A113300 Sum of even-indexed terms of tribonacci numbers.

0, 1, 3, 10, 34, 115, 389, 1316, 4452, 15061, 50951, 172366, 583110, 1972647, 6673417, 22576008, 76374088, 258371689, 874065163, 2956941266, 10003260650, 33840788379, 114482567053, 387291750188, 1310198605996, 4432370135229, 14994600761871, 50726371026838

Offset

0,3

Comments

A000073 is the tribonacci numbers. A099463 is the bisection of the tribonacci numbers.

Primes in this sequence include a(2) = 3, a(6) = 389, a(9) = 15061, a(10) = 50951. a(n) in this sequence is semiprime for n = 3, 4, 5, 11, 14, ...

Partial sums of A099463. a(n+1) gives row sums of Riordan array (1/(1-x)^2,(1+x)^2/(1-x)^2)). Congruent to 0,1,1,0,0,1,1,0,0,... modulo 2. - Paul Barry, Feb 07 2006

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (3,1,1).

Formula

a(n) = Sum_{i=0..n} A000073(2*n).

a(n) = Sum_{i=0..n} A099463(n).

a(n) + A113301(n) = A008937(n).

From Paul Barry, Feb 07 2006: (Start)

G.f.: x/(1 - 3*x - x^2 - x^3).

a(n) = 3*a(n-1) + a(n-2) + a(n-3). (End)

Mathematica

Accumulate[Take[LinearRecurrence[{1, 1, 1}, {0, 0, 1}, 60], {1, -1, 2}]] (* Harvey P. Dale, Nov 06 2011 *)
LinearRecurrence[{3, 1, 1}, {0, 1, 3}, 40] (* Vladimir Joseph Stephan Orlovsky, Jan 31 2012 *)
a[ n_] := Sum[ SeriesCoefficient[ SeriesCoefficient[ x / (1 - x - y - x y) , {x, 0, n - k}]^2 , {y, 0, k}], {k, 0, n}]; (* Michael Somos, Jun 27 2017 *)

Prog

(Magma) I:=[0, 1, 3]; [n le 3 select I[n] else 3*Self(n-1) +Self(n-2) +Self(n-3): n in [1..61]]; // G. C. Greubel, Nov 19 2021
(Sage)
@CachedFunction
def T(n): # T(n) = A000073(n)
    if (n<2): return 0
    elif (n==2): return 1
    else: return T(n-1) +T(n-2) +T(n-3)
def a(n): return sum( T(2*j) for j in (0..n) )
[a(n) for n in (0..60)] # G. C. Greubel, Nov 19 2021

Crossrefs

Cf. A000073, A008937, A099463, A113301.

Sequence in context: A289601 A291337 A255813 * A332872 A007052 A048580

Adjacent sequences: A113297 A113298 A113299 * A113301 A113302 A113303

Keyword

easy,nonn

Author

Jonathan Vos Post, Oct 24 2005

Status

approved