A303540 Number of ways to write n as a^2 + b^2 + binomial(2*c,c) + binomial(2*d,d), where a,b,c,d are nonnegative integers with a <= b and c <= d.

0, 1, 2, 3, 2, 2, 3, 4, 3, 2, 3, 6, 4, 2, 2, 4, 4, 2, 2, 5, 5, 5, 4, 4, 4, 4, 5, 6, 5, 5, 4, 5, 4, 4, 3, 4, 5, 5, 6, 5, 5, 5, 4, 7, 3, 4, 5, 6, 4, 2, 4, 6, 7, 4, 4, 5, 7, 6, 2, 5, 4, 6, 3, 2, 5, 5, 5, 4, 4, 3, 7, 9, 6, 5, 6, 11, 7, 3, 4, 8

Offset

1,3

Comments

Conjecture: a(n) > 0 for all n > 1. In other words, any integer n > 1 can be written as the sum of two squares and two central binomial coefficients.

It has been verified that a(n) > 0 for all n = 2..10^10.

See also A303539 and A303541 for related information.

Jiao-Min Lin (a student at Nanjing University) has verified a(n) > 0 for all 1 < n <= 10^11. - Zhi-Wei Sun, Jul 30 2022

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.

Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.

Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017-2018.

Example

a(2) = 1 since 2 = 0^2 + 0^2 + binomial(2*0,0) + binomial(2*0,0).
a(10) = 2 with 10 = 2^2 + 2^2 + binomial(2*0,0) + binomial(2*0,0) = 1^2 + 1^2 + binomial(2*1,1) + binomial(2*2,2).
a(2435) = 1 with 2435 = 32^2 + 33^2 + binomial(2*4,4) + binomial(2*5,5).

Maple

N:= 100: # for a(1)..a(N)
A:= Vector(N):
for b from 0 to floor(sqrt(N)) do
  for a from 0 to min(b, floor(sqrt(N-b^2))) do
    t:= a^2+b^2;
    for d from 0 do
      s:= t + binomial(2*d, d);
      if s > N then break fi;
      for c from 0 to d do
        u:= s + binomial(2*c, c);
        if u > N then break fi;
        A[u]:= A[u]+1;
od od od od:
convert(A, list); # Robert Israel, May 30 2018

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
c[n_]:=c[n]=Binomial[2n, n];
f[n_]:=f[n]=FactorInteger[n];
g[n_]:=g[n]=Sum[Boole[Mod[Part[Part[f[n], i], 1], 4]==3&&Mod[Part[Part[f[n], i], 2], 2]==1], {i, 1, Length[f[n]]}]==0;
QQ[n_]:=QQ[n]=(n==0)||(n>0&&g[n]);
tab={}; Do[r=0; k=0; Label[bb]; If[c[k]>n, Goto[aa]]; Do[If[QQ[n-c[k]-c[j]], Do[If[SQ[n-c[k]-c[j]-x^2], r=r+1], {x, 0, Sqrt[(n-c[k]-c[j])/2]}]], {j, 0, k}]; k=k+1; Goto[bb]; Label[aa]; tab=Append[tab, r], {n, 1, 80}]; Print[tab]

Crossrefs

Cf. A000290, A000984, A001481, A303233, A303234, A303338, A303363, A303389, A303393, A303399, A303428, A303401, A303432, A303434, A303539, A303541, A303543, A303601, A303639.

Sequence in context: A104377 A109337 A303539 * A137266 A062948 A365159

Adjacent sequences: A303537 A303538 A303539 * A303541 A303542 A303543

Keyword

nonn

Author

Zhi-Wei Sun, Apr 25 2018

Status

approved