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A214981 Number of terms in the greedy Lucas-and-Fibonacci representations of 1,2,...,n; partial sums of A214973.

%I A214981 #18 Feb 05 2021 18:18:44
%S A214981 1,2,3,4,5,7,8,9,11,13,14,16,17,19,21,23,25,26,28,30,31,33,35,37,39,
%T A214981 41,44,46,47,49,51,53,55,56,58,60,62,64,66,69,71,73,76,79,81,84,85,87,
%U A214981 89,91,93,95,98,100,101,103,105,107,109,111,114,116,118,121,124
%N A214981 Number of terms in the greedy Lucas-and-Fibonacci representations of 1,2,...,n; partial sums of A214973.
%C A214981 For comparison with Zeckendorf (Fibonacci) representations, it is conjectured that the limit of A179180(n)/A214981(n) exists and is between 1.2 and 1.4.
%H A214981 Clark Kimberling, <a href="/A214981/b214981.txt">Table of n, a(n) for n = 1..10000</a>
%H A214981 Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Kimberling/kimber12.html">Lucas Representations of Positive Integers</a>, J. Int. Seq., Vol. 23 (2020), Article 20.9.5.
%e A214981 The basis is B = (1,2,3,4,5,7,8,11,13,18,21,29,34,47,55,...), composed of Fibonacci numbers and Lucas numbers. Representations of positive integers using the greedy algorithm on B:
%e A214981    n  repres.   # terms  a(n)
%e A214981    1   1        1        1
%e A214981    2   2        1        2
%e A214981    3   3        1        3
%e A214981    4   4        1        4
%e A214981    5   5        1        5
%e A214981    6   5+1      2        7
%e A214981    7   7        1        8
%e A214981    8   8        1        9
%e A214981    9   8+1      2       11
%e A214981   10   8+2      2       13
%e A214981   27   21+5+1   3       44
%t A214981 (See the program at A214973.)
%Y A214981 Cf. A000032, A000045, A179180, A214973, A214977, A214979, A214980, A214981.
%K A214981 nonn
%O A214981 1,2
%A A214981 _Clark Kimberling_, Oct 22 2012
%E A214981 Edited by _Clark Kimberling_, Jun 13 2020