A060354 - OEIS

%I #80 Dec 30 2024 22:43:22

%S 0,1,2,6,16,35,66,112,176,261,370,506,672,871,1106,1380,1696,2057,

%T 2466,2926,3440,4011,4642,5336,6096,6925,7826,8802,9856,10991,12210,

%U 13516,14912,16401,17986,19670,21456,23347,25346,27456,29680,32021

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

%C Binomial transform of (0,1,0,3,0,0,0,...). - _Paul Barry_, Sep 14 2006

%C 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

%C Main diagonal of A317302. - _Omar E. Pol_, Aug 11 2018

%H Harry J. Smith, <a href="/A060354/b060354.txt">Table of n, a(n) for n = 0..1000</a>

%H D. W. Cranston, I. H. Sudborough, and D. B. West, <a href="http://dx.doi.org/10.1016/j.disc.2007.01.011">Short proofs for cut-and-paste sorting of permutations</a>, Discrete Math. 307, 22 (2007), 2866-2870.

%H Cheyne Homberger, <a href="http://arxiv.org/abs/1410.2657">Patterns in Permutations and Involutions: A Structural and Enumerative Approach</a>, arXiv preprint 1410.2657 [math.CO], 2014.

%H Homberger and Vatter, <a href="http://www.math.ufl.edu/~vatter/publications/poly-classes/">On the effective and automatic enumeration of polynomial permutation classes</a>. [Broken link]

%H C. Homberger, V. Vatter, <a href="http://arxiv.org/abs/1308.4946">On the effective and automatic enumeration of polynomial permutation classes</a>, arXiv preprint arXiv:1308.4946 [math.CO], 2013-2015.

%H <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

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

%F E.g.f.: exp(x)*x*(1+x^2/2). - _Paul Barry_, Sep 14 2006

%F G.f.: x*(1-2*x+4*x^2)/(1-x)^4. - _R. J. Mathar_, Sep 02 2008

%F a(n) = A057145(n,n). - _R. J. Mathar_, Jul 28 2016

%F a(n) = A000124(n-2) * n. - _Bruce J. Nicholson_, Jul 13 2018

%F a(n) = Sum_{i=0..n-1} (i*(n-2) + 1). - _Ivan N. Ianakiev_, Sep 25 2020

%p A060354 := proc(n)

%p     (n*(n-2)^2+n^2)/2 ;

%p end proc: # _R. J. Mathar_, Jul 28 2016

%t Table[(n (n-2)^2+n^2)/2,{n,0,50}] (* _Harvey P. Dale_, Aug 05 2011 *)

%t CoefficientList[Series[x (1 - 2 x + 4 x^2) / (1 - x)^4, {x, 0, 50}], x] (* _Vincenzo Librandi_, Feb 16 2015 *)

%t Table[PolygonalNumber[n,n],{n,0,50}] (* _Harvey P. Dale_, Mar 07 2016 *)

%t LinearRecurrence[{4,-6,4,-1},{0,1,2,6},50] (* _Harvey P. Dale_, Mar 07 2016 *)

%o (PARI) a(n) = { (n*(n - 2)^2 + n^2)/2 } \\ _Harry J. Smith_, Jul 04 2009

%o (Magma) [(n*(n-2)^2+n^2)/2: n in [0..50]]; // _Vincenzo Librandi_, Feb 16 2015

%Y First differences of A004255.

%Y Cf. A000124, A100177, A057145.

%K easy,nice,nonn,changed

%O 0,3

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