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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). 5
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
C. Dement, The Math Forum.
FORMULA
a(n)= 4*a(n-2)-16*a(n-4). G.f.: 4*(1-x-2*x^2-4*x^3)/(1-4*x^2+16*x^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
a(n) = 4*(-1)^floor((n+1)/2)*A138230(n). - R. J. Mathar, May 21 2019
MATHEMATICA
CoefficientList[Series[4(1-x-2x^2-4x^3)/(1-4x^2+16x^4), {x, 0, 40}], x] (* or *) LinearRecurrence[ {0, 4, 0, -16}, {4, -4, 8, -32}, 40] (* Harvey P. Dale, Feb 15 2024 *)
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
Sequence in context: A019122 A019202 A137717 * A102369 A298569 A281717
KEYWORD
sign,easy,changed
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 February 22 05:42 EST 2024. Contains 370240 sequences. (Running on oeis4.)