A104435 Number of ways to split 1, 2, 3, ..., 2n into 2 arithmetic progressions each with n terms.

1, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset

1,2

Comments

The common difference in an arithmetic progression must be a positive integer. - David A. Corneth, Apr 14 2024

Links

Table of n, a(n) for n=1..88.

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

Formula

a(1) = 1, a(2) = 3, a(n) = 2 for n >= 3. Proof of the latter: if the common difference in an aritmetic procression, starting with a number at least 1, is at least 3 then the largest term in that arithmetic progression is at least 1 + 3*(n-1) = 3*n - 2. But 3*n - 2 > 2*n for n > 2. - David A. Corneth, Apr 14 2024

G.f.: x*(1 + 2*x - x^2)/(1 - x). - Stefano Spezia, Apr 14 2024

Example

From R. J. Mathar, Apr 14 2024: (Start)
a(2)=3 offers 3 ways of splitting (1,2,3,4): {(1,2),(3,4)}, {(1,3),(2,4)}, {(1,4),(2,3)}.
a(n)=2 for n>=3 because there are at least the two ways of splitting (1,2,..,2n) into the even and odd numbers. (End)

Prog

(PARI) a(n) = if(n <= 2, [1, 3][n], 2) \\ David A. Corneth, Apr 14 2024

Crossrefs

Cf. A104429, A104430, A104431, A104432, A104433, A104434, A104436, A104437, A104438, A104439, A104440, A104441, A104442, A104443.

Sequence in context: A155677 A075791 A262626 * A178815 A248743 A085398

Adjacent sequences: A104432 A104433 A104434 * A104436 A104437 A104438

Keyword

nonn,easy,changed

Author

Jonas Wallgren, Mar 17 2005

Extensions

More terms from David A. Corneth, Apr 14 2024

Status

approved