On generalized Frame-Stewart numbers

For the multi-peg Tower of Hanoi problem with $k \geqslant 4$ pegs, so far the best solution is obtained by the Stewart's algorithm based on the the following recurrence relation: $\mathrm{S}\_k(n)=\min\_{1 \leqslant t \leqslant n} \left\{2 \cdot \mathrm{S}\_k(n-t) + \mathrm{S}\_{k-1}(t)\right\}$, $\mathrm{S}\_3(n) = 2^n -- 1$. In this paper, we generalize this recurrence relation to $\mathrm{G}\_k(n) = \min\_{1\leqslant t\leqslant n}\left\{ p\_k\cdot \mathrm{G}\_k(n-t) + q\_k\cdot \mathrm{G}\_{k-1}(t) \right\}$, $\mathrm{G}\_3(n) = p\_3\cdot \mathrm{G}\_3(n-1) + q\_3$, for two sequences of arbitrary positive integers $\left(p\_i\right)\_{i \geqslant 3}$ and $\left(q\_i\right)\_{i \geqslant 3}$ and we show that the sequence of differences $\left(\mathrm{G}\_k(n)- \mathrm{G}\_k(n-1)\right)\_{n \geqslant 1}$ consists of numbers of the form $\left(\prod\_{i=3}^{k}q\_i\right) \cdot \left(\prod\_{i=3}^{k}{p\_i}^{\alpha\_i}\right)$, with $\alpha\_i\geqslant 0$ for all $i$, arranged in nondecreasing order. We also apply this result to analyze recurrence relations for the Tower of Hanoi problems on several graphs.Comment: 13 pages ; 3 figure

THE DETERMINATION OF RAMANUJAN PAIRS.

We call two increasing sequences of positive integers {aᵢ}, {b(j)} a "Ramanujan Pair" if the following identity holds: (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI). The goal of this investigation is to determine all Ramanujan Pairs. Although this goal was not completely reached, we have determined all pairs for which the first term a₁ ≥ 5 and have proved that any Ramanujan Pair which begins with a₁ = m, where 1 ≤ m ≤ 4, aside from the known pairs, would have to branch off the first Euler identity with {aᵢ} = {i + m - 1}, {b(j)} = {j m}. A great deal of computing was done to discover the proofs given here. The search methods used and their programs are discussed in detail. Beyond these results, we have found all finite Ramanujan Pairs. Finally, modular Ramanujan Pairs (where the coefficients in the identity are reduced modulo n) are also examined

The Johannes Felbermeyer Collection at the Getty Research Institute

A version of a presentation at the annual College Art Association conference in Los Angeles (February 2012) given as part of the VRA affiliate session. Acknowledgements: I'd like to thank my colleage Tracey Schuster who co-presented on the VRA panel with me for her support, guidance, and unmatched Photo Archive expertise

Welcome to the electronic VRA Bulletin (e-VRAB): Editors' Welcome

The editors of the inaugural issue of the electronic VRA Bulletin (v. 38, n. 1) welcome the VRA membership to the new publishing platform of the Berkeley Electronic Press (bepress)

Article electronically published on August 14, 2002 LINEAR QUINTUPLE-PRODUCT IDENTITIES

Dedicated to our longtime friend John Selfridge Abstract. In the first part of this paper, series and product representations of four single-variable triple products T0, T1, T2, T3 and four single-variable quintuple products Q0, Q1, Q2, Q3 are defined. Reduced forms and reduction formulas for these eight functions are given, along with formulas which connect them. The second part of the paper contains a systematic computer search for linear trinomial Q identities. The complete set of such families is found to consist of two 2-parameter families, which are proved using the formulas in the first part of the paper. 1

Parity results for certain partition functions

partition generating functions of the following sort. 1 � (1 − q n) ≡ 1+ � q n2 q 3n