A047260 -id:A047260

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A113828

a(n) = Sum[2^(A047260(i)-1), {i,1,n}].

+20
3

1, 9, 25, 57, 121, 633, 1657, 3705, 7801, 40569, 106105, 237177, 499321, 2596473, 6790777, 15179385, 31956601, 166174329, 434609785, 971480697, 2045222521, 10635157113, 27815026297, 62174764665, 130894241401, 680650055289

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..26.

FORMULA

Empirical g.f.: x*(32*x^3+16*x^2+8*x+1) / ((x-1)*(8*x^2-1)*(8*x^2+1)). - Colin Barker, Sep 01 2013

EXAMPLE

a(2) = 2^(A047260(1)-1) + 2^(A047260(2)-1) = 2^0 + 2^3 = 9

MATHEMATICA

a = {}; s = 0; For[n = 1, n < 41, n++, If[Length[Intersection[{Mod[n, 6]}, {0, 1, 4, 5}]] > 0, s = s + 2^(n - 1); AppendTo[a, s]]]; a

CROSSREFS

Cf. A047260.

KEYWORD

nonn,less

AUTHOR

Artur Jasinski, Jan 27 2006

EXTENSIONS

Edited by Stefan Steinerberger, Jul 23 2007

STATUS

approved

A047243

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

+10
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

(list; graph; refs; listen; history; text; internal format)

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.

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Cino Hilliard, May 09 2003

STATUS

approved

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