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

1,2

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).

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

Mathematica

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

Prog

(Magma) [ n: n in [1..350] | n mod 12 eq 1 or n mod 12 eq 11 ];
(Haskell)
a091998 n = a091998_list !! (n-1)
a091998_list = 1 : 11 : map (+ 12) a091998_list
-- Reinhard Zumkeller, Jan 07 2012
(PARI) is(n)=n=n%12; n==11 || n==1 \\ Charles R Greathouse IV, Jul 02 2013

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.

Sequence in context: A192931 A002367 A160373 * A208296 A289696 A365313

Adjacent sequences: A091995 A091996 A091997 * A091999 A092000 A092001

Keyword

nonn,easy

Author

Ray Chandler, Feb 21 2004

Extensions

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

Status

approved