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A379739 Subdiagonal of the Hurt-Sada array.

%I A379739 #34 Jan 19 2025 14:26:26
%S A379739 2,3,4,3,7,8,9,6,7,13,14,9,10,18,19,12,13,23,24,25,16,28,29,30,19,20,
%T A379739 34,35,22,23,39,40,25,26,44,45,46,29,49,50,51,32,54,55,56,35,36,60,61,
%U A379739 38,39,65,66,67,42,70,71,72,45,75,76,77,48,49,81,82,51
%N A379739 Subdiagonal of the Hurt-Sada array.
%C A379739 The array is depicted in A368050.
%H A379739 Charles R Greathouse IV, <a href="/A379739/b379739.txt">Table of n, a(n) for n = 1..10000</a>
%H A379739 Jeffrey Shallit, <a href="/A379739/a379739.pdf">Automaton deciding whether a(n) >= n</a>
%H A379739 Jeffrey Shallit, <a href="https://arxiv.org/abs/2501.08823">The Hurt-Sada Array and Zeckendorf Representations</a>, arXiv:2501.08823 [math.NT], 2025. See p. 2.
%F A379739 If a(n) < n, then a(n) = floor((4-2*g)n + (5-3*g)), where g = (1+sqrt(5))/2, the golden ratio.  If a(n) >= n, then a(n) = floor((2*g-2)n) + 1.  There is an 8-state automaton (see the links section) that takes the Zeckendorf representation of n as input and decides whether a(n) >= n.
%o A379739 (PARI) zeckendorf(n)=my(f1=1,f2=2,fibs=List([1]),rep=List(),i); while(f2<=n,listput(fibs,f2);[f1,f2]=[f2,f1+f2];); i=#fibs; while(i,if(fibs[i]>n, i--, listput(rep,1); n-=fibs[i]; if(i==1,break); i-=2); listput(rep,0)); Vec(rep)
%o A379739 a(n)=my(g=quadgen(5), Z=zeckendorf(n), state=1, new0=[1, 3, 4, 6, 3, 8, 6, 8], new1=[2, 0, 5, 7, 0, 5, 0, 2]); for(i=1, #Z, state=if(Z[i], new1, new0)[state]); if(state!=5 && state!=8, 2*n/g+1, (4-2*g)*n + (5-3*g))\1 \\ _Charles R Greathouse IV_, Jan 14 2025
%Y A379739 Cf. A368050, A380079.
%K A379739 nonn
%O A379739 1,1
%A A379739 _Jeffrey Shallit_, Jan 14 2025