A090771 Numbers that are congruent to {1, 9} mod 10.

1, 9, 11, 19, 21, 29, 31, 39, 41, 49, 51, 59, 61, 69, 71, 79, 81, 89, 91, 99, 101, 109, 111, 119, 121, 129, 131, 139, 141, 149, 151, 159, 161, 169, 171, 179, 181, 189, 191, 199, 201, 209, 211, 219, 221, 229, 231, 239, 241, 249, 251, 259, 261, 269, 271, 279, 281

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, 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 10). - 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).

Formula

a(n) = sqrt(40*A057569(n) + 1). - Gary Detlefs, Feb 22 2010

From Bruno Berselli, Sep 16 2010 - Nov 17 2010: (Start)

G.f.: x*(1 + 8*x + x^2)/((1 + x)*(1 - x)^2).

a(n) = (10*n + 3*(-1)^n - 5)/2.

a(n) = -a(-n + 1) = a(n-1) + a(n-2) - a(n-3) = a(n-2) + 10.

a(n) = 10*A000217(n-1) + 1 - 2*Sum_{i=1..n-1} a(i) for n > 1. (End)

a(n) = 10*n - a(n-1) - 10 (with a(1) = 1). - Vincenzo Librandi, Nov 16 2010

a(n) = sqrt(10*A132356(n-1) + 1). - Ivan N. Ianakiev, Nov 09 2012

Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi/10)*cot(Pi/10) = A000796 * A019970 / 10 = sqrt(5 + 2*sqrt(5))*Pi/10. - Amiram Eldar, Dec 04 2021

E.g.f.: 1 + ((10*x - 5)*exp(x) + 3*exp(-x))/2. - David Lovler, Sep 03 2022

Mathematica

Flatten[Table[10n - {9, 1}, {n, 30}]] (* Alonso del Arte, Sep 02 2014 *)
LinearRecurrence[{1, 1, -1}, {1, 9, 11}, 60] (* Harvey P. Dale, Jul 05 2020 *)

Prog

(Haskell)
a090771 n = a090771_list !! (n-1)
a090771_list = 1 : 9 : map (+ 10) a090771_list
-- Reinhard Zumkeller, Jan 07 2012
(PARI) a(n)=n\2*10-(-1)^n \\ Charles R Greathouse IV, Sep 24 2015

Crossrefs

Cf. A000217, A000796, A090298, A005408, A047209, A007310, A019970, A047336, A047522, A091998, A113801, A175886, A175887.

Cf. A056020 (n = 1 or 8 mod 9), A175885 (n = 1 or 10 mod 11).

Cf. A045468 (primes), A195142 (partial sums).

Sequence in context: A346465 A263722 A299971 * A329279 A284295 A284294

Adjacent sequences: A090768 A090769 A090770 * A090772 A090773 A090774

Keyword

nonn,easy

Author

Giovanni Teofilatto, Feb 07 2004

Extensions

Edited and extended by Ray Chandler, Feb 10 2004

Status

approved