41, 41, 41, 41, 43, 43, 43, 43, 45, 48, 48, 50, 52, 52, 52, 52, 52, 54, 54, 54, 56, 56, 56, 58, 62, 62, 62, 64

Offset: 0

Alex Ratushnyak, Apr 01 2013

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A000925 (first differences).
Cf. A000606 (number of nonnegative solutions to x^2 + y^2 + z^2 <= n).
Cf. A057655 (number of integer solutions to x^2 + y^2 <= n).

nn = 68; t = Table[0, {nn}]; Do[d = x^2 + y^2; If[0 < d <= nn, t[[d]]++], {x, 0, nn}, {y, 0, nn}]; Accumulate[Join[{1}, t]] (* T. D. Noe, Apr 01 2013 *)

for n in range(99):
  k = 0
  for x in range(99):
    s = x*x
    if s>n: break
    for y in range(99):
        sy = s + y*y
        if sy>n: break
        k+=1
  print(str(k), end=', ')

G.f.: (1/(1 - x))*(Sum_{k>=0} x^(k^2))^2. - Ilya Gutkovskiy, Mar 14 2017

A240088 The number of ways of writing n as an ordered sum of a triangular number (A000217), a square (A000290) and a pentagonal number (A000326).

Original entry on oeis.org

1, 3, 3, 2, 3, 4, 4, 4, 3, 3, 5, 5, 5, 3, 3, 7, 7, 5, 2, 6, 5, 4, 8, 5, 6, 4, 8, 7, 5, 7, 4, 9, 6, 5, 4, 3, 9, 12, 9, 4, 7, 9, 8, 4, 6, 8, 7, 8, 4, 8, 9, 10, 9, 6, 10, 6, 7, 10, 9, 8, 7, 11, 7, 4, 10, 8, 10, 10, 7, 5, 10, 14, 11, 7, 6, 11, 10, 10, 4, 11, 10, 10, 13, 8, 7, 7, 13, 12, 8, 8, 6, 10, 17, 8, 10, 7, 16, 10, 3, 12, 9

Offset: 0

Robert G. Wilson v, Mar 31 2014

nonn

0 and 1 are triangular numbers, square numbers and pentagonal numbers.
It is conjectured that a(n) is always positive - this is one of the conjectures in Conjecture 1.1 of Sun (2009). - N. J. A. Sloane, Apr 01 2014
Note that both the conjecture in A160325 and the conjecture in A160324 imply that a(n) is always positive. - Zhi-Wei Sun, Apr 01 2014
a(n) > 0 for all n < 10^10. - Robert G. Wilson v, Aug 20 2016
Least number to be represented k ways, k >= 1: 0, 3, 1, 5, 10, 19, 15, 22, 31, 51, 61, 37, 82, 71, 126, 96, 92, 136, 162, 187, 206, 276, 191, 261, 236, 247, 317, 302, 401, 292, 422, 547, 456, 544, 551, 612, 591, 577, 521, 666, 742, 726, 682, 877, 796, 1052, 961, 1046, 1171, 1027, ..., . A275999.
Greatest number (conjectured) to be represented k ways, k >= 1: 0, 18, 168, 78, 243, 130, 553, 455, 515, 658, 865, 945, 633, 1918, 2258, 1385, 1583, 2828, 2135, 2335, 2785, 4533, 3168, 3478, 2790, 3868, 4193, 7328, 4953, 5278, 6390, 8148, 8015, 4585, 9160, 10485, 7613, 12333, 12025, 10178, 9923, 9720, 12558, 11340, 17420, 11753, 14893, 16155, 16415, 14343, ..., .
Conjectured lists of numbers that are represented in k >= 1 ways:
1: 0;
2: 3, 18;
3: 1, 2, 4, 8, 9, 13, 14, 35, 98, 168;
4: 5, 6, 7, 21, 25, 30, 34, 39, 43, 48, 63, 78;
5: 10, 11, 12, 17, 20, 23, 28, 33, 69, 193, 203, 230, 243;
6: 19, 24, 32, 44, 53, 55, 74, 90, 111, 130;
7: 15, 16, 27, 29, 40, 46, 56, 60, 62, 68, 73, 84, 85, 95, 108, 113, 123, 135, 139, 163, 165, 273, 553;
8: 22, 26, 42, 45, 47, 49, 59, 65, 83, 88, 89, 93, 112, 119, 125, 134, 140, 144, 186, 205, 233, 244, 320, 405, 455;
9: 31, 36, 38, 41, 50, 52, 58, 100, 109, 124, 160, 214, 249, 308, 358, 515; ..., .

