A303543 Number of ways to write n as a^2 + b^2 + C(k) + C(m) with 0 <= a <= b and 0 < k <= m, where C(k) denotes the Catalan number binomial(2k,k)/(k+1).

0, 1, 2, 3, 2, 3, 4, 4, 2, 3, 5, 5, 2, 3, 5, 5, 4, 3, 6, 8, 4, 3, 6, 6, 3, 3, 5, 7, 6, 3, 4, 8, 5, 2, 6, 7, 3, 4, 5, 5, 6, 4, 5, 10, 6, 4, 7, 8, 4, 2, 7, 9, 9, 5, 7, 11, 8, 2, 5, 11, 5, 4, 4, 8, 8, 4, 6, 11, 10, 3, 6, 8, 5, 5, 6, 7, 6, 6, 5, 9

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 Catalan numbers.

This is similar to the author's conjecture in A303540. It has been verified that a(n) > 0 for all n = 2..10^9.

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 with 2 = 0^2 + 0^2 + C(1) + C(1).
a(3) = 2 with 3 = 0^2 + 1^2 + C(1) + C(1) = 0^2 + 0^2 + C(1) + C(2).

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
c[n_]:=c[n]=Binomial[2n, n]/(n+1);
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=1; 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, 1, k}]; k=k+1; Goto[bb]; Label[aa]; tab=Append[tab, r], {n, 1, 80}]; Print[tab]

Crossrefs

Cf. A000108, A000290, A001481, A303233, A303234, A303338, A303363, A303389, A303393, A303399, A303428, A303401, A303432, A303434, A303539, A303540, A303541, A303601.

Sequence in context: A095161 A072106 A124524 * A124525 A106788 A123175

Adjacent sequences: A303540 A303541 A303542 * A303544 A303545 A303546

Keyword

nonn

Author

Zhi-Wei Sun, Apr 25 2018

Status

approved