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A113301
Sum of odd-indexed terms of tribonacci numbers.
2
0, 1, 5, 18, 62, 211, 715, 2420, 8188, 27701, 93713, 317030, 1072506, 3628263, 12274327, 41523752, 140473848, 475219625, 1607656477, 5438662906, 18398864822, 62242913851, 210566269283, 712340586524, 2409830942708, 8152399683933, 27579370581033, 93300342369742
OFFSET
0,3
COMMENTS
A000073 is the tribonacci numbers. A113300 is the sum of even-indexed terms of tribonacci numbers. A099463 is the bisection of the tribonacci numbers. A113300(n) + A113301(n) = cumulative sum of tribonacci numbers = A008937(n). Primes in A113300 include a(2) = 5, a(5) = 211, a(9) = 27701, .... A113300 is semiprime for n = 4, 10, 14, ...
FORMULA
a(n) = Sum_{j=0..n} A000073(2*j+1).
a(n) + A113300(n) = A008937(n).
a(n) = 4*a(n-1) - 2*a(n-2) - a(n-4), a(0)=0, a(1)=1, a(2)=5, a(3)=18. - Harvey P. Dale, Apr 12 2013
G.f.: x*(1+x) / ((1-x)*(1-3*x-x^2-x^3)). - Colin Barker, May 06 2013
EXAMPLE
a(0) = 0 = A000073(1);
a(1) = 0+1 = A000073(1) + A000073(3) = 1;
a(2) = 0+1+4 = A000073(1) + A000073(3) + A000073(5) = 5, prime;
a(3) = 0+1+4+13 = A000073(1) + A000073(3) + A000073(5) + A000073(7) = 18;
a(4) = 0+1+4+13+44 = A000073(1) + A000073(3) + A000073(5) + A000073(7) + A000073(9) = 62 = 2 * 31, semiprime;
a(5) = 0+1+4+13+44+149 = A000073(1) + A000073(3) + A000073(5) + A000073(7) + A000073(9) + A000073(11) = 211, prime.
MATHEMATICA
Accumulate[Take[LinearRecurrence[{1, 1, 1}, {0, 1, 1}, 40], {1, -1, 2}]] (* or *) LinearRecurrence[{4, -2, 0, -1}, {0, 1, 5, 18}, 30] (* Harvey P. Dale, Apr 12 2013 *)
PROG
(Magma) I:=[0, 1, 5, 18]; [n le 4 select I[n] else 4*Self(n-1) - 2*Self(n-2) -Self(n-4): n in [1..41]]; // G. C. Greubel, Nov 20 2021
(Sage)
@CachedFunction
def T(n): # A000073
if (n<2): return 0
elif (n==2): return 1
else: return T(n-1) +T(n-2) +T(n-3)
def A113301(n): return sum(T(2*j+1) for j in (0..n))
[A113301(n) for n in (0..40)] # G. C. Greubel, Nov 20 2021
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Oct 24 2005
EXTENSIONS
More terms from Colin Barker, May 06 2013
STATUS
approved