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A383426 Indices k such that A003849(k) = 1 and A383422(k) = 0.

%I A383426 #16 May 22 2025 17:50:19
%S A383426 4,14,22,25,33,40,43,51,61,69,72,80,90,98,101,108,116,119,127,137,145,
%T A383426 148,156,163,166,174,177,184,192,195,203,213,221,224,232,239,242,250,
%U A383426 260,268,271,279,286,289,297
%N A383426 Indices k such that A003849(k) = 1 and A383422(k) = 0.
%C A383426 The positive integers are partitioned by this sequence together with A383423, A383424, and A383425.
%C A383426 Conjecture: {a(n) - a(n-1), n>=2} = {3, 7, 8, 10}.
%H A383426 Chai Wah Wu, <a href="/A383426/b383426.txt">Table of n, a(n) for n = 1..10000</a>
%t A383426 wF = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0}}] &, {0}, 15];  (* A003849 *)
%t A383426 s[0] = "0"; s[1] = "011"; s[n_] := StringJoin[s[n - 1], s[n - 2]]; (* A383422 *)
%t A383426 wL = Join[{0}, IntegerDigits[FromDigits[s[15]]]];
%t A383426 -1+Select[Range[400], wF[[#]] == 1 && wL[[#]] == 0 &]
%o A383426 (Python)
%o A383426 from math import isqrt
%o A383426 from itertools import count, islice
%o A383426 def A383426_gen(): # generator of terms
%o A383426     for n in count(1):
%o A383426         if ((k:=(n+isqrt(5*n**2)&-2)-n)+1+isqrt(m:=5*(k+1)**2)>>1)-(k+isqrt(m-10*k-5)>>1)==1: yield k-1
%o A383426 A383426_list = list(islice(A383426_gen(),45)) # _Chai Wah Wu_, May 22 2025
%Y A383426 Cf. A003849, A383422, A383423, A383424, A383425, A383427.
%K A383426 nonn
%O A383426 1,1
%A A383426 _Clark Kimberling_, Apr 27 2025