A047336 Numbers that are congruent to {1, 6} mod 7.

1, 6, 8, 13, 15, 20, 22, 27, 29, 34, 36, 41, 43, 48, 50, 55, 57, 62, 64, 69, 71, 76, 78, 83, 85, 90, 92, 97, 99, 104, 106, 111, 113, 118, 120, 125, 127, 132, 134, 139, 141, 146, 148, 153, 155, 160, 162, 167, 169, 174, 176, 181, 183, 188, 190, 195, 197, 202, 204, 209

Offset: 1

Author

N. J. A. Sloane

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, n natural numbers), therefore ((2*h*n+(h-4)*(-1)^n-h)/4)^2-1 == 0 (mod h); in this case, a(n)^2-1 == 0 (mod 7). - Bruno Berselli, Nov 17 2010

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

Cf. A007310, A019674, A047522, A045472 (primes), A195041 (partial sums), A005408, A047209, A056020, A090771, A091998, A113801, A160389, A175885, A175886, A175887, A178818, A371858.

Programs

Haskell

a047336 n = a047336_list !! (n-1)
a047336_list = 1 : 6 : map (+ 7) a047336_list
-- Reinhard Zumkeller, Jan 07 2012

Magma

[n: n in [1..210]| n mod 7 in {1,6}]; // Bruno Berselli, Feb 22 2011

Mathematica

Rest[Flatten[Table[{7i-1,7i+1},{i,0,40}]]] (* Harvey P. Dale, Nov 20 2010 *)

PARI

a(n)=n\2*7-(-1)^n \\ Charles R Greathouse IV, May 02 2016

Formula

a(1) = 1; a(n) = 7(n-1) - a(n-1). - Rolf Pleisch, Jan 31 2008 (corrected by Jon E. Schoenfield, Dec 22 2008)

a(n) = (7/2)*(n-(1-(-1)^n)/2) - (-1)^n. - Rolf Pleisch, Nov 02 2010

From Bruno Berselli, Nov 17 2010: (Start)

G.f.: x*(1+5*x+x^2)/((1+x)*(1-x)^2).

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

a(n) = a(n-2)+7.

a(n) = 7*A000217(n-1)+1 - 2*Sum_{i=1..n-1} a(i) for n > 1. (End)

a(n) = 7*floor(n/2)+(-1)^(n+1). - Gary Detlefs, Dec 29 2011

Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi/7)*cot(Pi/7) = A019674 * A178818. - Amiram Eldar, Dec 04 2021

E.g.f.: 1 + ((14*x - 7)*exp(x) + 3*exp(-x))/4. - David Lovler, Sep 01 2022

From Amiram Eldar, Nov 22 2024: (Start)

Product_{n>=1} (1 - (-1)^n/a(n)) = 2*cos(Pi/7) (A160389).

Product_{n>=2} (1 + (-1)^n/a(n)) = (Pi/7) * cosec(Pi/7) (A371858). (End)

Extensions

More terms from Jon E. Schoenfield, Jan 18 2009