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A060354 The n-th n-gonal number: a(n) = n*(n^2-3*n+4)/2. 26
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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

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.

Homberger and Vatter, On the effective and automatic enumeration of polynomial permutation classes.

C. Homberger, 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).

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

a(n) = Sum_{j=0..n-1} (binomial(0,0*j) + binomial(n-1,2)). - Zerinvary Lajos, Sep 04 2006 [corrected by Jon E. Schoenfield, Aug 11 2018]

G.f.: x*(1-2x+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

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}] (* The program uses the PolygonalNumber function from Mathematica version 10 *) (* Harvey P. Dale, Mar 07 2016 *)

LinearRecurrence[{4, -6, 4, -1}, {0, 1, 2, 6}, 50] (* Harvey P. Dale, Mar 07 2016 *)

PROG

(PARI) { for (n=0, 1000, write("b060354.txt", n, " ", (n*(n - 2)^2 + n^2)/2); ) } \\ Harry J. Smith, Jul 04 2009

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

CROSSREFS

First differences of A004255.

Cf A000124, A100177, A057145.

Sequence in context: A248832 A280400 A199477 * A140131 A159938 A325743

Adjacent sequences:  A060351 A060352 A060353 * A060355 A060356 A060357

KEYWORD

easy,nice,nonn

AUTHOR

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

STATUS

approved

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Last modified September 19 23:59 EDT 2019. Contains 327207 sequences. (Running on oeis4.)