A266191 Sequence beginning with a(1)=1, a(2)=2, a(3)=3 and then after each new term a(n) is selected as the least unused number for which a(n)*a(n-1)*a(n-2) is a fibbinary number (A003714), i.e., has no adjacent 1's in its base-2 representation.

1, 2, 3, 6, 4, 7, 12, 8, 11, 15, 16, 22, 24, 5, 39, 14, 10, 59, 28, 20, 67, 31, 9, 63, 19, 18, 27, 38, 17, 54, 47, 34, 44, 51, 33, 88, 30, 32, 43, 48, 21, 86, 23, 42, 35, 46, 84, 70, 91, 168, 71, 55, 36, 75, 99, 40, 135, 110, 41, 60, 111, 80, 120, 62, 160, 134, 56, 141, 76, 112, 64, 78, 119, 113, 103, 183, 37, 167, 366, 73

Offset

1,2

Comments

It is conjectured that this sequence is not only injective, but also surjective on N, i.e., that it is a true permutation of natural numbers.

See also comments in related A266121.

Links

Antti Karttunen, Table of n, a(n) for n = 1..12000

Eric Angelini, a(n)*a(n+1) shows at least twice the same digit, Posting on SeqFan-list Dec 21 2015. [Source of inspiration for this sequence.]

Index entries for sequences that are permutations of the natural numbers

Example

For n=4, we start testing from the least so far unused number, which is 4, by multiplying it by a(3)*a(2) = 6. Because 6*4 = 24, which has two adjacent 1's in its binary representation "11000", 6 is disqualified. Next we try 5, and 6*5 = "11110", and 5 is also disqualified. Next we try 6, and 6*6 = "100100", with no adjacent 1's, and we have found the least unused number satisfying the required condition, thus we set a(4) = 6.

Prog

(Scheme, with defineperm1-macro from Antti Karttunen's IntSeq-library)
(defineperm1 (A266191 n) (cond ((<= n 3) n) (else (let ((prev1 (A266191 (- n 1))) (prev2 (A266191 (- n 2)))) (let loop ((k 1)) (cond ((and (not-lte? (A266192 k) (- n 1)) (isA003714? (* k prev1 prev2))) k) (else (loop (+ 1 k)))))))))
(define (isA003714? n) (= (* 3 n) (A003987bi n (* 2 n)))) ;; Where A003987bi implements bitwise-XOR (see A003987).
;; We consider a > b (i.e., not less than b) also in case a is #f.
;; (Because of the stateful caching system used by defineperm1-macro):
(define (not-lte? a b) (cond ((not (number? a)) #t) (else (> a b))))

Crossrefs

Left inverse: A266192 (also the right inverse if this sequence is a permutation of natural numbers).

Cf. A003714, A003987.

Cf. also A266117, A266121 and A266195 for similar permutations.

Sequence in context: A226615 A138608 A092283 * A273338 A099900 A193903

Adjacent sequences: A266188 A266189 A266190 * A266192 A266193 A266194

Keyword

nonn,base,changed

Author

Antti Karttunen, Dec 23 2015

Status

approved