A385106 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-4) with a(1) = 1, a(2) = 2, a(3) = 4, and a(4) = 7.

1, 2, 4, 7, 12, 21, 38, 70, 129, 236, 429, 778, 1412, 2567, 4672, 8505, 15478, 28158, 51217, 93160, 169465, 308290, 560852, 1020311, 1856132, 3376605, 6142582, 11174374, 20328113, 36980404, 67273829, 122382762, 222635316, 405011895, 736786328, 1340341377, 2438312358, 4435711166

Offset

1,2

Comments

a(n) the number of subsets of {3, 6, 9, 12, ..., 3*n} that are Schreier and contain 3*n.

Links

Table of n, a(n) for n=1..38.

Hùng Việt Chu and Zachary Louis Vasseur, Schreier sets of multiples of an integer, linear recurrence, and Pascal triangle, arXiv:2506.14312 [math.CO], 2025. See Table 1 p. 2.

Hùng Việt Chu and Zachary Louis Vasseur, Linear Recurrences of Generalized Schreier Sets Revisited, Journal of Integer Sequences, Vol. 29 (2026), Article 26.2.2. See p. 3.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1,1).

Formula

a(n) = 2 + Sum_{i=1..n-2} Sum_{j=0..3i-2} binomial(n-i-1,j), for n > 1.

a(n) = A079398(3*n).

G.f.: x*(1 - x + x^2)/(1 - 3*x + 3*x^2 - x^3 - x^4).

Mathematica

LinearRecurrence[{3, -3, 1, 1}, {1, 2, 4, 7}, 38 ] (* or *) Rest[CoefficientList[Series[x*(1 - x + x^2)/(1 - 3*x + 3*x^2 - x^3 - x^4), {x, 0, 38}], x]] (* or *) a[1]=1; a[n_]:=2 + Sum[Binomial[n-i-1, j], {i, n-2} , {j, 0, 3i-2} ]; Array[a, 38] (* James C. McMahon, Jun 24 2025 *)

Crossrefs

Cf. A079398, A385107.

Sequence in context: A218600 A000709 A054161 * A023433 A190168 A288133

Adjacent sequences: A385103 A385104 A385105 * A385107 A385108 A385109

Keyword

nonn,easy

Author

Hung Viet Chu, Jun 18 2025

Status

approved