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%I #14 Sep 08 2022 08:46:15
%S 5,7,9,21,37,67,125,229,421,775,1425,2621,4821,8867,16309,29997,55173,
%T 101479,186649,343301,631429,1161379,2136109,3928917,7226405,13291431,
%U 24446753,44964589,82702773,152114115,279781477,514598365,946493957
%N a(n) = a(n - 1) + a(n - 2) + a(n - 3) for n>2, a(0)=5, a(1)=7, a(2)=9.
%C Tribonacci sequence beginning 5, 7, 9.
%C In general, the ordinary generating function for the recurrence relation b(n) = b(n-1) + b(n-2) + b(n-3), with n>2 and b(0)=k, b(1)=m, b(2)=q, is (k + (m-k)*x + (q-m-k)*x^2)/(1 - x - x^2 - x^3).
%H G. C. Greubel, <a href="/A268410/b268410.txt">Table of n, a(n) for n = 0..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TribonacciNumber.html">Tribonacci Number</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,1)
%F G.f.: (5 + 2*x - 3*x^2)/(1 - x - x^2 - x^3).
%F a(n) = 3*K(n) - 4*T(n+1) + 8*T(n), where K(n) = A001644(n) and T(n) =A000073(n+1). - _G. C. Greubel_, Apr 23 2019
%t LinearRecurrence[{1, 1, 1}, {5, 7, 9}, 40]
%t RecurrenceTable[{a[0]==5, a[1]==7, a[2]==9, a[n]==a[n-1]+a[n-2]+a[n-3]}, a, {n, 40}]
%o (Magma) I:=[5,7,9]; [n le 3 select I[n] else Self(n-1)+Self(n-2)+Self(n-3): n in [1..40]]; // _Vincenzo Librandi_, Feb 04 2016
%o (PARI) my(x='x+O('x^40)); Vec((5+2*x-3*x^2)/(1-x-x^2-x^3)) \\ _G. C. Greubel_, Apr 23 2019
%o (Sage) ((5+2*x-3*x^2)/(1-x-x^2-x^3)).series(x, 40).coefficients(x, sparse=False) # _G. C. Greubel_, Apr 23 2019
%o (GAP) a:=[5,7,9];; for n in [4..40] do a[n]:=a[n-1]+a[n-2]+a[n-3]; od; a; # _G. C. Greubel_, Apr 23 2019
%Y Cf. similar sequences with initial values (p,q,r): A000073 (0,0,1), A081172 (1,1,0), A001590 (0,1,0; also 1,2,3), A214899 (2,1,2), A001644 (3,1,3), A145027 (2,3,4), A000213 (1,1,1), A141036 (2,1,1), A141523 (3,1,1), A214727 (1,2,2), A214825 (1,3,3), A214826 (1,4,4), A214827 (1,5,5), A214828 (1,6,6), A214829 (1,7,7), A214830 (1,8,8), A214831 (1,9,9).
%K nonn,easy,less
%O 0,1
%A _Ilya Gutkovskiy_, Feb 04 2016
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