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A277020
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Unary-binary representation of Stern polynomials: a(n) = A156552(A260443(n)).
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10
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0, 1, 2, 5, 4, 13, 10, 21, 8, 45, 26, 93, 20, 109, 42, 85, 16, 173, 90, 477, 52, 957, 186, 733, 40, 749, 218, 1501, 84, 877, 170, 341, 32, 685, 346, 3549, 180, 12221, 954, 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
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OFFSET
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0,3
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COMMENTS
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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.
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.
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LINKS
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FORMULA
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Other identities. For all n >= 0:
a(2n) = 2*a(n).
a(2^n) = 2^n.
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EXAMPLE
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n Stern polynomial Encoded as a(n)
"unary-binary" number (-> decimal)
----------------------------------------------------------------
0 B_0(x) = 0 "0" 0
1 B_1(x) = 1 "1" 1
2 B_2(x) = x "10" 2
3 B_3(x) = x + 1 "101" 5
4 B_4(x) = x^2 "100" 4
5 B_5(x) = 2x + 1 "1101" 13
6 B_6(x) = x^2 + x "1010" 10
7 B_7(x) = x^2 + x + 1 "10101" 21
8 B_8(x) = x^3 "1000" 8
9 B_9(x) = x^2 + 2x + 1 "101101" 45
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PROG
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(Scheme)
;; Another implementation, more practical to run:
(define (A277020 n) (list_of_numbers_to_unary_binary_representation (A260443as_index_lists n)))
(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))))))
(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)))))))
(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))))))
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CROSSREFS
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Cf. also A000079, A000120, A000225, A002450, A002487, A003754, A073491, A247648, A260442, A277010, A277012, A276081.
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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