A017173 a(n) = 9*n + 1.

1, 10, 19, 28, 37, 46, 55, 64, 73, 82, 91, 100, 109, 118, 127, 136, 145, 154, 163, 172, 181, 190, 199, 208, 217, 226, 235, 244, 253, 262, 271, 280, 289, 298, 307, 316, 325, 334, 343, 352, 361, 370, 379, 388, 397, 406, 415, 424, 433, 442, 451, 460, 469, 478

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Also all the numbers with digital root 1; A010888(a(n)) = 1. - Rick L. Shepherd, Jan 12 2009

A116371(a(n)) = A156144(a(n)); positions where records occur in A156144: A156145(n+1) = A156144(a(n)). - Reinhard Zumkeller, Feb 05 2009

If A=[A147296] 9*n^2+2*n (n>0, 11, 40, 87, ...); Y=[A010701] 3 (3, 3, 3, ...); X=[A017173] 9*n+1 (n>0, 10, 19, 28, ...), we have, for all terms, Pell's equation X^2 - A*Y^2 = 1. Example: 10^2 - 11*3^2 = 1; 19^2 - 40*3^2 = 1; 28^2 - 87*3^2 = 1. - Vincenzo Librandi, Aug 01 2010

Links

• Mohammed Yaseen, Table of n, a(n) for n = 0..10000
• Tanya Khovanova, Recursive Sequences.
• Index entries for linear recurrences with constant coefficients, signature (2,-1).

Crossrefs

Cf. A093644 ((9,1) Pascal, column m=1).

Cf. A010701, A010888, A016777, A116371, A147296, A156144, A156145.

Numbers with digital root m: this sequence (m=1), A017185 (m=2), A017197 (m=3), A017209 (m=4), A017221 (m=5), A017233 (m=6), A017245 (m=7), A017257 (m=8), A008591 (m=9).

Programs

Haskell

a017173 = (+ 1) . (* 9)
a017173_list = [1, 10 ..]  -- Reinhard Zumkeller, Feb 04 2014

Mathematica

Range[1, 1000, 9] (* Vladimir Joseph Stephan Orlovsky, May 28 2011 *)
LinearRecurrence[{2,-1},{1,10},60] (* Harvey P. Dale, Dec 27 2014 *)

PARI

forstep(n=1,500,9,print1(n", ")) \\ Charles R Greathouse IV, May 28 2011

Sage

[i+1 for i in range(480) if gcd(i,9) == 9] # Zerinvary Lajos, May 20 2009

Formula

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

a(n) = 2*a(n-1) - a(n-2) with a(0)=1, a(1)=10. - Vincenzo Librandi, Aug 01 2010

E.g.f.: exp(x)*(1 + 9*x). - Stefano Spezia, Apr 20 2023

a(n) = A016777(3*n). - Elmo R. Oliveira, Apr 12 2025