A000161 -id:A000161 - cp's OEIS frontend

A000925 Number of ordered ways of writing n as a sum of 2 squares of nonnegative integers.

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

Offset: 0

Jacques Haubrich (jhaubrich(AT)freeler.nl)

A. Das and A. C. Melissinos, Quantum Mechanics: A Modern Introduction, Gordon and Breach, 1986, p. 47.
E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985.

Cf. A000290, A010052, A000161, A247367.

a000925 n = sum $ map (a010052 . (n -)) $ takeWhile (<= n) a000290_list
-- Reinhard Zumkeller, Sep 14 2014

a[n_] := (pr = PowersRepresentations[n, 2, 2]; Count[Union[Join[pr, Reverse /@ pr]], {j_ /; j >= 0, k_ /; k >= 0}]); a /@ Range[0, 100] (* Jean-François Alcover, Apr 05 2011 *)
nn = 100; t = CoefficientList[Series[Sum[x^k^2, {k, 0, Sqrt[nn]}]^2, {x, 0, nn}], x] (* T. D. Noe, Apr 05 2011 *)
SquareQ[n_] := IntegerQ[Sqrt[n]]; Table[Count[FrobeniusSolve[{1, 1}, n], {?SquareQ}], {n, 0, 100}] (* Robert G. Wilson v, Apr 15 2017 *)

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

Coefficient of q^k in (1/4)*(1 + theta_3(0, q))^2.

a(A001481(n))>0; a(A022544(n))=0. - Benoit Cloitre, Apr 20 2003

A133388 Largest integer m such that n-m^2 is a square, or 0 if no such m exists.

Original entry on oeis.org

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

Offset: 1

M. F. Hasler, Nov 23 2007

nonn

The sequence could be extended to a(0) = 0.

We could have defined a(n) = -1 instead of 0 if n is not sum of two squares, and then include unambiguously a(0) = 0. At present, a(n) = 0 <=> A000161(n) = 0.

			a(3) = 0 since 3 cannot be written as sum of 2 perfect squares;
a(5) = 2 since 5 = 2^2 + 1^2.

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A000161, A001481.

a:= proc(n) local t, d;
      for t from 0 do d:= n-t^2;
        if d<0 then break elif issqr(d) then return isqrt(d) fi
      od; 0
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, May 14 2015

a[n_] := Module[{m, d, s}, For[m = 0, True, m++, d = n - m^2; If[d < 0, Break[], s = Sqrt[d]; If[IntegerQ[s], Return[s]]]]; 0];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, May 18 2018, after Alois P. Heinz *)

sum2sqr(n)={ if(n>1, my(L=List(), f, p=1); for(i=1, matsize(f=factor(n))[1], if(f[i,1]%4==1, listput(L, [qfbsolve(Qfb(1,0,1), f[i,1])*[1, I]~, f[i,2]] ),/*elseif*/ f[i,1]==2, p = (1+I)^f[i,2],/*elseif*/ bittest(f[i,2],0), return([]),/*else*/ p *= f[i,1]^(f[i,2]\2))); L=apply(s->vector(s[2]+1, j, s[1]^(s[2]+1-j)*conj(s[1])^(j-1)), L); my(S=List()); forvec(T=vector(#L, i, [1,#L[i]]), listput(S, prod( j=1, #T, L[j][T[j]] ))); Set(apply(f->vecsort(abs([real(f), imag(f)])), Set(S)*p)), if(n<0, [], [[0, n]]))} \\ updated by M. F. Hasler, May 12 2018. (If PARI version 2.12.x returns an error, append [1] to qfbsolve(...) above. - M. F. Hasler, Dec 12 2019)
apply( A133388=n->if(n=sum2sqr(n),vecmax(Mat(n~))), [1..50]) \\ This sequence: maximum

from sympy.solvers.diophantine.diophantine import diop_DN
def A133388(n): return max((a for a, b in diop_DN(-1,n)),default=0) # Chai Wah Wu, Sep 08 2022

a(n) = max( sup { max(a,b) | a^2+b^2 = n ; a,b in Z }, 0 )
a(A022544(j))=0, j>0. - R. J. Mathar, Jun 17 2009
a(n^2) = a(n^2 + 1) = n, for all n. Conversely, whenever a(n) = a(n+1), then n = k^2. - M. F. Hasler, Sep 02 2018

A000174 Number of partitions of n into 5 squares.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

nonn

E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 84.

