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%I #26 Oct 08 2016 17:01:57
%S 0,1,2,5,4,13,10,21,8,45,26,93,20,109,42,85,16,173,90,477,52,957,186,
%T 733,40,749,218,1501,84,877,170,341,32,685,346,3549,180,12221,954,
%U 7133,104,14269,1914,49021,372,28605,1466,5853,80,5869,1498,30685,436,61373,3002,23517,168,12013,1754,24029,340,7021,682,1365
%N Unary-binary representation of Stern polynomials: a(n) = A156552(A260443(n)).
%C Sequence encodes Stern polynomials (see A125184, A260443) with "unary-binary method" where any nonzero coefficient c > 0 is encoded as a run of c 1-bits, separated from neighboring 1-runs by exactly one zero (this follows because A260442 is a subsequence of A073491). See the examples.
%C Terms which are not multiples of 4 form a subset of A003754, or in other words, each term is 2^k * {a term from a certain subsequence of A247648}, which subsequence remains to be determined.
%H Antti Karttunen, <a href="/A277020/b277020.txt">Table of n, a(n) for n = 0..8191</a>
%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F a(n) = A156552(A260443(n)).
%F Other identities. For all n >= 0:
%F A087808(a(n)) = n.
%F A000120(a(n)) = A002487(n).
%F a(2n) = 2*a(n).
%F a(2^n) = 2^n.
%F a(A000225(n)) = A002450(n).
%e n Stern polynomial Encoded as a(n)
%e "unary-binary" number (-> decimal)
%e ----------------------------------------------------------------
%e 0 B_0(x) = 0 "0" 0
%e 1 B_1(x) = 1 "1" 1
%e 2 B_2(x) = x "10" 2
%e 3 B_3(x) = x + 1 "101" 5
%e 4 B_4(x) = x^2 "100" 4
%e 5 B_5(x) = 2x + 1 "1101" 13
%e 6 B_6(x) = x^2 + x "1010" 10
%e 7 B_7(x) = x^2 + x + 1 "10101" 21
%e 8 B_8(x) = x^3 "1000" 8
%e 9 B_9(x) = x^2 + 2x + 1 "101101" 45
%o (Scheme)
%o (define (A277020 n) (A156552 (A260443 n)))
%o ;; Another implementation, more practical to run:
%o (define (A277020 n) (list_of_numbers_to_unary_binary_representation (A260443as_index_lists n)))
%o (definec (A260443as_index_lists n) (cond ((zero? n) (list)) ((= 1 n) (list 1)) ((even? n) (cons 0 (A260443as_index_lists (/ n 2)))) (else (add_two_lists (A260443as_index_lists (/ (- n 1) 2)) (A260443as_index_lists (/ (+ n 1) 2))))))
%o (define (add_two_lists nums1 nums2) (let ((len1 (length nums1)) (len2 (length nums2))) (cond ((< len1 len2) (add_two_lists nums2 nums1)) (else (map + nums1 (append nums2 (make-list (- len1 len2) 0)))))))
%o (define (list_of_numbers_to_unary_binary_representation nums) (let loop ((s 0) (nums nums) (b 1)) (cond ((null? nums) s) (else (loop (+ s (* (A000225 (car nums)) b)) (cdr nums) (* (A000079 (+ 1 (car nums))) b))))))
%Y Cf. A087808 (a left inverse), A156552, A260443, A277189 (odd bisection).
%Y Cf. also A000079, A000120, A000225, A002450, A002487, A003754, A073491, A247648, A260442, A277010, A277012, A276081.
%K nonn,base
%O 0,3
%A _Antti Karttunen_, Oct 07 2016
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