(MAGMAMagma) 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

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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

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).

G. C. Greubel, <a href="/A268410/b268410.txt">Table of n, a(n) for n = 0..1000</a>

a(n) = 3*K(n) - 4*T(n+1) + 8*T(n), where K(n) = A001644(n) and T(n) =A000073(n). - G. C. Greubel, Apr 23 2019

LinearRecurrence[{1, 1, 1}, {5, 7, 9}, 3340]

RecurrenceTable[{a[0] == 5, a[1] == 7, a[2] == 9, a[n] == a[n - 1] + a[n - 2] + a[n - 3]}, a, {n, 3240}]

(MAGMA) I:=[5, 7, 9]; [n le 3 select I[n] else Self(n-1)+Self(n-2)+Self(n-3): n in [1..3540]]; // Vincenzo Librandi, Feb 04 2016

(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

(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

(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

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nonn,easy,less,changed

a(n) = a(n - 1) + a(n - 2) + a(n - 3) for n>2, a(0)=5, a(1)=7, a(2)=9.

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).

Cf. A000073, A000213, A001590, A001644, A007486, A020992, A073728, A081172, A086192, A086213, A100683, A101757, A101758, A135491, A141036, A141523, A145027, A186830, A213665, A214727, A214825, A214826, A214827, A214828, A214829, A214830, A214831, A214899, A248959.

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).

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