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The distribution of second order linear recurrence sequences mod $2^m$

100%

Morgan M.

Acta Arithmetica

1998

tom 83

nr 2

181-195

Sums of Powered Characteristic Roots Count Distance-Independent Circular Sets

75%

Skupień Z.

Discussiones Mathematicae Graph Theory

2013

tom 33

nr 1

217-229

Significant values of a combinatorial count need not fit the recurrence for the count. Consequently, initial values of the count can much outnumber those for the recurrence. So is the case of the count, Gl(n), of distance-l independent sets on the cycle Cn, studied by Comtet for l ≥ 0 and n ≥ 1 [sic]. We prove that values of Gl(n) are nth power sums of the characteristic roots of the corresponding recurrence unless 2 ≤ n ≤ l. Lucas numbers L(n) are thus generalized since L(n) is the count in question if l = 1. Asymptotics of the count for 1 ≤ l ≤ 4 involves the golden ratio (if l = 1) and three of the four smallest Pisot numbers inclusive of the smallest of them, plastic number, if l = 4. It is shown that the transition from a recurrence to an OGF, or back, is best presented in terms of mutually reciprocal (shortly: coreciprocal) polynomials. Also the power sums of roots (i.e., moments) of a polynomial have the OGF expressed in terms of the co-reciprocal polynomial.

Some alternating sums of Lucas numbers

75%

Čerin Z.

Open Mathematics

2005

tom 3

nr 1

1-13

We consider alternating sums of squares of odd and even terms of the Lucas sequence and alternating sums of their products. These alternating sums have nice representations as products of appropriate Fibonacci and Lucas numbers.

Ograniczanie wyników

1 Acta Arithmetica

1 Discussiones Mathematicae Graph Theory

1 Open Mathematics

1 Morgan M.

1 Skupień Z.

1 Čerin Z.

1 2013

1 2005

1 1998

Wyniki wyszukiwania

The distribution of second order linear recurrence sequences mod $2^m$

Sums of Powered Characteristic Roots Count Distance-Independent Circular Sets

Some alternating sums of Lucas numbers