A056020 Numbers that are congruent to +-1 mod 9.

1, 8, 10, 17, 19, 26, 28, 35, 37, 44, 46, 53, 55, 62, 64, 71, 73, 80, 82, 89, 91, 98, 100, 107, 109, 116, 118, 125, 127, 134, 136, 143, 145, 152, 154, 161, 163, 170, 172, 179, 181, 188, 190, 197, 199, 206, 208, 215, 217, 224, 226, 233, 235, 242, 244, 251, 253

Offset

1,2

Comments

Or, numbers k such that k^2 == 1 (mod 9).

Or, numbers k such that the iterative cycle j -> sum of digits of j^2 when started at k contains a 1. E.g., 8 -> 6+4 = 10 -> 1+0+0 = 1 and 17 -> 2+8+9 = 19 -> 3+6+1 = 10 -> 1+0+0 = 1. - Asher Auel, May 17 2001

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Formula

a(1) = 1; a(n) = 9(n-1) - a(n-1). - Rolf Pleisch, Jan 31 2008 [Offset corrected by Jon E. Schoenfield, Dec 22 2008]

From R. J. Mathar, Feb 10 2008: (Start)

O.g.f.: 1 + 5/(4(x+1)) + 27/(4(-1+x)) + 9/(2(-1+x)^2).

a(n+1) - a(n) = A010697(n). (End)

a(n) = (9*A132355(n) + 1)^(1/2). - Gary Detlefs, Feb 22 2010

From Bruno Berselli, Nov 17 2010: (Start)

a(n) = a(n-2) + 9, for n > 2.

a(n) = 9*A000217(n-1) + 1 - 2*Sum_{i=1..n-1} a(i), n > 1. (End)

Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi/9)*cot(Pi/9) = A019676 * A019968. - Amiram Eldar, Dec 04 2021

E.g.f.: 1 + ((18*x - 9)*exp(x) + 5*exp(-x))/4. - David Lovler, Sep 04 2022

Mathematica

Select[ Range[ 300 ], PowerMod[ #, 2, 3^2 ]==1& ]

Prog

(PARI) a(n)=9*(n>>1)+if(n%2, 1, -1) \\ Charles R Greathouse IV, Jun 29 2011
(PARI) for(n=1, 40, print1(9*n-8, ", ", 9*n-1, ", ")) \\ Charles R Greathouse IV, Jun 29 2011
(Haskell)
a056020 n = a056020_list !! (n-1)
a05602_list = 1 : 8 : map (+ 9) a056020_list
-- Reinhard Zumkeller, Jan 07 2012

Crossrefs

Cf. A007953, A047522 (n=1 or 7 mod 8), A090771 (n=1 or 9 mod 10).

Cf. A129805 (primes), A195042 (partial sums).

Cf. A005408, A019676, A019968, A047209, A007310, A047336, A175885, A091998, A175886, A113801, A175887.

Sequence in context: A038209 A319293 A061908 * A049510 A121846 A059094

Adjacent sequences: A056017 A056018 A056019 * A056021 A056022 A056023

Keyword

nonn,easy

Author

Robert G. Wilson v, Jun 08 2000

Status

approved