OFFSET
0,8
COMMENTS
This is to tribonacci numbers A000073 as A052011 is to Fibonacci numbers and as A052012 is to Lucas numbers A000204. It is mere coincidence that all values until a(12) = 38 are themselves Fibonacci. The formula plus the known asymptotic prime distribution gives the asymptotic approximation of this sequence, which is the same even if we use one of the alternative definitions of tribonacci with different initial values.
FORMULA
a(n) = A000720(A000073(n+1) - 1) - A000720(A000073(n)) for n >= 3. [formula edited Andrew Howroyd, Jan 02 2020]
EXAMPLE
Between Trib(8)=24 and Trib(9)=44 we find the following primes: 29, 31, 37, 41, 43, hence a(8)=5.
MAPLE
A131354 := proc(n)
a := 0 ;
if isprime(k) then
a := a+1 ;
end if;
end do;
a ;
end proc: # R. J. Mathar, Dec 14 2011
MATHEMATICA
trib[n_] := SeriesCoefficient[x^2/(1 - x - x^2 - x^3), {x, 0, n}];
a[n_] := PrimePi[trib[n + 1] - 1] - PrimePi[trib[n]];
a /@ Range[0, 42] (* Jean-François Alcover, Apr 10 2020 *)
PROG
(PARI) \\ here b(n) is A000073(n).
b(n)={polcoef(x^2/(1-x-x^2-x^3) + O(x*x^n), n)}
a(n)={primepi(b(n+1)-1) - primepi(b(n))} \\ Andrew Howroyd, Jan 02 2020
CROSSREFS
KEYWORD
nonn
AUTHOR
Jonathan Vos Post, Oct 21 2007
EXTENSIONS
Terms a(26) and beyond from Andrew Howroyd, Jan 02 2020
STATUS
approved