TY - JOUR

T1 - Geometry of Figurate Numbers and Sums of Powers of Consecutive Natural Numbers

AU - Marko, František

AU - Litvinov, Semyon

PY - 2020/1/2

Y1 - 2020/1/2

N2 - First, we give a geometric proof of Fermat’s fundamental formula for figurate numbers. Then we use geometrical reasoning to derive weighted identities with figurate numbers and observe some of their applications. Next, we utilize figurate numbers to provide a matrix formulation for the closed forms of the sums (Formula presented.) thus generating Bernoulli numbers. Finally, we present a formula—motivated by the inclusion-exclusion principle—for (Formula presented.) as a linear combination of figurate numbers.

AB - First, we give a geometric proof of Fermat’s fundamental formula for figurate numbers. Then we use geometrical reasoning to derive weighted identities with figurate numbers and observe some of their applications. Next, we utilize figurate numbers to provide a matrix formulation for the closed forms of the sums (Formula presented.) thus generating Bernoulli numbers. Finally, we present a formula—motivated by the inclusion-exclusion principle—for (Formula presented.) as a linear combination of figurate numbers.

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U2 - 10.1080/00029890.2020.1671129

DO - 10.1080/00029890.2020.1671129

M3 - Article

AN - SCOPUS:85076896240

VL - 127

SP - 4

EP - 22

JO - American Mathematical Monthly

JF - American Mathematical Monthly

SN - 0002-9890

IS - 1

ER -