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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

%I #28 Jan 14 2024 08:59:18

%S 1,2,0,204,30840,3743473440,400814250895866480,

%T 192435610587299441243182587501623263200,

%U 2911899996313975217187797869354128351340558818020188112521784134070351919360

%N 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.

%C This sequence can be also computed with a recurrence that does not explicitly refer to 3^n. See the C program.

%C 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))).

%D S. Wolfram, A New Kind of Science, Wolfram Media Inc., (2002), p. 119.

%H Antti Karttunen, <a href="/A037096/b037096.txt">Table of n, a(n) for n = 0..11</a>

%H Antti Karttunen, <a href="/A036284/a036284.c.txt">C program for computing this sequence</a>.

%H S. Wolfram, <a href="http://www.wolframscience.com/nksonline/page-119">A New Kind of Science, Wolfram Media Inc., (2002), p. 119</a>.

%F a(n) = Sum_{k=0 .. A000225(n)} (floor(A000244(k)/(2^n)) mod 2) * 2^k.

%F Other identities and observations:

%F For n >= 2, a(n) = A000215(n-1)*A037097(n) = A048720(A037097(n), A048723(3, A000079(n-1))).

%e When powers of 3 are written in binary (see A004656), under each other as:

%e 000000000001 (1)

%e 000000000011 (3)

%e 000000001001 (9)

%e 000000011011 (27)

%e 000001010001 (81)

%e 000011110011 (243)

%e 001011011001 (729)

%e 100010001011 (2187)

%e 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.

%p a(n) := sum( 'bit_n(3^i, n)*(2^i)', 'i'=0..(2^(n))-1);

%p bit_n := (x, n) -> `mod`(floor(x/(2^n)), 2);

%Y Cf. A036284, A037095, A037097, A136386 for related sequences.

%Y Cf. A000079, A000215, A000225, A000244, A004656, A007088, A048720, A048723, A055010.

%Y Cf. also A004642, A265209, A265210 (for 2^n written in base 3).

%K nonn,base

%O 0,2

%A _Antti Karttunen_, Jan 29 1999

%E Entry revised by _Antti Karttunen_, Dec 29 2007

%E Name changed and the example corrected by _Antti Karttunen_, Dec 05 2015

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Last modified March 28 13:21 EDT 2024. Contains 371254 sequences. (Running on oeis4.)