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A105726
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Involution of nonnegative integers. A deeply recursive variant of A056539.
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2
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0, 1, 2, 3, 6, 5, 4, 7, 14, 9, 10, 13, 12, 11, 8, 63, 126, 17, 22, 25, 26, 21, 18, 29, 28, 19, 20, 27, 24, 23, 64, 31, 62, 129, 46, 49, 54, 41, 38, 57, 58, 37, 42, 53, 50, 45, 34, 253, 252, 35, 44, 51, 52, 43, 36, 59, 56, 39, 40, 55, 192, 191, 32, 15, 30, 65, 382, 385, 110
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OFFSET
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0,3
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LINKS
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EXAMPLE
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By definition a(0)=0 and a(1)=1.
a(2)=2, as 2 = 10 in binary, thus its run counts are (1 1), which reversed yields the same (1 1), from which we collect a binary number 10, i.e. 2 in decimal.
a(4)=6, as 4 = 100 in binary, thus its run counts are (1 2), which reversed and each element mapped through a itself yields (2 1), which are run counts of binary number 110, i.e. 6 in decimal.
a(16)=126, as 16 = 10000 in binary, thus its run counts are (1 4), which reversed yields (4 1) and a(4)=6 and a(1)=1, thus we find that the run counts (6 1) form a binary 1111110 i.e. 126 in decimal.
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PROG
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(Scheme): (define (A105726 n) (cond ((< n 2) n) (else (runcount1list->binexp (map A105726 (reverse! (binexp->runcount1list n))))))) (define (binexp->runcount1list n) (if (zero? n) (list) (let loop ((n n) (rc (list)) (count 0) (prev-bit (modulo n 2))) (if (zero? n) (cons count rc) (if (eq? (modulo n 2) prev-bit) (loop (floor->exact (/ n 2)) rc (1+ count) (modulo n 2)) (loop (floor->exact (/ n 2)) (cons count rc) 1 (modulo n 2))))))) (define (runcount1list->binexp lista) (let loop ((lista lista) (s 0) (state 1)) (cond ((null? lista) s) (else (loop (cdr lista) (+ (* s (expt 2 (car lista))) (* state (- (expt 2 (car lista)) 1))) (- 1 state))))))
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CROSSREFS
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Differs from its "shallow" version A056539 first time at n=15, where A056539(15)=15, but here a(15)=63.
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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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