OFFSET
0,3
COMMENTS
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).
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
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Oct 24 2005
STATUS
approved