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

0,3

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, ...

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).

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

Cf. A000073, A008937, A099463, A113300.

Sequence in context: A350782 A255837 A122234 * A111567 A121050 A029869

Adjacent sequences: A113298 A113299 A113300 * A113302 A113303 A113304

Keyword

easy,nonn

Author

Jonathan Vos Post, Oct 24 2005

Extensions

More terms from Colin Barker, May 06 2013

Status

approved