NNTDM 15 (2009) 2, 1-44
A remark on ψ-function and Pell-Padovan’s sequence 1
Krassimir Atanassov1 , Dimitar Dimitrov1 and Anthony Shannon2 CLBME - Bulg. Academy of Sci., P.O.Box 12, Sofia-1113, Bulgaria, e-mail:
[email protected],
[email protected] 2 Warrane College, University of New South Wales, Kensington, 1465, Australia, email:
[email protected]
1
Introduction
The Pell-Padovan sequence is defined [4, 6] by P (0) = P (1) = P (2) = 1, P (n + 3) = 2P (n + 1) + P (n), for every natural number n ≥ 1. We shall investigate a property of an extended form of this sequence [6, 7] P (1) = a, P (2) = b, P (3) = c, P (n + 3) = pP (n + 1) + q(n), for every natural number n ≥ 1, where a, b, c, p, q are fixed natural numbers. The paper is a continuation of [2]
2
Remarks on ψ-function
A digital arithmetic function will be described, following [1, 3, 5]. Let n=
k X
ai .10k−i ≡ a1 a2 ...ak ,
i=1
where ai is a natural number and 0 ≤ ai ≤ 9 (1 ≤ i ≤ k). Let for n = 0 : ϕ(n) = 0 and for n > 0: ϕ(n) =
k X
ai .
i=1
We shall use the decimal count system everywhere hereafter. Let us define a sequence of functions ϕ0 , ϕ1 , ϕ2 , ..., where (l is a natural number) ϕ0 (n) = n, 1
ϕl+1 = ϕ(ϕl (n)). Since for k > 1 ϕ(n) =
k X
ai
