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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

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.

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. 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

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 December 10 20:48 EST 2017. Contains 295856 sequences.