Equivalent sequences for other numbers of squares: A010052 (1), A000161 (2), A000164 (3), A002635 (4), A000177 (6), A025422 (7), A025423 (8), A025424 (9), A025425 (10).

Cf. A025429, A295160 (largest number k with a(k) = n).

Table[PowersRepresentations[n, 5, 2] // Length, {n, 0, 100}] (* Jean-François Alcover, Feb 04 2016 *)

A300667 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and z <= w such that 3x or y is a square and x + 2y is also a square.

Original entry on oeis.org

1, 2, 2, 1, 2, 4, 3, 1, 2, 4, 3, 1, 1, 3, 3, 1, 2, 5, 6, 3, 4, 5, 4, 2, 2, 5, 6, 3, 1, 4, 5, 2, 2, 4, 5, 3, 4, 4, 3, 1, 2, 6, 5, 3, 2, 4, 3, 1, 1, 3, 7, 4, 4, 5, 7, 4, 2, 4, 5, 3, 1, 2, 3, 3, 2, 6, 8, 4, 7, 7, 5, 1, 3, 4, 4, 4, 3, 4, 3, 3, 4

Offset: 0

Zhi-Wei Sun, Mar 10 2018

nonn

Conjecture 1: a(n) > 0 for all n >= 0, and a(n) = 1 only for n = 16^k*m with k = 0,1,2,... and m = 0, 3, 7, 11, 12, 15, 28, 39, 47, 60, 71, 92, 119, 172, 232, 253, 263, 316, 347, 515.
Conjecture 2: Each n = 0,1,2,... can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that 3*x or y is a square and 2*x - y is also a square.
By the author's 2017 JNT paper, any nonnegative integer can be written as the sum of a fourth power and three squares.
See also A281976, A300666, A300708 and A300712 for similar conjectures.
a(n) > 0 for all n = 0..10^8. Also, Conjecture 2 holds for all n = 0..10^8. In a 2018 paper Y.-C. Sun and Z.-W. Sun proved that any nonnegative integer can be written as x^2 + y^2 + z^2 + w^2 with x + 2*y a square, where x,y,z,w are nonnegative integers. - Zhi-Wei Sun, Oct 04 2020

			a(12) = 1 since 12 = 0^2 + 2^2 + 2^2 + 2^2 with 3*0 = 0^2 and 0 + 2*2 = 2^2.
a(39) = 1 since 39 = 2^2 + 1^2 + 3^2 + 5^2 with 1 = 1^2 and 2 + 2*1 = 2^2.
a(172) = 1 since 172 = 7^2 + 1^2 + 1^2 + 11^2 with 1 = 1^2 and 7 + 2*1 = 3^2.
a(232) = 1 since 232 = 0^2 + 0^2 + 6^2 + 14^2 with 0 = 0^2 and 0 + 2*0 = 0^2.
a(253) = 1 since 253 = 8^2 + 4^2 + 2^2 + 13^2 with 4 = 2^2 and 8 + 2*4 = 4^2.
a(263) = 1 since 263 = 3^2 + 3^2 + 7^2 + 14^2 with 3*3 = 3^2 and 3 + 2*3 = 3^2.
a(515) = 1 since 515 = 1^2 + 0^2 + 15^2 + 17^2 with 0 = 0^2 and 1 + 2*0 = 1^2.

Yu-Chen Sun and Zhi-Wei Sun, Some variants of Lagrange's four squares theorem, Acta Arith. 183(2018), 339-356.

