A091998 Numbers that are congruent to {1, 11} mod 12.

1, 11, 13, 23, 25, 35, 37, 47, 49, 59, 61, 71, 73, 83, 85, 95, 97, 107, 109, 119, 121, 131, 133, 143, 145, 155, 157, 167, 169, 179, 181, 191, 193, 203, 205, 215, 217, 227, 229, 239, 241, 251, 253, 263, 265, 275, 277, 287, 289, 299, 301, 311, 313, 323, 325, 335

Offset: 1

Author

Ray Chandler, Feb 21 2004

Keywords

• nonn
• easy

Comments

Cf. property described by Gary Detlefs in A113801: more generally, these numbers are of the form (2*h*n + (h-4)*(-1)^n-h)/4 (h and n in A000027), then ((2*h*n + (h-4)*(-1)^n - h)/4)^2 - 1 == 0 (mod h); in our case, a(n)^2 - 1 == 0 (mod 12). Also a(n)^2 - 1 == 0 (mod 24).

Links

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

Crossrefs

First row of A092260.

Cf. A175885 (n == 1 or 10 (mod 11)), A175886 (n == 1 or 12 (mod 13)).

Cf. A097933 (primes), A195143 (partial sums).

Cf. A000217, A005408, A047209, A007310, A047336, A047522, A056020, A090771, A113801, A175887, A188887.

Programs

Haskell

a091998 n = a091998_list !! (n-1)
a091998_list = 1 : 11 : map (+ 12) a091998_list
-- Reinhard Zumkeller, Jan 07 2012

Magma

[ n: n in [1..350] | n mod 12 eq 1 or n mod 12 eq 11 ];

Mathematica

LinearRecurrence[{1,1,-1},{1,11,13},100] (* Harvey P. Dale, Jul 26 2017 *)

PARI

is(n)=n=n%12;n==11 || n==1 \\ Charles R Greathouse IV, Jul 02 2013

Formula

a(n) = 12*n - a(n-1) - 12 (with a(1)=1). - Vincenzo Librandi, Nov 16 2010

a(n) = 6*n + 2*(-1)^n - 3.

G.f.: x*(1+10*x+x^2)/((1+x)*(1-x)^2).

a(n) - a(n-1) - a(n-2) + a(n-3) = 0 for n > 3.

a(n) = a(n-2) + 12 for n > 2.

a(n) = 12*A000217(n-1) + 1 - 2*Sum_{i=1..n-1} a(i) for n > 1.

Sum_{n>=1} (-1)^(n+1)/a(n) = (2 + sqrt(3))*Pi/12. - Amiram Eldar, Dec 04 2021

E.g.f.: 1 + (6*x - 3)*exp(x) + 2*exp(-x). - David Lovler, Sep 04 2022

From Amiram Eldar, Nov 23 2024: (Start)

Product_{n>=1} (1 - (-1)^n/a(n)) = sqrt(2 + sqrt(3)) = 2*cos(Pi/12) (A188887).

Product_{n>=2} (1 + (-1)^n/a(n)) = (Pi/3)*cos(Pi/12). (End)

Extensions

Formulae and comment added by Bruno Berselli, Nov 17 2010 - Nov 18 2010