Robert G. Wilson v and Robert Israel, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, On universal sums of polygonal numbers, Science China Mathematics, Vol. 58, No. 7 (2015), 1367-1396; arXiv:0905.0635 [math.NT], 2009-2015 [Edited, _Felix Fröhlich_, Aug 24 2016].
Zhi-Wei Sun, Various new conjectures involving polygonal numbers and primes, a message to the Number Theory List, May 8 2009.

Cf. A000925, A008443, A101428, A115171, A115172, A115173, A115174, A115175, A115176, A115177, A144642.

Cf. A160324, A160325, A160326. - Zhi-Wei Sun, Apr 01 2014

# requires Maple 17 and up
with(SignalProcessing):
N:= 10000;  # to get terms up to a(N)
A:= Array(0..N,datatype=float);
B:= Array(0..N,datatype=float);
C:= Array(0..N,datatype=float);
for i from 0 to floor(sqrt(N)) do A[i^2]:= 1 od:
for i from 0 to floor((1+sqrt(1+8*N))/2) do B[i*(i-1)/2]:= 1 od:
for i from 0 to floor((1+sqrt(1+24*N))/6) do C[i*(3*i-1)/2]:= 1 od:
R:= Convolution(Convolution(A,B),C);
R:= evalhf(map(round,R));
# Note that a(i) = R[i+1] for i from 0 to N
# Robert Israel, Apr 01 2014

p = Table[n (3n - 1)/2, {n, 0, 26}]; s = Table[n^2, {n, 0, 32}]; t = Table[n (n + 1)/2, {n, 0, 45}]; a = Sort@ Flatten@ Table[ p[[i]] + s[[j]] + t[[k]], {i, 26}, {j, 32}, {k, 45}]; Table[ Count[a, n], {n, 0, 105}]

A283305 List points (x,y) having integer coordinates with x >= 0, y >= 0, sorted first by x^2+y^2 and in case of a tie, by x-coordinate. Sequence gives x-coordinates.

Original entry on oeis.org

0, 0, 1, 1, 0, 2, 1, 2, 2, 0, 3, 1, 3, 2, 3, 0, 4, 1, 4, 3, 2, 4, 0, 3, 4, 5, 1, 5, 2, 5, 4, 3, 5, 0, 6, 1, 6, 2, 6, 4, 5, 3, 6, 0, 7, 1, 5, 7, 4, 6, 2, 7, 3, 7, 5, 6, 0, 8, 1, 4, 7, 8, 2, 8, 6, 3, 8, 5, 7, 4, 8, 0, 9, 1, 9, 2, 6, 7, 9, 5, 8, 3, 9, 4, 9, 7, 0, 6, 8, 10, 1, 10, 2, 10, 5, 9, 3, 10, 7, 8

Offset: 1

N. J. A. Sloane, Mar 04 2017, following a suggestion from Ahmet Arduç

nonn

			The first few points (listing [x^2+y^2,x,y]) are: [0, 0, 0], [1, 0, 1], [1, 1, 0], [2, 1, 1], [4, 0, 2], [4, 2, 0], [5, 1, 2], [5, 2, 1], [8, 2, 2], [9, 0, 3], [9, 3, 0], [10, 1, 3], [10, 3, 1], [13, 2, 3], [13, 3, 2], [16, 0, 4], [16, 4, 0], [17, 1, 4], [17, 4, 1], [18, 3, 3], [20, 2, 4], [20, 4, 2], [25, 0, 5], [25, 3, 4], [25, 4, 3], ...

For the y coordinates see A283306.

See also A000925, A283303, A283304, A283307, A283308.

