TRIGONOMETRIC-TYPE IDENTITIES AND THE PARITY OF BALANCING AND LUCAS-BALANCING NUMBERS

Ngô Văn Định

doi:10.34238/tnu-jst.4525

TRIGONOMETRIC-TYPE IDENTITIES AND THE PARITY OF BALANCING AND LUCAS-BALANCING NUMBERS

About this article

Received: 20/05/21 Revised: 11/11/21 Published: 24/11/21

Authors

Ngo Van Dinh , TNU - University of Sciences

Abstract

Balancing numb ers n are originally defined as the solution of the Diophantine equation 1+2+· · ·+(n−1) = (n+1)+· · ·+(n+r), where r is called the balancer corresponding to the balancing number n. By slightly modifying, n is the cobalancing number with the cobalancer r if 1+2+· · ·+n = (n+1)+· · ·+(n+r). Let B n denote the nth balancing number and bn denote the nth cobalancing number. Then 8B 2 n +1 and 8b 2n +8bn +1 are perfect squares. The nth Lucasbalancing number Cn and the nth Lucas-cobalancing number cn are the positive roots of 8Bn2 +1 and 8b 2n +8bn +1, respectively. In this paper, we establish some trigonometric-type identities and some arithmetic properties concerning the parity of balancing, cobalancing, Lucas-balancing and Lucas-cobalancing numbers.

Keywords

Balancing number; Cobalancing number; Lucas-balancing number; trigonometric-type identity; Parity

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DOI: https://doi.org/10.34238/tnu-jst.4525

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