A213705 a(n)=n if n <= 3, otherwise a(n) = A007477(n-1) + A007477(n).

1, 2, 3, 5, 9, 17, 33, 66, 134, 277, 579, 1224, 2610, 5609, 12135, 26408, 57770, 126962, 280192, 620674, 1379586, 3075943, 6877611, 15417934, 34646156, 78027146, 176087292, 398143230, 901827322, 2046112299, 4649558191, 10581041518, 24112473412, 55019560650, 125696393844, 287494670302

Offset: 1

Author

Antti Karttunen, Sep 14 2012

Keywords

• nonn
• easy

Comments

a(n) gives the number of "plausible parsings" of the sentence "Etsivät^(n+1)" in Finnish (with the most common word order, SV & SVO), that is, sentences which consist only of n+1 copies of the word "etsivät". See the OEIS Wiki page.

See A007477 for the number of plausible parsings of "Buffalo^n" sentences in English.

In my view the value of a(0) should be 0 in this context (single word "Etsivät." is not a valid Finnish sentence, except as an answer to a question), although this is arguable. However, it is probably that this sequence occurs in other (combinatorial) contexts as well, and there a(0) might be something else than 0, so I left it off, and made the sequence start from offset 1.

Examples

			G.f. = x + 2*x^2 + 3*x^3 + 5*x^4 + 9*x^5 + 17*x^6 + 33*x^7 + ... - _Michael Somos_, Nov 07 2019
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..1000
• Antti Karttunen, Etsivät etsivät etsivät..., OEIS Wiki.

Crossrefs

Cf. A007477, A007853.

Programs

Maple

b:= n-> coeff(series(RootOf(A=(A*x)^2+x+1, A), x, n+1), x, n):
a:= n-> `if`(n<2, n, b(n-1) +b(n)):
seq(a(n), n=1..40);  # Alois P. Heinz, Sep 14 2012

Mathematica

(* b = A007477 *) b[n_] := Sum[Binomial[2*k+2, n-k-2]*Binomial[n-k-2, k]/(k + 1), {k, 0, n-2}]; a[n_] := b[n-1] + b[n]; a[1] = 1; a[2] = 2; Array[a, 40] (* Jean-François Alcover, Mar 04 2016 *)

PARI

b(n) = sum(k=0, n - 2, binomial(2*k + 2, n - k - 2)*binomial(n - k - 2, k)/(k + 1));
a(n) = if(n<3, n, b(n - 1) + b(n)); \\ Indranil Ghosh, Apr 11 2017

PARI

{a(n) = polcoeff( (1 + x) * (1 - 2*x^2 - sqrt(1 - 4*x^2 - 4*x^3 + x^3 * O(x^n))) / (2*x^2), n)}; /* Michael Somos, Nov 07 2019 */

Python

from sympy import binomial
def b(n): return sum([binomial(2*k + 2, n - k - 2)*binomial(n - k - 2, k)//(k + 1) for k in range(n - 1)])
def a(n): return n if n<3 else b(n - 1) + b(n)
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Apr 11 2017

Scheme

: (define (A213705 n) (if (< n 2) n (+ (A007477 (- n 1)) (A007477 n))))

Formula

Given the g.f. A(x) and the g.f. of A007853 B(x), then -x = A(-B(x)). - Michael Somos, Nov 07 2019