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017-2018.
Yu-Chen Sun and Zhi-Wei Sun, Some variants of Lagrange's four squares theorem, arXiv:1605.03074 [math.NT], 2016-2018.

Cf. A000118, A000290, A271518, A281976, A300666, A300708, A300712.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[(SQ[3(m^2-2y)]||SQ[y])&&SQ[n-(m^2-2y)^2-y^2-z^2],r=r+1],{m,0,(5n)^(1/4)},{y,0,Min[m^2/2,Sqrt[n]]},{z,0,Sqrt[Max[0,(n-(m^2-2y)^2-y^2)/2]]}];tab=Append[tab,r],{n,0,80}];Print[tab]

A300667(n)=sum(x=0,sqrtint(n),sum(y=0,sqrtint(n-x^2),if(issquare(x+2*y)&&(issquare(y)||issquare(3*x)),if(n>x^2+y^2,A000161(n-x^2-y^2),1)))) \\ M. F. Hasler, Mar 11 2018

A118882 Numbers which are the sum of two squares in two or more different ways.

Original entry on oeis.org

25, 50, 65, 85, 100, 125, 130, 145, 169, 170, 185, 200, 205, 221, 225, 250, 260, 265, 289, 290, 305, 325, 338, 340, 365, 370, 377, 400, 410, 425, 442, 445, 450, 481, 485, 493, 500, 505, 520, 530, 533, 545, 565, 578, 580, 585, 610, 625, 629, 650, 676, 680

Offset: 1

Franklin T. Adams-Watters, May 03 2006

nonn

Numbers whose prime factorization includes at least two primes (not necessarily distinct) congruent to 1 mod 4 and any prime factor congruent to 3 mod 4 has even multiplicity. Products of two values in A004431.

Squares of distances that are the distance between two points in the square lattice in two or more nontrivially different ways. A quadrilateral with sides a,b,c,d has perpendicular diagonals iff a^2+c^2 = b^2+d^2. This sequence is the sums of the squares of opposite sides of such quadrilaterals, excluding kites (a=b,c=d), but including right triangles (the degenerate case d=0).

			50 = 7^2 + 1^2 = 5^2 + 5^2, so 50 is in the sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sums of squares

Cf. A004431, A009177, A085265.

Cf. A007692, A001481, A022544.

import Data.List (findIndices)
a118882 n = a118882_list !! (n-1)
a118882_list = findIndices (> 1) a000161_list
-- Reinhard Zumkeller, Aug 16 2011

Select[Range[1000], Length[PowersRepresentations[#, 2, 2]] > 1&] (* Jean-François Alcover, Mar 02 2019 *)

from itertools import count, islice
from math import prod
from sympy import factorint
def A118882_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        f = factorint(n)
        if 1>1):
            yield n
A118882_list = list(islice(A118882_gen(),30)) # Chai Wah Wu, Sep 09 2022

A000161(a(n)) > 1. [Reinhard Zumkeller, Aug 16 2011]

A025435 Number of partitions of n into 2 distinct squares.

Original entry on oeis.org

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

Offset: 0

David W. Wilson

nonn

a(A004435(n)) = 0; a(A001983(n)) > 0. - Reinhard Zumkeller, Dec 20 2013

			G.f. = x + x^4 + x^5 + x^9 + x^10 + x^13 + x^16 + x^17 + x^20 + 2*x^25 + ...

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

Cf. A010052, A000290, A000161, A025441.

a025435 0 = 0
a025435 n = a010052 n + sum
   (map (a010052 . (n -)) $ takeWhile (< n `div` 2) $ tail a000290_list)
-- Reinhard Zumkeller, Dec 20 2013

