Equivalence of OEIS A007729 and A174868

TL;DR

This paper confirms that the sixth binary partition function matches the partial sums of the Stern-Brocot sequence, establishing an equivalence between two mathematical sequences.

Contribution

It proves the conjecture that these two sequences are equal, providing a new link between binary partitions and the Stern-Brocot sequence.

Findings

Confirmed the conjecture relating the sixth binary partition function to Stern-Brocot partial sums.

Established the equivalence of OEIS sequences A007729 and A174868.

Validated the conjecture through mathematical verification.

Abstract

We verify the conjecture that the sixth binary partition function is equal (aside from the initial zero term) to the partial sums of the Stern-Brocot sequence.

Equations

$$0,1,2,4,5,8,10,13,14,18,21,26,28,33,36,40,41,46\dots$$

$$k=\sum _{i\ge 0}{\epsilon }_i{2}^i$$

$$b_{2k}^{\prime }=b_k^{\prime }+b_{k-1}^{\prime }+b_{k-2}^{\prime }$$

$$b_{2k}^{\prime }$$

$$b_{2k}=2b_{2k-1}^{\prime }+b_{2k}^{\prime }$$

$$b_{2k+1}=2b_{2k}^{\prime }+b_{2k-1}^{\prime }$$

$${\hat{b}}_{2k}$$

$${\hat{b}}_{2k+1}$$

$$s_{2k}$$

$$s_{2k+1}$$

$${\sigma }_{2k}=\sum _{1\le j\le k}{\ell }_j=2{\sigma }_k+{\sigma }_{k-1}$$

$${\sigma }_{2k+1}=\sum _{0\le j\le k}{\ell }_j^{\prime }=2{\sigma }_k+{\sigma }_{k+1}.$$

Full text

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Let $b^{\prime}_{k}$ be the sixth binary partition function, which is the number of ways to write $k$ as a sum

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with $\varepsilon_{i}\in\{0,1,2,3,4,5\}$. We obtain

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by counting the number of representations of $2k$ with $\varepsilon_{0}=0,2$ or $4$; in each case we have a representation $\hat{\varepsilon}$ of $\frac{2k-\varepsilon_{0}}{2}$ by taking $\hat{\varepsilon}_{i}=\varepsilon_{i+1}$, and the correspondence is clearly one-to-one. Also $b^{\prime}_{2k+1}=b^{\prime}_{2k}$, since we can get a representation of $2k+1$ only by taking a representation of $2k$ and adding 1 to $\varepsilon_{0}$. Thus

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since $b^{\prime}_{k-1}$ equals either $b^{\prime}_{k}$ or $b^{\prime}_{k-2}$.

We eliminate the even/odd repetition by defining $b_{k}=b^{\prime}_{2k}$. Then $b_{0}=1,b_{1}=2$ and

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This is A007729. If we prepend a zero, defining $\hat{b}_{0}=0$ and $\hat{b}_{k}=b_{k-1}$ we obtain

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The same recurrence with the same initial conditions gives A174868, the partial sums of the Stern-Brocot sequence [1]. The Stern-Brocot sequence itself can be defined by $s_{0}=0,s_{1}=1$, and

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The partial sums are $\sigma_{k}=\sum_{0\leq i\leq k}s_{k}$. Letting $\ell_{j}=s_{2j-1}+s_{2j}=2s_{j}+s_{j-1}$ we get

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and similarly with $\ell^{\prime}_{j}=s_{2j}+s_{2j+1}=2s_{j}+s_{j+1}$,

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