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
Links
- Hung Viet 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.
- Index entries for linear recurrences with constant coefficients, signature (3,-3,1,1)
Programs
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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 *)
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).
Comments