A017233 a(n) = 9*n + 6.

6, 15, 24, 33, 42, 51, 60, 69, 78, 87, 96, 105, 114, 123, 132, 141, 150, 159, 168, 177, 186, 195, 204, 213, 222, 231, 240, 249, 258, 267, 276, 285, 294, 303, 312, 321, 330, 339, 348, 357, 366, 375, 384, 393, 402, 411, 420, 429, 438, 447, 456, 465, 474, 483

Offset: 0

Author

David J. Horn and Laura Krebs Gordon (lkg615(AT)verizon.net), 1985

Keywords

• nonn
• easy

Comments

General form: (q*n-1)*q, cf. A017233 (q=3), A098502 (q=4). - Vladimir Joseph Stephan Orlovsky, Feb 16 2009

Numbers whose digital root is 6; that is, A010888(a(n)) = 6. (Ball essentially says that Iamblichus (circa 350) announced that a number equal to the sum of three integers 3*n, 3*n - 1, and 3*n - 2 has 6 as what is now called the number's digital root.) - Rick L. Shepherd, Apr 01 2014

References

• W. W. R. Ball, A Short Account of the History of Mathematics, Sterling Publishing Company, Inc., 2001 (Facsimile Edition) [orig. pub. 1912], pages 110-111.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..5000
• Tanya Khovanova, Recursive Sequences.
• Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
• Index entries for linear recurrences with constant coefficients, signature (2,-1).

Crossrefs

Cf. A008585, A008591, A010888, A016789, A017209, A017221, A017233, A019557, A098502.

Programs

Magma

[9*n+6: n in [0..60]]; // Vincenzo Librandi, Jul 24 2011

Mathematica

Range[6, 1000, 9] (* Vladimir Joseph Stephan Orlovsky, May 28 2011 *)
LinearRecurrence[{2,-1},{6,15},60] (* Harvey P. Dale, Feb 01 2014 *)

PARI

a(n)=9*n+6 \\ Charles R Greathouse IV, Oct 07 2015

Formula

G.f.: 3*(2+x)/(x-1)^2. - R. J. Mathar, Mar 20 2018

Sum_{n>=0} (-1)^n/a(n) = sqrt(3)*Pi/27 - log(2)/9. - Amiram Eldar, Dec 12 2021

E.g.f.: 3*exp(x)*(2 + 3*x). - Stefano Spezia, Dec 07 2024

From Elmo R. Oliveira, Apr 10 2025: (Start)

a(n) = 3*A016789(n) = A019557(n+1)/2.

a(n) = 2*a(n-1) - a(n-2). (End)