L:=[];
M:=30;
for i from 0 to M do
for j from 0 to M do
L:=[op(L),[i^2+j^2,i,j]]; od: od:
t4:= sort(L,proc(a,b) evalb(a[1]<=b[1]); end);
t4x:=[seq(t4[i][2],i=1..100)]; # A283305
t4y:=[seq(t4[i][3],i=1..100)]; # A283306

nt = 105; (* number of terms to produce *)
S[m_] := S[m] = Table[{x, y}, {x, 0, m}, {y, 0, m}] // Flatten[#, 1]& // SortBy[#, {#.#&, #[[1]]&}]& // #[[All, 1]]& // PadRight[#, nt]&;
S[m = 2];
S[m = 2m];
While[S[m] =!= S[m/2], m = 2m];
S[m] (* Jean-François Alcover, Mar 05 2023 *)

for(r2=0,113,for(x=0,round(sqrt(r2)),y2=r2-x^2;if(issquare(y2),print1(x,", ")))) \\ Hugo Pfoertner, Jun 18 2018

A101428 Number of ways to write n as an ordered sum of a triangular number (A000217) and a square (A000290).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jan 03 2009

0 is both a triangular number and a square number.
First occurrence of k beginning at 0: 8, 2, 1, 10, 37, 136, 235, 1549, 631, 2314, 2116, 11026, 3997, 148240, 19045, 20827, 25876, 893116, 67951, ?19?, 35974, 187444, 1542655, 354061, 131905, ?25?, ?26?, 835399, 323767, ?29?, 611560, ?31?, 515629, ?33?, ?34?, ?35?, 1187146, ?37?, ?38?, ?39?, 1474939, ..., . - Robert G. Wilson v, Mar 30 2014
Variant of A082660 (which allows only positive triangular numbers). - R. J. Mathar, Apr 28 2020
a(n) is the number of representations of 8*n + 1 as 2*A^2 + B^2 with A even and B odd and both nonnegative integers. - Vladimir Pletser, Aug 30 2025

			Examples: n=1 gives the a(1)=2 cases 1=1+0=0+1; a(26)=2 because 26=25+1=16+10.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000925, A115171, A115172, A115173, A115174, A115175, A115176, A115177, A144642.

A000217 := proc(n) n*(n+1)/2 ; end:
A101428 := proc(n)
local a,y,t ;
a := 0 ;
for y from 0 do
t := A000217(y) ;
if n-t < 0 then
RETURN(a) ;
else
if issqr(n-t) then
a := a+1 ;
fi;
fi;
od:
end:
for n from 0 to 100 do printf("%a,",A101428(n)) ; od:

t = FoldList[#1 + #2 &, 0, Range@ 15]; s = Range[0, 10]^2, a = Sort@ Flatten@ Table[ s[[j]] + t[[k]], {j, 15}, {k, 11}]; Table[Count[a, n], {n, 0, 104}] (* or *)
triQ[n_] := IntegerQ@ Sqrt[8n + 1]; f[n_] := Block[{c = k = 0, lmt = 2 + Floor[Sqrt[n]]}, While[k < lmt, If[ triQ[n - k^2], c++]; k++]; c]; Array[f, 105, 0] (* Robert G. Wilson v, Mar 30 2014 *)

G.f.: sum(i>=0, x^(i^2) ) * sum(i>=0, x^(i*(i+1)/2) ). - Ralf Stephan, May 17 2014

A295819 Number of nonnegative solutions to (x,y) = 1 and x^2 + y^2 = n.

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Nov 28 2017

nonn

			a(1) = 2;
(1,0) = 1 and 1^2 + 0^2 =  1.
(0,1) = 1 and 0^2 + 1^2 =  1.
a(2) = 1;
(1,1) = 1 and 1^2 + 1^2 =  2. ->  1^2 +  1^2 == 1^2 + 1 == 0 mod  2.
a(5) = 2;
(2,1) = 1 and 2^2 + 1^2 =  5. ->  2^2 +  1^2 == 2^2 + 1 == 0 mod  5.
(1,2) = 1 and 1^2 + 2^2 =  5. ->  3^2 +  6^2 == 3^2 + 1 == 0 mod  5.
a(10) = 2;
(3,1) = 1 and 3^2 + 1^2 = 10. ->  3^2 +  1^2 == 3^2 + 1 == 0 mod 10.
(1,3) = 1 and 1^2 + 3^2 = 10. ->  7^2 + 21^2 == 7^2 + 1 == 0 mod 10.
a(13) = 2;
(3,2) = 1 and 3^2 + 2^2 = 13. -> 21^2 + 14^2 == 8^2 + 1 == 0 mod 13.
(2,3) = 1 and 2^2 + 3^2 = 13. -> 18^2 + 27^2 == 5^2 + 1 == 0 mod 13.

