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a(n) = a(n-1) + 8a(n-2) - 3a(n-3) - 17a(n-4) + a(n-5) + 13a(n-6) + 4a(n-7) - 6a(n-8) - a(n-9).
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%I #10 Apr 06 2018 11:48:30

%S 0,0,0,1,4,9,33,79,250,647,1914,5157,14797,40663,115036,319098,896896,

%T 2498663,7002446,19546383,54706009,152839003,427510168,1194858679,

%U 3341293220,9340298995,26116085175,73010960431,204132824544,570699926244,1595594838888

%N a(n) = a(n-1) + 8a(n-2) - 3a(n-3) - 17a(n-4) + a(n-5) + 13a(n-6) + 4a(n-7) - 6a(n-8) - a(n-9).

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Proton">Proton</a>.

%F G.f.: x^2(x - 1)(x^4 + 4x^3 - x^2 - 4x - 1)/((x^3 + x^2 - 1)(x^6 + 5x^5 - 9x^4 - 3x^3 + 7x^2 + x - 1))

%F a(n) = a(n-1) + 8a(n-2) - 3a(n-3) - 17a(n-4) + a(n-5) + 13a(n-6) + 4a(n-7) - 6a(n-8) - a(n-9). [M. F. Hasler, Nov 24 2010]

%F M = {{0, 1, 0, 1, 0, 0, 0, 0, 0},

%F {0, 0, 1, 0, 1, 0, 0, 0, 0},

%F {1, 1, 0, 0, 0, 1, 0, 0, 0},

%F {1, 0, 0, 0, 0, 1, 1, 0, 0},

%F {0, 1, 0, 1, 0, 0, 0, 1, 0},

%F {0, 0, 1, 0, 1, 1, 0, 0, 1},

%F {0, 0, 0, 1, 0, 0, 0, 1, 0},

%F {0, 0, 0, 0, 1, 0, 0, 0, 1},

%F {0, 0, 0, 0, 0, 1, 1, 1, 0}};

%F v(n)=M.v(n-1) ; a(n) is the first component of v(n).

%t M = {{0, 1, 0, 1, 0, 0, 0, 0, 0},

%t {0, 0, 1, 0, 1, 0, 0, 0, 0},

%t {1, 1, 0, 0, 0, 1, 0, 0, 0},

%t {1, 0, 0, 0, 0, 1, 1, 0, 0},

%t {0, 1, 0, 1, 0, 0, 0, 1, 0},

%t {0, 0, 1, 0, 1, 1, 0, 0, 1},

%t {0, 0, 0, 1, 0, 0, 0, 1, 0},

%t {0, 0, 0, 0, 1, 0, 0, 0, 1},

%t {0, 0, 0, 0, 0, 1, 1, 1, 0}};

%t v[0] = {0, 0, 0, 0, 0, 0, 0, 0, 1};

%t v[n_] := v[n] = M.v[n - 1]

%t Table[v[n][[1]], {n, 0, 30}]

%Y Cf. A000931, A123589.

%K nonn,easy,less

%O 0,5

%A _Roger L. Bagula_, Nov 24 2010

%E Better name (using recurrence) from _Joerg Arndt_, Apr 06 2018