A060354 The n-th n-gonal number: a(n) = n*(n^2 - 3*n + 4)/2.

0, 1, 2, 6, 16, 35, 66, 112, 176, 261, 370, 506, 672, 871, 1106, 1380, 1696, 2057, 2466, 2926, 3440, 4011, 4642, 5336, 6096, 6925, 7826, 8802, 9856, 10991, 12210, 13516, 14912, 16401, 17986, 19670, 21456, 23347, 25346, 27456, 29680, 32021

Offset: 0

Author

Hareendra Yalamanchili (hyalaman(AT)mit.edu), Apr 01 2001

Keywords

• easy
• nice
• nonn

Comments

Binomial transform of (0,1,0,3,0,0,0,...). - Paul Barry, Sep 14 2006

Also the number of permutations of length n which can be sorted by a single cut-and-paste move (in the sense of Cranston, Sudborough, and West). - Vincent Vatter, Aug 21 2013

Main diagonal of A317302. - Omar E. Pol, Aug 11 2018

a(n) is the number of ternary strings of length n that contain exactly one 1, zero or two 2's and have no restriction on the number of 0's. For example, a(5) = 35 since the strings are 12200 (30 of this type) and 10000 (5 of this type). - Enrique Navarrete, May 08 2025

Links

• Harry J. Smith, Table of n, a(n) for n = 0..1000
• D. W. Cranston, I. H. Sudborough, and D. B. West, Short proofs for cut-and-paste sorting of permutations, Discrete Math. 307, 22 (2007), 2866-2870.
• Cheyne Homberger, Patterns in Permutations and Involutions: A Structural and Enumerative Approach, arXiv preprint 1410.2657 [math.CO], 2014.
• C. Homberger and V. Vatter, On the effective and automatic enumeration of polynomial permutation classes. [Broken link]
• C. Homberger and V. Vatter, On the effective and automatic enumeration of polynomial permutation classes, arXiv preprint arXiv:1308.4946 [math.CO], 2013-2015.
• Index to sequences related to polygonal numbers
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

First differences of A004255.

Cf. A000124, A100177, A057145.

Programs

Magma

[(n*(n-2)^2+n^2)/2: n in [0..50]]; // Vincenzo Librandi, Feb 16 2015

Maple

A060354 := proc(n)
    (n*(n-2)^2+n^2)/2 ;
end proc: # R. J. Mathar, Jul 28 2016

Mathematica

Table[(n (n-2)^2+n^2)/2,{n,0,50}] (* Harvey P. Dale, Aug 05 2011 *)
CoefficientList[Series[x (1 - 2 x + 4 x^2) / (1 - x)^4, {x, 0, 50}], x] (* Vincenzo Librandi, Feb 16 2015 *)
Table[PolygonalNumber[n,n],{n,0,50}] (* Harvey P. Dale, Mar 07 2016 *)
LinearRecurrence[{4,-6,4,-1},{0,1,2,6},50] (* Harvey P. Dale, Mar 07 2016 *)

PARI

a(n) = { (n*(n - 2)^2 + n^2)/2 } \\ Harry J. Smith, Jul 04 2009

Formula

a(n) = (n*(n-2)^2 + n^2)/2.

E.g.f.: exp(x)*x*(1+x^2/2). - Paul Barry, Sep 14 2006

G.f.: x*(1-2*x+4*x^2)/(1-x)^4. - R. J. Mathar, Sep 02 2008

a(n) = A057145(n,n). - R. J. Mathar, Jul 28 2016

a(n) = A000124(n-2) * n. - Bruce J. Nicholson, Jul 13 2018

a(n) = Sum_{i=0..n-1} (i*(n-2) + 1). - Ivan N. Ianakiev, Sep 25 2020