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Let M(r,m,n) be the number of unrestricted partitions of n with crank congruent to r modulo m. Here the relations between the numbers M(r,13,13n+k) when 0≤r<13 are given. All of the results are proved by elementary methods. © 2012 Springer Science+Business Media, LLC.

Publication

The generalized order-k Narayana's cows numbers

(2016-08-01) Bilgici G.; Bilgici, G

Narayana's cows problem is a problem similar to the Fibonacci's rabbit problem. We define the numbers which are the solutions of this problem as Narayana's cows numbers. Narayana's cows sequence satisfies the third order recurrence relation Nr = Nr-1+Nr-3 with initial condition N0 = N1 = N2 = 1. In this paper, we define a new recurrence which is called generalized order-k Narayana's cows sequence and by using this generalization and some matrix properties, we give some identities related to the Narayana's cows numbers.

Web of Science

THE GENERALIZED ORDER-k NARAYANA'S COWS NUMBERS

(2016.01.01) Bilgici, G

Publication

Two generalizations of Lucas sequence

(2014-10-15) Bilgici G.; Bilgici, G

We define a generalization of Lucas sequence by the recurrence relation lm=blm-1+lm-2 (if m is even) or l m=alm-1+lm-2 (if m is odd) with initial conditions l0=2 and l1=a. We obtain some properties of the sequence {lm}m=0 and give some relations between this sequence and the generalized Fibonacci sequence {qm}m=0 which is defined in Edson and Yayenie (2009). Also, we give corresponding generalized Lucas sequence with the generalized Fibonacci sequence given in Yayenie (2011). © 2014 Elsevier Inc. All rights reserved.

Web of Science

Two generalizations of Lucas sequence

(2014.01.01) Bilgici, G