A061790 a(n) = A000217(n) - A061786(n).

0, 0, 0, 0, 0, 0, 1, 2, 3, 3, 5, 6, 8, 11, 12, 14, 18, 20, 25, 27, 31, 35, 42, 46, 50, 55, 61, 67, 74, 78, 87, 94, 101, 111, 118, 124, 133, 143, 153, 159, 172, 181, 193, 206, 214, 227, 240, 251, 265, 277, 290, 303, 322, 337, 350, 363, 378, 392, 410, 421, 440, 461

Offset: 1

Author

Labos Elemer, Jun 22 2001

Keywords

• nonn

Comments

If the {s+t} sums are generated by addition 2 terms of an S set consisting of n different entries, then at least 1 and at most n(n+1)/2=A000217(n) distinct values can be obtained. The set of first n squares gives results falling between these two extremes.

Original name: "Number of sums i^2 + j^2 that occur more than once for 1 <= i <= j <= n." This was incorrect because sums that occur more than twice are overcounted. - Robert Israel, Jun 26 2025

Examples

			S={1,4,9,...,100,121} provides 61 different sums of two (not necessarily different) squares: {2,5,8,..,202,221,242}. Only 5 of these sums arise more than once:
   50 = 1 +  49 = 25 +  25;
   65 = 1 +  64 = 16 +  49;
   85 = 4 +  81 = 36 +  49;
  125 = 4 + 121 = 25 + 100;
  130 = 9 + 121 = 49 +  81.
Therefore a(11) = (12*11/2) - 61 = 5.
		

Crossrefs

Cf. A000217, A385352.

Programs

Maple

N:= 100: # for a(1) .. a(N)
V:= Vector(2*N^2, datatype=integer[4]):
R:= Vector(N):
count:= 0:
for n from 1 to N do
  for i from 1 to n do
    t:= i^2 + n^2;
    V[t]:= V[t]+1;
    if V[t] = 1 then count:= count+1 fi;
  od;
  R[n]:= n*(n+1)/2 - count
od:
convert(R,list); # Robert Israel, Jun 26 2025

Mathematica

f[x_] := x^2 t0=Table[Length[Union[Flatten[Table[f[u]+f[w], {w, 1, m}, {u, 1, m}]]]], {m, 1, 75}] t1=Table[(w*(w+1)/2)-Part[t0, w], {w, a, b}]

Python

def A061790(n): return (n*(n+1)>>1)-len({i**2+j**2 for i in range(1,n+1) for j in range(1,i+1)}) # Chai Wah Wu, Jun 27 2025

Extensions

Definition corrected by Robert Israel, Jun 26 2025