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A037096
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Periodic vertical binary vectors computed for powers of 3: a(n) = Sum_{k=0 .. (2^n)-1} (floor((3^k)/(2^n)) mod 2) * 2^k.
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7
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1, 2, 0, 204, 30840, 3743473440, 400814250895866480, 192435610587299441243182587501623263200, 2911899996313975217187797869354128351340558818020188112521784134070351919360
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OFFSET
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0,2
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COMMENTS
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This sequence can be also computed with a recurrence that does not explicitly refer to 3^n. See the C-program.
Conjecture: For n>=3, each term a(n), when considered as a GF(2)[X]-polynomial, is divisible by the GF(2)[X]-polynomial (x + 1) ^ A055010(n-1). If this holds, then for n>=3, a(n) = A048720(A136386(n), A048723(3,A055010(n-1))).
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REFERENCES
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S. Wolfram, A New Kind of Science, Wolfram Media Inc., (2002), p. 119.
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LINKS
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A. Karttunen, Table of n, a(n) for n = 0..11
A. Karttunen, C program for computing this sequence
S. Wolfram, A New Kind of Science, Wolfram Media Inc., (2002), p. 119.
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FORMULA
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a(n) = Sum_{k=0 .. A000225(n)} (floor(A000244(k)/(2^n)) mod 2) * 2^k.
Other identities and observations:
For n>=2, a(n) = A000215(n-1)*A037097(n) = A048720(A037097(n), A048723(3, A000079(n-1))).
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EXAMPLE
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When powers of 3 are written in binary (see A004656), under each other as:
000000000001 (1)
000000000011 (3)
000000001001 (9)
000000011011 (27)
000001010001 (81)
000011110011 (243)
001011011001 (729)
100010001011 (2187)
it can be seen that the bits in the n-th column from the right can be arranged in periods of 2^n: 1, 2, 4, 8, ... This sequence is formed from those bits: 1, is binary for 1, thus a(0) = 1. 01, reversed is 10, which is binary for 2, thus a(1) = 2, 0000 is binary for 0, thus a(2)=0, 000110011, reversed is 11001100 = A007088(204), thus a(3) = 204.
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MAPLE
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a(n) := sum( 'bit_n(3^i, n)*(2^i)', 'i'=0..(2^(n))-1);
bit_n := (x, n) -> `mod`(floor(x/(2^n)), 2);
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CROSSREFS
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Cf. A036284, A037095, A037097, A136386 for related sequences.
Cf. A000079, A000215, A000225, A000244, A004656, A007088, A048720, A048723, A055010.
Cf. also A004642, A265209, A265210 (for 2^n written in base 3).
Sequence in context: A269905 A012335 A012331 * A111814 A036938 A221750
Adjacent sequences: A037093 A037094 A037095 * A037097 A037098 A037099
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KEYWORD
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nonn,base
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AUTHOR
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Antti Karttunen, Jan 29 1999
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EXTENSIONS
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Entry revised by Antti Karttunen, Dec 29 2007
Name changed and the example corrected by Antti Karttunen, Dec 05 2015
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STATUS
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approved
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