A027468 9 times the triangular numbers A000217.

0, 9, 27, 54, 90, 135, 189, 252, 324, 405, 495, 594, 702, 819, 945, 1080, 1224, 1377, 1539, 1710, 1890, 2079, 2277, 2484, 2700, 2925, 3159, 3402, 3654, 3915, 4185, 4464, 4752, 5049, 5355, 5670, 5994, 6327, 6669, 7020, 7380, 7749, 8127, 8514, 8910, 9315

Offset: 0

Author

Olivier Gérard

Keywords

• nonn
• easy

Comments

Staggered diagonal of triangular spiral in A051682, between (0,1,11) spoke and (0,8,25) spoke. - Paul Barry, Mar 15 2003

Number of permutations of n distinct letters (ABCD...) each of which appears thrice with n-2 fixed points. - Zerinvary Lajos, Oct 15 2006

Number of n permutations (n>=2) of 4 objects u, v, z, x with repetition allowed, containing n-2=0 u's. Example: if n=2 then n-2 =zero (0) u, a(1)=9 because we have vv, zz, xx, vx, xv, zx, xz, vz, zv. A027465 formatted as a triangular array: diagonal: 9, 27, 54, 90, 135, 189, 252, 324, ... . - Zerinvary Lajos, Aug 06 2008

a(n) is also the least weight of self-conjugate partitions having n different parts such that each part is a multiple of 3. - Augustine O. Munagi, Dec 18 2008

Also sequence found by reading the line from 0, in the direction 0, 9, ..., and the same line from 0, in the direction 0, 27, ..., in the square spiral whose vertices are the generalized hendecagonal numbers A195160. Axis perpendicular to A195147 in the same spiral. - Omar E. Pol, Sep 18 2011

Sum of the numbers from 4*n to 5*n. - Wesley Ivan Hurt, Nov 01 2014

Examples

			The first such self-conjugate partitions, corresponding to a(n)=1,2,3,4 are 3+3+3, 6+6+6+3+3+3, 9+9+9+6+6+6+3+3+3, 12+12+12+9+9+9+6+6+6+3+3+3. - _Augustine O. Munagi_, Dec 18 2008
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Augustine O. Munagi, Pairing conjugate partitions by residue classes, Discrete Math., Vol. 308, No. 12 (2008), pp. 2492-2501.
• Enrique Navarrete and Daniel Orellana, Finding Prime Numbers as Fixed Points of Sequences, arXiv:1907.10023 [math.NT], 2019.
• Leo Tavares, Illustration: Centroid Triangles.
• D. Zvonkine, Counting ramified coverings and intersection theory on Hurwitz spaces II (local structure of Hurwitz spaces and combinatorial results), Moscow Mathematical Journal, Vol. 7, No. 1 (2007), pp. 135-162.
• D. Zvonkine, Home Page.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
• Index entries for two-way infinite sequences.

Crossrefs

Third diagonal of A027465.

Cf. A000459, A002378, A008585, A024966, A028895, A028896, A038764, A033996, A045943, A046092, A049598, A059073, A080855, A134171, A283394.

Cf. A060544.

Programs

Magma

[9*n*(n+1)/2: n in [0..50]]; // Vincenzo Librandi, Dec 29 2012

Maple

[seq(9*binomial(n+1,2), n=0..50)]; # Zerinvary Lajos, Nov 24 2006

Mathematica

Table[(9/2)*n*(n+1), {n,0,50}] (* G. C. Greubel, Aug 22 2017 *)

PARI

a(n)=9*n*(n+1)/2

Sage

[9*binomial(n+1, 2) for n in (0..50)] # G. C. Greubel, May 20 2021

Formula

Numerators of sequence a[n, n-2] in (a[i, j])^2 where a[i, j] = binomial(i-1, j-1)/2^(i-1) if j<=i, 0 if j>i.

a(n) = (9/2)*n*(n+1).

a(n) = 9*C(n, 1) + 9*C(n, 2) (binomial transform of (0, 9, 9, 0, 0, ...)). - Paul Barry, Mar 15 2003

G.f.: 9*x/(1-x)^3.

a(-1-n) = a(n).

a(n) = 9*C(n+1,2), n>=0. - Zerinvary Lajos, Aug 06 2008

a(n) = a(n-1) + 9*n (with a(0)=0). - Vincenzo Librandi, Nov 19 2010

a(n) = A060544(n+1) - 1. - Omar E. Pol, Oct 03 2011

a(n) = A218470(9*n+8). - Philippe Deléham, Mar 27 2013

E.g.f.: (9/2)*x*(x+2)*exp(x). - G. C. Greubel, Aug 22 2017

a(n) = A060544(n+1) - 1. See Centroid Triangles illustration. - Leo Tavares, Dec 27 2021

From Amiram Eldar, Feb 15 2022: (Start)

Sum_{n>=1} 1/a(n) = 2/9.

Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/9 - 2/9. (End)

From Amiram Eldar, Feb 21 2023: (Start)

Product_{n>=1} (1 - 1/a(n)) = -(9/(2*Pi))*cos(sqrt(17)*Pi/6).

Product_{n>=1} (1 + 1/a(n)) = 9*sqrt(3)/(4*Pi). (End)

Extensions

More terms from Patrick De Geest, Oct 15 1999