A347838 - OEIS

%I #16 Dec 30 2021 09:21:31

%S 2,5,11,14,17,23,26,29,35,38,41,47,50,53,59,62,65,71,74,77,83,86,89,

%T 95,98,101,107,110,113,119,122,125,131,134,137,143,146,149,155,158,

%U 161,167,170,173,179,182,185,191,194,197,203,206,209,215,218,221,227,230,233,239

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

%C 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.

%C 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.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,-1).

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

%F 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.

%F 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.

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

%F 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).

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

%F 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

%F 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

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

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

%K nonn,easy

%O 1,1

%A _Wolfdieter Lang_, Oct 21 2021