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%I #17 Mar 03 2016 22:49:09
%S 3,6,7,12,11,15,22,24,14,19,27,38,28,23,46,48,43,30,39,35,59,44,31,75,
%T 62,87,51,83,56,47,88,54,76,55,96,86,60,71,67,70,78,112,79,107,102,91,
%U 120,139,118,140,119,142,131,134,155,240,156,135,152,108,95,92,103,179,184,115,147,224,94,175,123,150,111,158,214,163,203
%N Lexicographically first injection of natural numbers beginning with a(1)=3 such that for all n >= 1, a(n)*a(n+1) is a fibbinary number (A003714), i.e., has no adjacent 1's in its base-2 representation.
%C The sequence is conjectured to be a permutation of A091067.
%C The scatter plot is quite interesting (essentially the same as A269367). Compare also to the graph of A269361.
%H Antti Karttunen, <a href="/A269363/b269363.txt">Table of n, a(n) for n = 1..65537</a>
%t fibbinaryQ[n_] := BitAnd[n, 2 n]==0; a[1]=3; a[n_] := a[n] = For[k=1, True, k++, If[Mod[k, 4] != 1, If[fibbinaryQ[a[n-1] k], If[FreeQ[Array[a, n-1], k], Return[k]]]]]; Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Mar 02 2016 *)
%o (Scheme, with defineperm1-macro from _Antti Karttunen_'s IntSeq-library)
%o (defineperm1 (A269363 n) (cond ((= 1 n) 3) (else (let ((prev (A269363 (- n 1)))) (let loop ((k 1)) (cond ((and (not-lte? (A269363inv_cache k) (- n 1)) (isa003714? (* k prev))) k) (else (loop (+ 1 k)))))))))
%o (define (A269363inv_cache n) (A269363 (- n)))
%o ;; We consider a > b (i.e. not less than b) also in case a is #f.
%o ;; (Because of the stateful caching system used by defineperm1-macro):
%o (define (not-lte? a b) (cond ((not (number? a)) #t) (else (> a b))))
%o (define (isA003714? n) (= (* 3 n) (A003987bi n (* 2 n)))) ;; Where A003987bi implements bitwise-XOR (see A003987).
%Y Cf. A003714, A003987, A091067, A266121, A266191, A269361, A269365, A269366.
%Y Cf. A269367 (the terms ranked with A255070).
%K nonn,base,look
%O 1,1
%A _Antti Karttunen_, Feb 25 2016
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