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A037096 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. 7
1, 2, 0, 204, 30840, 3743473440, 400814250895866480, 192435610587299441243182587501623263200, 2911899996313975217187797869354128351340558818020188112521784134070351919360 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
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))).
REFERENCES
S. Wolfram, A New Kind of Science, Wolfram Media Inc., (2002), p. 119.
LINKS
FORMULA
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))).
EXAMPLE
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.
MAPLE
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);
CROSSREFS
Cf. A036284, A037095, A037097, A136386 for related sequences.
Cf. also A004642, A265209, A265210 (for 2^n written in base 3).
Sequence in context: A269905 A012335 A012331 * A111814 A036938 A344463
KEYWORD
nonn,base
AUTHOR
Antti Karttunen, Jan 29 1999
EXTENSIONS
Entry revised by Antti Karttunen, Dec 29 2007
Name changed and the example corrected by Antti Karttunen, Dec 05 2015
STATUS
approved

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Last modified March 28 08:02 EDT 2024. Contains 371236 sequences. (Running on oeis4.)