A025435 := proc(n)
    local i, j, ans;
    ans := 0;
    for i from 0 to n do
        for j from i+1 to n do
            if i^2+j^2=n then
                ans := ans+1
            fi
        end do
    end do;
    ans ;
end proc: # R. J. Mathar, Aug 04 2018

a[ n_] := If[ n < 0, 0, Sum[ Boole[ n == i^2 + j^2], {i, Sqrt[n]}, {j, 0, i - 1}]]; (* Michael Somos, Jun 24 2015 *)
a[ n_] := Length@ PowersRepresentations[ n, 2, 2] - Boole @ IntegerQ @ Sqrt[2 n]; (* Michael Somos, Jun 24 2015 *)
a[ n_] := SeriesCoefficient[ With[ {f = (EllipticTheta[ 3, 0, x] + 1)/2, g = (EllipticTheta[ 3, 0, x^2] + 1)/2}, f f - g] / 2, {x, 0, n}]; (* Michael Somos, Jun 24 2015 *)

{a(n) = if( n<0, 0, sum(i=1, sqrtint(n), sum(j=0, i-1, n == i^2 + j^2)))}; /* Michael Somos, Jun 24 2015 */

A025435(n)=sum(k=sqrtint((n-1+!n)\2)+1, sqrtint(n), issquare(n-k^2))-issquare(n/2) \\ or A000161(n)-issquare(n/2). - M. F. Hasler, Aug 05 2018

from math import prod
from sympy import factorint
def A025435(n):
    f = factorint(n)
    return int(not any(e&1 for p,e in f.items() if p>2))*(1-((f.get(2,0)&1)<<1)) + (((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 0 # Chai Wah Wu, Sep 08 2022

a(n) = A000161(n) - A010052(2*n). - M. F. Hasler, Aug 05 2018

a(n) = Sum_{i=1..n} c(i) * c(2*n-i), where c is the square characteristic (A010052). - Wesley Ivan Hurt, Nov 26 2020

A063014 Number of solutions to n^2 = b^2 + c^2 (with c >= b >= 0).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 3, 2, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 1, 5, 1, 1, 2, 1, 2, 1, 1, 2, 2, 3, 1, 1, 2, 1, 2, 1, 2, 1, 1, 5, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 1, 2, 1, 1, 3, 2, 2

Offset: 0

Henry Bottomley, Jul 26 2001

nonn

			a(0)=1 since 0^2 = 0^2 + 0^2;
a(5)=2 since 5^2 = 0^2 + 5^2 = 3^2 + 4^2;
a(25)=3 since 25^2 = 0^2 + 25^2 = 7^2 + 24^2 = 15^2 + 20^2.

Felix Huber, Table of n, a(n) for n = 0..10000
Michael Gilleland, Some Self-Similar Integer Sequences

Cf. A000161, A046080.

Column k=2 of A255212.

with(NumberTheory):
A063014 := n -> nops(SumOfSquares(n^2));
seq(A063014(n), n = 0 .. 100); # Felix Huber, Jun 01 2024

a(0) = 1; a(n) = A046080(n) + 1 for n > 0. [amended by Georg Fischer, Jan 25 2020]

a(n) = A000161(n^2). - Christian Krause, Dec 08 2022

A234300 Number of unit squares, aligned with a Cartesian grid, partially encircled along the edge of the first quadrant of a circle centered at the origin ordered by increasing radius.

Original entry on oeis.org

0, 1, 1, 3, 2, 3, 3, 5, 3, 5, 4, 5, 5, 7, 5, 7, 5, 7, 7, 9, 7, 9, 8, 9, 7, 9, 7, 11, 9, 11, 9, 11, 10, 11, 9, 11, 11, 13, 11, 13, 11, 13, 11, 13, 11, 13, 13, 15, 12, 15, 13, 15, 13, 15, 13, 15, 13, 15, 15, 17, 13, 17, 15, 17, 16, 17, 15, 17, 15, 17, 15, 17, 17, 19, 17, 19, 15, 19, 17, 19, 17, 19, 17, 19, 18, 19, 17, 21, 19, 21, 19, 21, 19, 21, 19, 21, 19, 21, 19, 21