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A006278.
Similar sequences: A000010, A000925, A295820, A295848, A295976.
A000089 is essentially the same sequence.

a[n_] := Sum[j = Sqrt[n - i^2] // Floor; Boole[GCD[i, j] == 1 && i^2 + j^2 == n], {i, 0, Sqrt[n]}];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jul 05 2018, after Andrew Howroyd *)

a(n) = {sum(i=0, sqrtint(n), my(j=sqrtint(n-i^2)); gcd(i,j)==1 && i^2+j^2==n)} \\ Andrew Howroyd, Dec 12 2017

a(n) = A000089(n) for n >= 2.

a(A006278(n)) = 2^n for n >= 1.

A321380 Expansion of 1/2 * Product_{i>=0, j>=0} (1 + x^(i^2 + j^2)).

Original entry on oeis.org

1, 2, 2, 2, 3, 6, 8, 8, 8, 12, 19, 22, 22, 26, 37, 48, 53, 58, 72, 92, 108, 118, 136, 166, 195, 222, 254, 298, 346, 394, 453, 524, 600, 672, 762, 884, 1011, 1126, 1255, 1438, 1652, 1850, 2047, 2302, 2626, 2960, 3274, 3636, 4095, 4602, 5112, 5662, 6319, 7056, 7834

Offset: 0

Seiichi Manyama, Nov 08 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000925, A033461, A321381.

G.f.: Product_{k>0} (1 + x^k)^A000925(k).

A247367 Number of ways to write n as a sum of a square and a nonsquare.

Original entry on oeis.org

0, 0, 1, 2, 1, 1, 3, 3, 2, 2, 2, 4, 4, 2, 4, 4, 3, 3, 4, 5, 3, 5, 5, 5, 5, 2, 4, 6, 6, 4, 6, 6, 5, 6, 4, 6, 5, 5, 7, 7, 5, 5, 7, 7, 7, 5, 7, 7, 7, 6, 5, 8, 6, 6, 8, 8, 8, 8, 6, 8, 8, 6, 8, 8, 7, 5, 9, 9, 7, 9, 9, 9, 8, 7, 7, 9, 9, 9, 9, 9, 7, 8, 8, 10, 10, 6

Offset: 0

Reinhard Zumkeller, Sep 14 2014

nonn

			a(10) = #{0+10, 4+6} = 2;
a(11) = #{0+11, 1+10, 4+7, 9+2} = 4;
a(12) = #{0+12, 1+11, 4+8, 9+3} = 4;
a(13) = #{0+13, 1+12} = 2;
a(14) = #{0+14, 1+13, 4+10, 9+5} = 4;
a(15) = #{0+15, 1+14, 4+11, 9+6} = 4;
a(16) = #{1+15, 4+12, 9+7} = 3;
a(17) = #{0+17, 4+13, 9+8} = 3;
a(18) = #{0+18, 1+17, 4+14, 16+2} = 4;
a(19) = #{0+19, 1+18, 4+15, 9+10, 16+3} = 5;
a(20) = #{0+20, 1+19, 9+11} = 3.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A000037, A000290, A010052, A000925, A000161.

a247367 n = sum $ map ((1 -) . a010052 . (n -)) $
                  takeWhile (<= n) a000290_list

sQ[n_] := sQ[n] = IntegerQ[Sqrt[n]];
a[n_] := Sum[Boole[sQ[k] && !sQ[n-k] || !sQ[k] && sQ[n-k]], {k, 0, Quotient[n, 2]}];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Aug 10 2022 *)

cp's OEIS Frontend

A000161 Number of partitions of n into 2 squares.

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A045847 Matrix whose (i,j)-th entry is number of representations of j as a sum of i squares of nonnegative integers; read by diagonals.

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A014110 Number of ordered ways of writing n as a sum of 4 squares of nonnegative integers.

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A224212 Number of nonnegative solutions to x^2 + y^2 <= n.

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A240088 The number of ways of writing n as an ordered sum of a triangular number (A000217), a square (A000290) and a pentagonal number (A000326).

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A283305 List points (x,y) having integer coordinates with x >= 0, y >= 0, sorted first by x^2+y^2 and in case of a tie, by x-coordinate. Sequence gives x-coordinates.

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A101428 Number of ways to write n as an ordered sum of a triangular number (A000217) and a square (A000290).

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