Euler, in this publication with Eneström number E326, provides an induction fallacy which arises from analyzing a particular sequence. Euler wrote this work in 1763, one of only two papers he wrote on sequences and/or series in the 1760’s, out of a total of 79 papers on series during his career. His goal in E326 is to investigate the middle terms in the expansion of powers of quadratic trinomial expressions, beginning with the specific simple quadratic , before considering the general quadratic .
The induction fallacy shows up during the analysis of the simple case when Euler first finds an explicit formula for the middle terms, now known as central trinomial coefficients (see the Online Encyclopedia of Integer Sequences, https://oeis.org/A002426 ). He then investigates a recursive formula which involves pronic and Fibonacci numbers, resulting in two integer sequences which agree for the first nine terms and then disagree from the tenth term onward. [C. Edward Sandifer, How Euler Did It, Mathematical Association of America, 2007, p. 143-146.]
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Huffman, Cynthia Ph.D.
"Analytical Observations (Translation of E326),"
Euleriana: 3(1), pp.3-22.
Available at: https://scholarlycommons.pacific.edu/euleriana/vol3/iss1/2