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%I #23 Jan 17 2024 15:04:27
%S 2,3,2,2,2,3,2,2,3,5,5,3,2,2,3,2,2,3,5,5,3,2,2,3,3,4,5,5,4,3,3,2,2,3,
%T 5,5,3,2,2,3,2,2,3,5,5,3,2,2,3,3,4,5,5,4,3,3,2,2,3,5,5,3,2,3,5,5,4,5,
%U 7,8,5,4,5,8,7,5,4,5,5,3,2,3,5,5,3,2,2,3,3,4,5,5,4,3,3,2,2,3,5,5
%N Number of representations of n in the Fibonacci-squared base system. The columns are ..., 64, 25, 9, 4, 1, 1 = ..., 8^2, 5^2, 3^2, 2^2, 1^2, 1^2, i.e., the Fibonacci numbers A000045 squared. The 'digits' are 0, 1 or 2.
%C Note there are two columns labeled 1.
%H David A. Corneth, <a href="/A147561/b147561.txt">Table of n, a(n) for n = 1..10000</a>
%H Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibrep.html#fibsqrep">Fibonacci Bases: The Fibonacci^2 Base System</a>
%e a(2) = 3 since 2 is 02, 20 and 11 using both columns labeled 1;
%e a(10) = 5 because 10 = 9 + 1 with 2 Fib-sq reps 1010, 1001; 10 = 2*4 + 2 with 3 Fib-sq reps 220, 211 and 202; so there are in total 5 Fib-sq representations for 10.
%o (PARI) first(n) = {my(fib2list = List(), fib2 = 1, t = 1, res = vector(n)); while(fib2 <= n, listput(fib2list, fib2); t++; fib2 = fibonacci(t)^2); for(i=1,3^#fib2list-1, b = digits(i,3); b = concat(vector(#fib2list-#b),b); s = sum(i=1,#b, b[i]*fib2list[i]); if(s<=n, res[s]++));res} \\ _David A. Corneth_, Jul 24 2017
%Y Cf. A000045, A007598.
%K nonn,base,look
%O 1,1
%A _Ron Knott_, Nov 07 2008
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