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

1,1

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).

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

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 *)

Prog

(PARI) vector(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
(Sage) [3*n-2-(-1)^n for n in (1..70)] # G. C. Greubel, Jun 30 2019
(GAP) List([1..70], n-> 3*n-2-(-1)^n) # G. C. Greubel, Jun 30 2019

Crossrefs

Cf. A030531. Complement of A047260.

Sequence in context: A281111 A301917 A327312 * A277094 A099148 A309423

Adjacent sequences: A047240 A047241 A047242 * A047244 A047245 A047246

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Cino Hilliard, May 09 2003

Status

approved