A017617 -id:A017617 - cp's OEIS frontend

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

Wolfdieter Lang, Oct 21 2021

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.

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

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

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

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

A126562 Number of intersections of at least four edges in a cube of n X n X n smaller cubes.

Original entry on oeis.org

0, 7, 32, 81, 160, 275, 432, 637, 896, 1215, 1600, 2057, 2592, 3211, 3920, 4725, 5632, 6647, 7776, 9025, 10400, 11907, 13552, 15341, 17280, 19375, 21632, 24057, 26656, 29435, 32400, 35557, 38912, 42471, 46240, 50225, 54432, 58867, 63536, 68445

Offset: 1

Jonathan R. Love (japanada11(AT)yahoo.ca), Mar 12 2007

a(n-1) is the number of points in a cubic lattice of n^3 equally spaced points from which all the 12*n-16 points on the 12 edges are removed. - Luciano Ancora, Jun 25 2015

			On a cube made of 3 X 3 X 3 smaller cubes, each of the 6 sides has 4 intersections of four edges and in the center, there are 8 intersections of six edges. 6 * 4 + 8 = 32, which is a(3).

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A017617.

[6*(n-1)^2 + (n-1)^3: n in [1..40]]; // Vincenzo Librandi, Jun 27 2015

LinearRecurrence[{4, -6, 4, -1}, {0, 7, 32, 81}, 50] (* Vincenzo Librandi, Jun 27 2015 *)

concat(0, Vec(x^2*(7+4*x-5*x^2)/(1-x)^4 + O(x^50))) \\ Michel Marcus, Jun 26 2015

def a(n): return (n**3+3*n**2-9*n+5) # Torlach Rush, May 01 2024

a(n) = 6*(n-1)^2 + (n-1)^3.
G.f.: x^2*(7+4*x-5*x^2)/(1-x)^4. - Colin Barker, Jul 29 2012
a(n-1) = n^3 - (12*n-16). - Luciano Ancora, Jun 25 2015

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

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