OFFSET
0,2
COMMENTS
From M. F. Hasler, Mar 20 2018: (Start)
The matrix M is given by (leaving "," for easier use in CAS software):
[1, 2, 3, 4, 5, 6, 7, 8, 0, 0, 0, 0, 0, 0, 0, 0],
[2, 0, 4, 0, 6, 0, 0, 7, 0, 9, 0, 11, 0, 13, 16, 0],
[3, 0, 0, 2, 7, 8, 0, 0, 0, 12, 9, 0, 0, 13, 14, 0],
[4, 3, 0, 0, 8, 0, 6, 0, 0, 0, 10, 9, 0, 15, 0, 13],
[5, 0, 0, 0, 0, 2, 3, 4, 0, 14, 15, 16, 9, 0, 0, 0],
[6, 5, 0, 7, 0, 0, 0, 3, 0, 0, 16, 0, 10, 9, 12, 0],
[7, 8, 5, 0, 0, 4, 0, 0, 0, 0, 0, 14, 11, 0, 9, 10],
[8, 0, 6, 5, 0, 0, 2, 0, 0, 15, 0, 0, 12, 11, 0, 9],
[0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8],
[0, 9, 0, 11, 0, 13, 16, 0, 2, 9, 4, 0, 6, 0, 0, 7],
[0, 12, 9, 0, 0, 13, 14, 0, 3, 0, 0, 2, 7, 8, 0, 0],
[0, 0, 10, 9, 0, 15, 0, 13, 4, 3, 0, 0, 8, 0, 6, 0],
[0, 14, 15, 16, 9, 0, 0, 0, 5, 0, 0, 0, 0, 2, 3, 4],
[0, 0, 16, 0, 10, 9, 12, 0, 6, 5, 0, 7, 0, 0, 0, 3],
[0, 0, 0, 14, 11, 0, 9, 10, 7, 8, 5, 0, 0, 4, 0, 0],
[0, 15, 0, 0, 12, 11, 0, 9, 8, 0, 6, 5, 0, 0, 2, 0].
It has a block structure [A, B; B, C], where B has its nonzero entries in [9..16] and A and C have elements in [0..8]. It has 10 complex, 3 negative and 3 positive eigenvalues, the largest of which is ~ 66.4, equal to the growth rate of the sequence. (End)
LINKS
Geoffrey M. Dixon, Octonion Tutorial
Wikipedia, Octonion
Index entries for linear recurrences with constant coefficients, order 16, signature (11, 1370, 77929, 3643587, 60643963, 1886144770, 9489622853, 321733140518, -1184682856228, 17176558617832, -180204018965968, -1936508818801952, -5294214919102464, -11553018771853312, 548121408581582848, 1868826208968376320).
FORMULA
G.f.: (1 + 25*x + 650*x^2 + 6131*x^3 - 901216*x^4 - 10037472*x^5 - 763524904*x^6 - 1431773954*x^7 - 165714608004*x^8 + 940706179040*x^9 - 12149482438608*x^10 + 131168170705952*x^11 + 385574909641728*x^12 + 4510688027342848*x^13 + 14569369451429888*x^14 - 467206552242094080*x^15)/(1 - 11*x - 1370*x^2 - 77929*x^3 - 3643587*x^4 - 60643963*x^5 - 1886144770*x^6 - 9489622853*x^7 - 321733140518*x^8 + 1184682856228*x^9 - 17176558617832*x^10 + 180204018965968*x^11 + 1936508818801952*x^12 + 5294214919102464*x^13 + 11553018771853312*x^14 - 548121408581582848*x^15 - 1868826208968376320*x^16). - M. F. Hasler, Mar 20 2018
MATHEMATICA
M = {{1, 2, 3, 4, 5, 6, 7, 8, 0, 0, 0, 0, 0, 0, 0, 0}, {2, 0, 4, 0, 6, 0, 0, 7, 0, 9, 0, 11, 0, 13, 16, 0}, {3, 0, 0, 2, 7, 8, 0, 0, 0, 12, 9, 0, 0, 13, 14, 0}, {4, 3, 0, 0, 8, 0, 6, 0, 0, 0, 10, 9, 0, 15, 0, 13}, {5, 0, 0, 0, 0, 2, 3, 4, 0, 14, 15, 16, 9, 0, 0, 0}, {6, 5, 0, 7, 0, 0, 0, 3, 0, 0, 16, 0, 10, 9, 12, 0}, {7, 8, 5, 0, 0, 4, 0, 0, 0, 0, 0, 14, 11, 0, 9, 10}, {8, 0, 6, 5, 0, 0, 2, 0, 0, 15, 0, 0, 12, 11, 0, 9}, {0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8}, {0, 9, 0, 11, 0, 13, 16, 0, 2, 9, 4, 0, 6, 0, 0, 7}, {0, 12, 9, 0, 0, 13, 14, 0, 3, 0, 0, 2, 7, 8, 0, 0}, {0, 0, 10, 9, 0, 15, 0, 13, 4, 3, 0, 0, 8, 0, 6, 0}, {0, 14, 15, 16, 9, 0, 0, 0, 5, 0, 0, 0, 0, 2, 3, 4}, {0, 0, 16, 0, 10, 9, 12, 0, 6, 5, 0, 7, 0, 0, 0, 3}, {0, 0, 0, 14, 11, 0, 9, 10, 7, 8, 5, 0, 0, 4, 0, 0}, {0, 15, 0, 0, 12, 11, 0, 9, 8, 0, 6, 5, 0, 0, 2, 0}}; v[1] = {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}; v[n_] := v[n] = M.v[n - 1]; a = Table[Floor[v[n][[1]]], {n, 1, 25}]
CharacteristicPolynomial[M, x] (* Roots are the eigenvalues, largest of which is ~ 66.4. - M. F. Hasler, Mar 20 2018 *)
PROG
(PARI) M=Mat([/*paste here the list of rows from comments*/]~); a(n)=vecsum((M^n)[1, ]) \\ Use mateigen(M, 1)[1] to see the eigenvalues, vecmax(abs(%)) shows the largest is ~ 66.4. M. F. Hasler, Mar 20 2018
CROSSREFS
KEYWORD
nonn,easy,less
AUTHOR
Roger L. Bagula, Jul 02 2006
EXTENSIONS
Edited by M. F. Hasler, Mar 20 2018
STATUS
approved