A047243 Numbers that are congruent to {2, 3} mod 6.

2, 3, 8, 9, 14, 15, 20, 21, 26, 27, 32, 33, 38, 39, 44, 45, 50, 51, 56, 57, 62, 63, 68, 69, 74, 75, 80, 81, 86, 87, 92, 93, 98, 99, 104, 105, 110, 111, 116, 117, 122, 123, 128, 129, 134, 135, 140, 141, 146, 147, 152, 153, 158, 159, 164, 165, 170, 171, 176, 177, 182, 183

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Solutions to 3^x - 2^x == 5 (mod 7). - Cino Hilliard, May 09 2003

References

• Emil Grosswald, Topics From the Theory of Numbers, 1966, p. 65, problem 23.

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Crossrefs

Cf. A030531. Complement of A047260.

Programs

GAP

List([1..70], n-> 3*n-2-(-1)^n) # G. C. Greubel, Jun 30 2019

Magma

[3*n-2-(-1)^n: n in [1..70]]; // G. C. Greubel, Jun 30 2019

Mathematica

Select[Range[0, 210], MemberQ[{2, 3}, Mod[#, 6]] &] (* or *)
Fold[Append[#1, 6 #2 - Last@ #1 - 7] &, {2}, Range[2, 70]] (* or *)
Rest@ CoefficientList[Series[x(2+x+3x^2)/((1+x)(1-x)^2), {x, 0, 70}], x] (* Michael De Vlieger, Jan 12 2018 *)

PARI

vector(70, n, 3*n-2-(-1)^n) \\ G. C. Greubel, Jun 30 2019

Sage

[3*n-2-(-1)^n for n in (1..70)] # G. C. Greubel, Jun 30 2019

Formula

a(n) = 6*n - 7 - a(n-1), with a(1)=2. - Vincenzo Librandi, Aug 05 2010

G.f.: x*(2+x+3*x^2) / ( (1+x)*(1-x)^2 ). - R. J. Mathar, Oct 08 2011

From Guenther Schrack, Jun 21 2019: (Start)

a(n) = a(n-2) + 6 with a(1)=2, a(2)=3 for n > 2;

a(n) = 3*n - 2 - (-1)^n. (End)

E.g.f.: 3 - 3*(1-x)*cosh(x) - (1-3*x)*sinh(x). - G. C. Greubel, Jun 30 2019

E.g.f.: 3 + (3*x-3)*exp(x) + 2*sinh(x). - David Lovler, Jul 16 2022

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(12*sqrt(3)) + log(3)/4 - log(2)/3. - Amiram Eldar, Dec 13 2021

Extensions

More terms from Cino Hilliard, May 09 2003