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A255071
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Number of steps required to reach (2^n)-2 from 2^(n+1)-2 by iterating the map x -> x - (number of runs in binary representation of x).
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16
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1, 2, 3, 5, 9, 16, 29, 53, 97, 178, 328, 608, 1134, 2126, 4001, 7552, 14292, 27115, 51565, 98274, 187657, 358982, 687944, 1320793, 2540702, 4896919, 9456143, 18291753, 35435799, 68731296, 133436379, 259238717, 503912508, 979923792, 1906297165, 3709809375, 7222584181
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OFFSET
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1,2
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LINKS
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FORMULA
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Other identities and observations:
It seems that a(n) <= A213709(n) for all n >= 1. A254119 gives the difference between these two sequences.
Here secondmsb is implemented by the starting offset 2 version of A079944, and effectively gives the second most significant bit in the binary expansion of n. The formula follows from the semi-regular nature of number-of-runs beanstalk, as in the upper half of any next higher range [A255062(n+1) .. A255061(n+2)] of its infinite trunk (A255056), the beanstalk imitates its behavior in the range [A255062(n) .. A255061(n+1)].
(End)
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PROG
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(PARI)
A005811(n) = hammingweight(bitxor(n, n\2));
A255071(n) = { my(k, i); k = (2^(n+1))-2; i = 1; while(1, k = k - A005811(k); if(!bitand(k+1, k+2), return(i), i++)); };
for(n=1, 48, write("b255071.txt", n, " ", A255071(n)));
(Scheme)
(define (A079944off2 n) (A000035 (floor->exact (/ n (A072376 n))))) ;; Cf.
;; Shifted variant gives: (map A255071shifted (iota 16)) --> (0 1 2 3 5 9 16 29 53 97 178 328 608 1134 2126 4001)
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CROSSREFS
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A255069 gives the first differences of this sequence.
Cf. A005811, A079944, A236840, A255056, A255120, A255121, A255063, A255064, A255072, A255125, A255126, A254119.
a(n) differs from A192804(n+1) for the first time at n=11, where a(11) = 328, while A192804(12) = 327.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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