A347838 Positive numbers that are congruent to 2, 5, or 11 modulo 12.

2, 5, 11, 14, 17, 23, 26, 29, 35, 38, 41, 47, 50, 53, 59, 62, 65, 71, 74, 77, 83, 86, 89, 95, 98, 101, 107, 110, 113, 119, 122, 125, 131, 134, 137, 143, 146, 149, 155, 158, 161, 167, 170, 173, 179, 182, 185, 191, 194, 197, 203, 206, 209, 215, 218, 221, 227, 230, 233, 239

Offset: 1

Author

Wolfdieter Lang, Oct 21 2021

Keywords

• nonn
• easy

Comments

This sequence follows from the first column sequence of the array A347834, namely A047529 ({1,3,7} (mod 8)), as given in the formula below.

Together with A017617, the positive integers congruent to 8 modulo 12, one obtains A016789, the positive integers congruent to 2 modulo 3. See the array A347839.

Links

• Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Crossrefs

Cf. A002264, A016921, A016789, A017617, A017653, A040117(k+1), A047261, A047529, A102283, A347834, A347839.

Programs

Mathematica

Map[(3 # + 1)/2 &, LinearRecurrence[{1, 0, 1, -1}, {1, 3, 7, 9}, 60]] (* Michael De Vlieger, Oct 21 2021 *)

Formula

a(n) = (3*A047529(n) + 1)/2.

Trisection: a(3*k+1) = 2 + 12*k, a(3*k+2) = 5 + 12*k, a(3*k+3) = 11 + 12*k, or with a(3*k) = -1 + 12*k for k >= 0.

O.g.f. with a(0) =-1: G(x) = (-1 + 3*x + 3*x^2 + 7*x^3)/((1 - x)*(1 - x^3)) = -6/(1-x) + 4/(1-x)^2 + (1 + x)/(1 + x + x^2). Note that (1 - x)*(1 - x^3) = (1-x)^2*(1 + x + x^2) = 1 - x - x^3 + x^4.

a(n) = a(n-1) + a(n-3) - a(n-4), for n >= 4, given a(n) for 0..3, with a(0) = -1.

a(n) = 2*b(n) + 3*b(n-1) + 6*b(n-2) + b(n-3), with b(n) = floor((n+2)/3) = A002264(n+2).

a(n) = -1 + 3*n + 3*floor(n/3) (from the partial fraction decomposition of G).

E.g.f.: 1 + 2*exp(x)*(2*x - 1) + exp(-x/2)*(3*cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2))/3. - Stefano Spezia, Dec 08 2021

Sum_{n>=1} (-1)^(n+1)/a(n) = ((sqrt(2)+1)*Pi + sqrt(3)*log(sqrt(3)+2) + sqrt(6)*log(5-2*sqrt(6)))/12. - Amiram Eldar, Dec 30 2021