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).

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

Offset: 0

Author

Creighton Dement, Jul 31 2004

Keywords

• sign
• easy

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.
• Index entries for linear recurrences with constant coefficients, signature (0,4,0,-16).

Crossrefs

Cf. A048473, A094015.

Programs

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 *)

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

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

Extensions

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