cp's OEIS Frontend

A277020 Unary-binary representation of Stern polynomials: a(n) = A156552(A260443(n)).

%I A277020 #26 Oct 08 2016 17:01:57
%S A277020 0,1,2,5,4,13,10,21,8,45,26,93,20,109,42,85,16,173,90,477,52,957,186,
%T A277020 733,40,749,218,1501,84,877,170,341,32,685,346,3549,180,12221,954,
%U A277020 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 A277020 Unary-binary representation of Stern polynomials: a(n) = A156552(A260443(n)).
%C A277020 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 A277020 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 A277020 Antti Karttunen, <a href="/A277020/b277020.txt">Table of n, a(n) for n = 0..8191</a>
%H A277020 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F A277020 a(n) = A156552(A260443(n)).
%F A277020 Other identities. For all n >= 0:
%F A277020 A087808(a(n)) = n.
%F A277020 A000120(a(n)) = A002487(n).
%F A277020 a(2n) = 2*a(n).
%F A277020 a(2^n) = 2^n.
%F A277020 a(A000225(n)) = A002450(n).
%e A277020 n    Stern polynomial       Encoded as              a(n)
%e A277020                             "unary-binary" number   (-> decimal)
%e A277020 ----------------------------------------------------------------
%e A277020 0    B_0(x) = 0                     "0"               0
%e A277020 1    B_1(x) = 1                     "1"               1
%e A277020 2    B_2(x) = x                    "10"               2
%e A277020 3    B_3(x) = x + 1               "101"               5
%e A277020 4    B_4(x) = x^2                 "100"               4
%e A277020 5    B_5(x) = 2x + 1             "1101"              13
%e A277020 6    B_6(x) = x^2 + x            "1010"              10
%e A277020 7    B_7(x) = x^2 + x + 1       "10101"              21
%e A277020 8    B_8(x) = x^3                "1000"               8
%e A277020 9    B_9(x) = x^2 + 2x + 1     "101101"              45
%o A277020 (Scheme)
%o A277020 (define (A277020 n) (A156552 (A260443 n)))
%o A277020 ;; Another implementation, more practical to run:
%o A277020 (define (A277020 n) (list_of_numbers_to_unary_binary_representation (A260443as_index_lists n)))
%o A277020 (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 A277020 (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 A277020 (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 A277020 Cf. A087808 (a left inverse), A156552, A260443, A277189 (odd bisection).
%Y A277020 Cf. also A000079, A000120, A000225, A002450, A002487, A003754, A073491, A247648, A260442, A277010, A277012, A276081.
%K A277020 nonn,base
%O A277020 0,3
%A A277020 _Antti Karttunen_, Oct 07 2016