A168489 Numbers that are congruent to {7,11} mod 12.

7, 11, 19, 23, 31, 35, 43, 47, 55, 59, 67, 71, 79, 83, 91, 95, 103, 107, 115, 119, 127, 131, 139, 143, 151, 155, 163, 167, 175, 179, 187, 191, 199, 203, 211, 215, 223, 227, 235, 239, 247, 251, 259, 263, 271, 275, 283, 287, 295, 299, 307, 311, 319, 323, 331, 335

Offset: 1

Author

Vincenzo Librandi, Nov 27 2009

Keywords

• nonn
• easy

Comments

From Arkadiusz Wesolowski, Mar 16 2014: (Start)

Odd numbers m for which 2/m is not equal to 1/x + 1/y with x = 2*floor((m + 1)/4) + 1 and an integer y > x.

The primes together with 3 are in A002145. (End)

Odd numbers not of the form (4j+1)*3^k, {j,k>=0}. - Bob Selcoe, Aug 30 2015

Nonnegative k for which k == 3 (mod 4) and k^2 == 1 (mod 3). - Bruno Berselli, Apr 26 2018

Numbers that are not divisible by their digital root in base 5. - Amiram Eldar, Nov 24 2022

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Crossrefs

Cf. A002145, A239233.

Programs

Magma

[6*n-(-1)^n: n in [1..60]]; // Vincenzo Librandi, Aug 10 2012

Mathematica

LinearRecurrence[{1, 1, -1}, {7, 11, 19}, 60] (* or *) Select[Range[350],  MemberQ[{7, 11}, Mod[#, 12]]&] (* Harvey P. Dale, Nov 10 2011 *)
Rest[CoefficientList[Series[x (7 + 4 x + x^2)/((1 + x) (x - 1)^2), {x, 0, 56}], x] ] (* Ray Chandler, Jul 07 2015 *)
RecurrenceTable[{a[n] == 12 n - 6 - a[n-1], a[1]==7}, a, {n, 1, 100}] (* G. C. Greubel, Aug 30 2015 *)

PARI

x='x+O('x^100); Vec(x*(7+4*x+x^2)/((1+x)*(x-1)^2)) \\ Altug Alkan, Oct 22 2015

Formula

a(n) = 12*n - a(n-1) - 6 for n>1, a(1)=7.

From R. J. Mathar, Mar 21 2010: (Start)

a(n) = 6*n - (-1)^n.

a(n) = a(n-1) + a(n-2) - a(n-3).

G.f.: x*(7 + 4*x + x^2)/ ((1+x) * (x-1)^2). (End)

E.g.f.: 1 + 6*x*exp(x) - exp(-x). - G. C. Greubel, Aug 30 2015

Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(3)*Pi/12 - log(2+sqrt(3))/(2*sqrt(3)). - Amiram Eldar, Dec 30 2021