Sums of Polygonal Numbers

Basic Properties of the Polygonal Sequence

10.20944/preprints202102.0385.v1 ◽

2021 ◽

Author(s):

Kunle Adegoke

Keyword(s):

Generating Functions ◽

Recurrence Relations ◽

Partial Sums ◽

Basic Properties ◽

Polygonal Numbers

We study various properties of the polygonal numbers; such as their recurrence relations, fundamental identities, weighted binomial and ordinary sums and the partial sums and generating functions of their powers. A feature of our results is that they are presented naturally in terms of the polygonal numbers themselves and not in terms of arbitrary integers as is the case in most literature.

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Basic Properties of the Polygonal Sequence

10.20944/preprints202102.0385.v3 ◽

2021 ◽

Author(s):

Kunle Adegoke

Keyword(s):

Generating Functions ◽

Continued Fraction ◽

Recurrence Relations ◽

Partial Sums ◽

Basic Properties ◽

Polygonal Numbers

We study various properties of the polygonal numbers; such as their recurrence relations; fundamental identities; weighted binomial and ordinary sums; partial sums and generating functions of their powers; and a continued fraction representation for them. A feature of our results is that they are presented naturally in terms of the polygonal numbers themselves and not in terms of arbitrary integers; unlike what obtains in most literature.

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I. Continuation of the subject of a paper read Dec. 22, 1853, the supplement to which was read Jan. 12, 1854, by Sir Frederick Pollock, &c.; with a proof of Fermat's first and second theorems of the polygonal numbers, viz. that every odd number is composed of four square numbers or less, and of three triangular numbers or less

Proceedings of the Royal Society of London ◽

10.1098/rspl.1854.0002 ◽

1856 ◽

Vol 7 ◽

pp. 1-4

Keyword(s):

Triangular Numbers ◽

The Subject ◽

Polygonal Numbers ◽

Four Square

The object of this paper is in the first instance to prove the truth of a theorem stated in the supplement to a former paper, viz. “that every odd number can be divided into four squares (zero being considered an even square) the algebraic sum of whose roots (in some form or other) will equal 1, 3, 5, 7, &c. up to the greatest possible sum of the roots.” The paper also contains a proof, that if every odd number 2 n + 1 can be divided into four square numbers, the algebraic sum of whose roots is equal to 1, then any number n is composed of not exceeding three triangular numbers.

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