Those integers $\binom{2n}n\ (n=0,1,2,\ldots)$ are called central binomial coefficients. By Stirling's formula, $$\binom{2n}n\sim \frac{4^n}{\sqrt{n\pi}}\ \ \ \ (n\to+\infty).$$ Of course, the central binomial coefficients are much sparser than the squares of integers.

Motivated by Lagrange's four-square theorem, here I ask the following question.

QUESTION: Is every integer $n>1$ the sum of two squares and two central binomial coefficients?

I conjecture that the answer is affirmative. I have verified this for $n$ up to $6\times10^9$; for example, $$2435=32^2+33^2+\binom{2\times4}4+\binom{2\times5}5.$$ For the number of ways to write a positive integer $n$ as $a^2+b^2+\binom{2c}c+\binom{2d}d$ with $a,b,c,d\in\mathbb N=\{0,1,2,\ldots\}$, $a\leqslant b$ and $c\leqslant d$, one may consult http://oeis.org/A303540.

A set $A\subseteq \mathbb N$ is called an additive basis of order two if $\{a+b:\ a,b\in A\}=\mathbb N$. My above conjecture essentially says that $$\left\{k^2+\binom{2m}m-1:\ k,m=0,1,2,\ldots\right\}$$ is an additive basis of order two.

I have some similar questions. I conjecture that each integer $n>1$ can be written as the sum of two squares and two Catalan numbers, and that $$\left\{a^2+b^2+\binom{2k+1}k+\binom{2m+1}m:\ a,b,k,m\in\mathbb N\right\}=\{2,3,\ldots\}.$$ See http://oeis.org/A303543 and http://oeis.org/A303639 for related data.

My above question looks quite challenging. Any helpful ideas?