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A050407 a(n) = n*(n^2 - 6*n + 11)/6. 13
0, 1, 1, 1, 2, 5, 11, 21, 36, 57, 85, 121, 166, 221, 287, 365, 456, 561, 681, 817, 970, 1141, 1331, 1541, 1772, 2025, 2301, 2601, 2926, 3277, 3655, 4061, 4496, 4961, 5457, 5985, 6546, 7141, 7771, 8437, 9140, 9881, 10661, 11481, 12342, 13245, 14191 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Number of invertible shuffles of n-2 cards. - Adam C. McDougall (mcdougal(AT)stolaf.edu) and David Molnar (molnar(AT)stolaf.edu), Apr 09 2002

If Y is a 3-subset of an n-set X then, for n>=3, a(n-2) is the number of (n-3)-subsets of X which have neither one element nor two elements in common with Y. - Milan Janjic, Dec 28 2007

Let A be the Hessenberg n X n matrix defined by: A[1,j]=j mod 2, A[i,i]:=1, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=4, a(n+1)=-coeff(charpoly(A,x),x^(n-3)). - Milan Janjic, Jan 24 2010

Starting with offset 3: (1, 2, 5, 11, 21, ...) = triangle A144257 * [1,2,3,...]. - Gary W. Adamson, Feb 18 2010

(1 + 2x + 5x^2 + 11x^3 + ...) = (1 + 2x + 3x^2 + ...)*(1 + 2x^2 + 3x^3 + ...). - Gary W. Adamson, Jul 26 2010

Starting (1, 2, 5, 11, ...) = binomial transform of [1, 1, 2, 1, 0, 0, 0, ...]. - Gary W. Adamson, Aug 25 2010

For n > 1: abs(abs(a(n+2) - a(n+1)) - abs(a(n+1) - a(n))) = n - 1; see also A086283. - Reinhard Zumkeller, Oct 17 2014

For n > 0, a(n) is the number of valid hook configurations of permutations of [n-1] that avoid the patterns 132 and 321. - Colin Defant, Apr 28 2019

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Colin Defant, Motzkin intervals and valid hook configurations, arXiv preprint arXiv:1904.10451 [math.CO], 2019.

Robert DiSario, Problem 10931, Amer. Math. Monthly, 109 (No. 3, 2002), 298.

Nurul Hilda Syani Putri, Mashadi, Sri Gemawati, Sequences from heptagonal pyramid corners of integer, International Mathematical Forum, Vol. 13, 2018, no. 4, 193-200.

Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.

Amit Kumar Singh, Akash Kumar and Thambipillai Srikanthan, Accelerating Throughput-aware Run-time Mapping for Heterogeneous MPSoCs, ACM Transactions on Design Automation of Electronic Systems, 2012. - From N. J. A. Sloane, Dec 25 2012

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

FORMULA

From Paul Barry, Jul 21 2003: (Start)

Diagonal sums of square array A086460 (starting 1, 1, 2, ...).

a(n+2) = 1 + n*(n+1)*(n-1)/6 = Sum_{k=0..n} (0^k + (n-k)*k). (End)

a(n) = binomial(n-1,3) + binomial(n-1,0), n>=0. - Zerinvary Lajos, Jul 24 2006

G.f.: x*(1-3*x+3*x^2)/(1-x)^4. - Colin Barker, May 06 2012

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jun 22 2012

a(n) = A000292(n-3) + 1, n > 2. - Ivan N. Ianakiev, Apr 27 2014

E.g.f.: x*(6 - 3*x + x^2)*exp(x)/6. - G. C. Greubel, Oct 30 2019

MAPLE

seq(binomial(n-1, 3) + 1, n = 0..46); # Zerinvary Lajos, Jul 24 2006

MATHEMATICA

Table[n*(n^2-6*n+11)/6, {n, 0, 50}] (* Vladimir Joseph Stephan Orlovsky, Dec 17 2008 *)

LinearRecurrence[{4, -6, 4, -1}, {0, 1, 1, 1}, 50] (* Vincenzo Librandi, Jun 22 2012 *)

Join[{0, 1, 1}, Nest[Accumulate, Range[0, 50], 2]+1] (* Harvey P. Dale, Sep 23 2017 *)

PROG

(MAGMA) I:=[0, 1, 1, 1]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Jun 22 2012

(Haskell)

a050407 n = n * (n ^ 2 - 6 * n + 11) `div` 6

-- Reinhard Zumkeller, Oct 17 2014

(PARI) a(n)=n*(n^2-6*n+11)/6 \\ Charles R Greathouse IV, Oct 07 2015

(Python) for n in range(0, 50): print(n*(n**2 - 6*n + 11)/6, end=', ') # Stefano Spezia, Jan 05 2019

(Sage) [n*(n^2-6*n+11)/6 for n in (0..50)] # G. C. Greubel, Oct 30 2019

(GAP) List([0..50], n-> n*(n^2-6*n+11)/6); # G. C. Greubel, Oct 30 2019

CROSSREFS

Apart from initial terms, one more than the tetrahedral numbers A000292.

Cf. A144257, A086283.

Sequence in context: A328670 A294745 A332063 * A113032 A100134 A137356

Adjacent sequences:  A050404 A050405 A050406 * A050408 A050409 A050410

KEYWORD

easy,nonn

AUTHOR

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 22 1999

STATUS

approved

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Last modified May 31 22:45 EDT 2020. Contains 334756 sequences. (Running on oeis4.)