A008588 Nonnegative multiples of 6.

0, 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96, 102, 108, 114, 120, 126, 132, 138, 144, 150, 156, 162, 168, 174, 180, 186, 192, 198, 204, 210, 216, 222, 228, 234, 240, 246, 252, 258, 264, 270, 276, 282, 288, 294, 300, 306, 312, 318, 324, 330, 336, 342, 348

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

For n > 3, the number of squares on the infinite 3-column half-strip chessboard at <= n knight moves from any fixed point on the short edge.

Second differences of A000578. - Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 15 2004

A008615(a(n)) = n. - Reinhard Zumkeller, Feb 27 2008

A157176(a(n)) = A001018(n). - Reinhard Zumkeller, Feb 24 2009

These numbers can be written as the sum of four cubes (i.e., 6*n = (n+1)^3 + (n-1)^3 + (-n)^3 + (-n)^3). - Arkadiusz Wesolowski, Aug 09 2013

A122841(a(n)) > 0 for n > 0. - Reinhard Zumkeller, Nov 10 2013

Surface area of a cube with side sqrt(n). - Wesley Ivan Hurt, Aug 24 2014

a(n) is representable as a sum of three but not two consecutive nonnegative integers, e.g., 6 = 1 + 2 + 3, 12 = 3 + 4 + 5, 18 = 5 + 6 + 7, etc. (see A138591). - Martin Renner, Mar 14 2016 (Corrected by David A. Corneth, Aug 12 2016)

Numbers with three consecutive divisors: for some k, each of k, k+1, and k+2 divide n. - Charles R Greathouse IV, May 16 2016

Numbers k for which {phi(k),phi(2k),phi(3k)} is an arithmetic progression. - Ivan Neretin, Aug 12 2016

References

• John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 81.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Tanya Khovanova, Recursive Sequences
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 318
• Index entries for linear recurrences with constant coefficients, signature (2,-1).

Crossrefs

Essentially the same as A008458.

Cf. A016921, A016933, A016945, A016957, A016969, A138591.

Cf. A044102 (subsequence).

Programs

Haskell

a008588 = (* 6)
a008588_list = [0, 6 ..]  -- Reinhard Zumkeller, Nov 10 2013

Magma

[6*n: n in [0..60] ]; // Vincenzo Librandi, Jul 16 2011

Maple

[ seq(6*n,n=0..45) ];

Mathematica

Range[0, 500, 6] (* Vladimir Joseph Stephan Orlovsky, May 26 2011 *)
NestList[6+#&,0,60] (* Harvey P. Dale, Nov 17 2024 *)

Maxima

makelist(6*n,n,0,30); /* Martin Ettl, Nov 12 2012 */

PARI

a(n)=6*n \\ Charles R Greathouse IV, Feb 08 2012

Sage

[6*n for n in range(59)] # Stefano Spezia, Mar 09 2025

Formula

From Vincenzo Librandi, Dec 24 2010: (Start)

a(n) = 6*n = 2*a(n-1) - a(n-2).

G.f.: 6*x/(1-x)^2. (End)

a(n) = Sum_{k>=0} A030308(n,k)*6*2^k. - Philippe Deléham, Oct 24 2011

a(n) = Sum_{k=2n-1..2n+1} k. - Wesley Ivan Hurt, Nov 22 2015

From Ilya Gutkovskiy, Aug 12 2016: (Start)

E.g.f.: 6*x*exp(x).

Convolution of A010722 and A057427.

Sum_{n>=1} (-1)^(n+1)/a(n) = log(2)/6 = A002162*A020793. (End)

a(n) = 6 * A001477(n). - David A. Corneth, Aug 12 2016