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A096252 Array read by rows, starting with n=0: row n lists A057077(n+1)*8^(n+1)/2, A057077(n+2)*8^(n+1)/2, A057077(n+1)*8^(n+1). 2
4, -4, 8, -32, -32, -64, -256, 256, -512, 2048, 2048, 4096, 16384, -16384, 32768, -131072, -131072, -262144, -1048576, 1048576, -2097152, 8388608, 8388608, 16777216, 67108864, -67108864, 134217728, -536870912, -536870912, -1073741824 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

a(n) = ves( ('i + 'ii' + 'ij' + 'ik')^n ) a(n) = ves( ('j + 'jj' + 'ji' + 'jk')^n ) a(n) = ves( ('k + 'kk' + 'ki' + 'kj')^n ).

The elements x = 'i + 'ii' + 'ij' + 'ik'; y = 'j + 'jj' + 'ji' + 'jk'; and z = 'k + 'kk' + 'ki' + 'kj' are elements of the ring generated from the quaternion factor space Q X Q / {(1,1), (-1,-1)}. Each is represented by a gray shaded area of "Floret's cube". The elements x/2, y/2, z/2 are members of a group, itself a subset of the real algebra generated from Q X Q / {(1,1), (-1,-1)}, which is isomorphic to Q X C_3 (order 24).

This sequence is the term-wise sum of three sequences: a(n) = ves(x^n) = jes(x^n) + les(x^n) + tes(x^n), where jes(x^n)=(1, -6, 8, -24, 16, 0, -64, 384, -512, 1536, -1024, 0, 4096, -24576, 32768, -98304, ...), les(x^n)=(3, 0, 0, 0, -48, 0 -192, 0, 0, 0, 3072, 0, 12288, 0, 0, 0, ...), tes(x^n)=(0, 2, 0, -8, 0, -64, 0, -128, 0, 512, 0, 4096, 0, 8192, 0, -32768, ...). Concerning "les"- notice that if (..., s, 0, 0, 0, t, ...), then t = -16s and if (..., s, 0, t, ...), then t = 4s.

LINKS

Danny Rorabaugh, Table of n, a(n) for n = 0..1000

C. Dement, The Math Forum.

FORMULA

Conjectures: a(n)= 4*a(n-2)-16*a(n-4). G.f.: 4(1-x-2x^2-4x^3)/(1-4x^2+16x^4). - R. J. Mathar, Nov 26 2008

a(n) = (-1)^(floor((floor(n/3)+((n mod 3) mod 2)+1)/2)) * 8^(floor(n/3)+1) / 2^(((n+1)^2) mod 3). - Danny Rorabaugh, May 13 2016

PROG

(Sage)

[(-1)^(floor((floor(n/3)+((n%3)%2)+1)/2)) * 8^(floor(n/3)+1) / 2^(((n+1)^2)%3) for n in range(30)]

# Danny Rorabaugh, May 13 2016

CROSSREFS

Cf. A048473, A094015.

Sequence in context: A019122 A019202 A137717 * A102369 A281717 A205971

Adjacent sequences:  A096249 A096250 A096251 * A096253 A096254 A096255

KEYWORD

sign,easy

AUTHOR

Creighton Dement, Jul 31 2004

EXTENSIONS

Edited with clearer definition by Omar E. Pol, Dec 29 2008

STATUS

approved

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Last modified September 23 21:28 EDT 2017. Contains 292391 sequences.