A113301 Sum of odd-indexed terms of tribonacci numbers.

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

Author

Jonathan Vos Post, Oct 24 2005

Keywords

• easy
• nonn

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, ...

Examples

			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.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-2,0,-1).

Crossrefs

Cf. A000073, A008937, A099463, A113300.

Programs

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

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 *)

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

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

Extensions

More terms from Colin Barker, May 06 2013