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A050407 n*(n^2 - 6*n + 11)/6. 8
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 by 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

LINKS

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

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

Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081, 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

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}. - Paul Barry, Jul 21 2003

a(n) = C(n-1,3) + C(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

MAPLE

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

MATHEMATICA

a[n_]:=n*(n^2-6*n+11)/6; ...and/or...a=1; lst={0, 1, 1, a}; k=1; e=1; Do[a+=k; AppendTo[lst, a]; e++; k+=e, {n, 0, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Dec 17 2008 *)

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

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

CROSSREFS

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

Cf. A144257, A086283.

Sequence in context: A256310 A026390 A005575 * 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 December 10 03:42 EST 2016. Contains 278993 sequences.