Offset: 0

Rajan Murthy, Dec 22 2013

nonn

The first decrease from a(4) = 3 to a(5) = 2 occurs when the radius squared increases from an arbitrary position between 1 and 2 (when 3 squares are on the edge) to exactly 2 (when only 2 squares are on the edge because the circle of square radius 2 passes through the upper right corner on the y=x line). Similar decreases occur when the circle passes through other upper right corners. At least some (if not all) adjacent duplicates occur when the square radius corresponds to a perfect square, that is a corner which is only a lower right corner, i.e., on the y = 0 line. For example, a(6)=a(7)=3 occurs when, for n = 6 , a(n) corresponding to the interval between 2 and 4; and, for n=7, a(n) corresponding to the exact square radius of 4. Some of the confusion may come from the fact that for odd n, there is a unique circle corresponding to elements of a(n) (passing through the corner of specific square(s) on the grid), while for even n, there is a set of circles with a range of radii (which do not pass through corners) corresponding to the elements of a(n). It seems easier to organize the concept in terms of intervals and corners for the sake of consistency.

a(n) is even when the radius squared corresponds to an element of A024517.

			At radius 0, there are no partially filled squares.  At radius >1 but < sqrt(2), there are 3 partially filled squares along the edge of the circle.  At radius = sqrt(2), there are 2 partially filled squares along the edge of the circle.

Rajan Murthy, Table of n, a(n) for n = 0..4999

Cf. A001481 (corresponds to the square radius of alternate entries), A232499 (number of completely encircled squares when the radii are indexed by A000404), A235141 (first differences), A024517.

A237708 is the analog for the 3-dimensional Cartesian lattice and A239353 for the 4-dimensional Cartesian lattice.

function index = n_edgeindex (N)
    if N < 1 then
        N = 1
    end
    N = floor(N)
    i = 0:ceil(N/2)
    i = i^2
    index = i
    for j = 1:length(i)
       index = [index i+ i(j).*ones(i)]
    end
    index = unique(index)
    index = index(1:ceil(N))
    d = diff(index)/2
    d = d +  index(1:length(d))
    index = gsort([index d],"g","i")
    index = index(1:N)
endfunction
function l = n_edge_n (i)
        l=0
        h=0
        while (i > (2*h^2))
            h=h+1
        end
        if i < (2*h^2) then
                l = l+1
        end
        if i >1 then
            t=[0 1]
           while (i>max(t))
               t = [t (sqrt(max(t))+1)^2]
           end
        for j = 1:h
           b=t
           t=[2*(j)^2 (j+1)^2 + (j)^2]
           while (i>max(t))
               t = [t (sqrt(max(t)-(j)^2)+1)^2 + (j)^2]
           end
           l = l+ 2*(length(b)-length(t))
           if max(t) == i then
               l = l-2
           end
        end
       end
endfunction
function a =n_edge (N)
    if N <1 then
        N =1
    end
    N = floor(N)
    a= []
    index = n_edgeindex(N)
    for i = index
        a = [a n_edge_n(i)]
    end
endfunction

a(2k+1) = a(2k) + 2*A000161(A001481(k+1)) - A010052(A001481(k+1)/2). - Rajan Murthy, Jan 14 2013

a(2k) = a(2k-1) - 2*(A000161(A001481(k+1)) - A010052(A001481(k+1))) + A010052(A001481(k+1)/2). - Rajan Murthy, Jan 14 2013

A069011 Triangle with T(n,k) = n^2 + k^2.

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 9, 10, 13, 18, 16, 17, 20, 25, 32, 25, 26, 29, 34, 41, 50, 36, 37, 40, 45, 52, 61, 72, 49, 50, 53, 58, 65, 74, 85, 98, 64, 65, 68, 73, 80, 89, 100, 113, 128, 81, 82, 85, 90, 97, 106, 117, 130, 145, 162, 100, 101, 104, 109, 116, 125, 136, 149, 164, 181, 200

