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

Author

Jonathan Vos Post, Oct 24 2005

Keywords

• easy
• nonn

Comments

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).

Crossrefs

Cf. A000073, A008937, A099463, A113301.

Programs

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

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 *)

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

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)