Offset: 0

Henry Bottomley, Apr 02 2002

For any i,j >=0 a(i)*a(j) is a member of this sequence, since (a^2 + b^2)*(c^2 + d^2) = (a*c + b*d)^2 + (a*d - b*c)^2. - Boris Putievskiy, May 05 2013
A227481(n) = number of squares in row n. - Reinhard Zumkeller, Oct 11 2013
Norm of the complex numbers n +- i*k and k +- i*n, where i denotes the imaginary unit. - Stefano Spezia, Aug 07 2025

			Triangle T(n,k) begins:
    0;
    1,   2;
    4,   5,   8;
    9,  10,  13,  18;
   16,  17,  20,  25,  32;
   25,  26,  29,  34,  41,  50;
   36,  37,  40,  45,  52,  61,  72;
   49,  50,  53,  58,  65,  74,  85,  98;
   64,  65,  68,  73,  80,  89, 100, 113, 128;
   81,  82,  85,  90,  97, 106, 117, 130, 145, 162;
  100, 101, 104, 109, 116, 125, 136, 149, 164, 181, 200;
  ...

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened

Cf. A001481 for terms in this sequence, A000161 for number of times each term appears, A048147 for square array.
Column k=0 gives A000290.
Main diagonal gives A001105.
Row sums give A132124.
T(2n,n) gives A033429.

a069011 n k = a069011_tabl !! n !! k
a069011_row n = a069011_tabl !! n
a069011_tabl = map snd $ iterate f (1, [0]) where
   f (i, xs@(x:_)) = (i + 2, (x + i) : zipWith (+) xs [i + 1, i + 3 ..])
-- Reinhard Zumkeller, Oct 11 2013

Table[n^2 + k^2, {n, 0, 12}, {k, 0, n}] (* Paolo Xausa, Aug 07 2025 *)

T(n+1,k+1) = T(n,k) + 2*(n+k+1), k=0..n; T(n+1,0) = T(n,0) + 2*n + 1. - Reinhard Zumkeller, Oct 11 2013

G.f.: x*(1 + 2*y + 5*x^3*y^2 - x^2*y*(2 + 5*y) + x*(1 - 4*y + 2*y^2))/((1 - x)^3*(1 - x*y)^3). - Stefano Spezia, Aug 04 2025

A125022 Numbers with a unique partition as the sum of 2 squares x^2 + y^2.

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20, 26, 29, 32, 34, 36, 37, 40, 41, 45, 49, 52, 53, 58, 61, 64, 68, 72, 73, 74, 80, 81, 82, 89, 90, 97, 98, 101, 104, 106, 109, 113, 116, 117, 121, 122, 128, 136, 137, 144, 146, 148, 149, 153, 157, 160, 162, 164, 173, 178, 180, 181

Offset: 1

Artur Jasinski, Nov 16 2006

nonn

A000161(a(n)) = 1. [Reinhard Zumkeller, Aug 16 2011]

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A002145, A124982, A125021, A034026.

Cf. A025284, A081324, A022544, A001481, A118882.

import Data.List (elemIndices)
a125022 n = a125022_list !! (n-1)
a125022_list = elemIndices 1 a000161_list
-- Reinhard Zumkeller, Aug 16 2011

Select[Range[0,200],Length@PowersRepresentations[#,2,2]==1&] (* Giorgos Kalogeropoulos, Mar 21 2021 *)

a(n) = A125021(n)/2.

Name edited by Giorgos Kalogeropoulos, Mar 21 2021

cp's OEIS Frontend

A000925 Number of ordered ways of writing n as a sum of 2 squares of nonnegative integers.

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A133388 Largest integer m such that n-m^2 is a square, or 0 if no such m exists.

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A000174 Number of partitions of n into 5 squares.

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A300667 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and z <= w such that 3*x or y is a square and x + 2*y is also a square.

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A118882 Numbers which are the sum of two squares in two or more different ways.

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A025435 Number of partitions of n into 2 distinct squares.

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A063014 Number of solutions to n^2 = b^2 + c^2 (with c >= b >= 0).

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A300667 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and z <= w such that 3x or y is a square and x + 2y is also a square.