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A006331 a(n) = n*(n+1)*(2n+1)/3.
(Formerly M1963)
34
0, 2, 10, 28, 60, 110, 182, 280, 408, 570, 770, 1012, 1300, 1638, 2030, 2480, 2992, 3570, 4218, 4940, 5740, 6622, 7590, 8648, 9800, 11050, 12402, 13860, 15428, 17110, 18910, 20832, 22880, 25058, 27370, 29820, 32412, 35150, 38038, 41080, 44280 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Triangles in rhombic matchstick arrangement of side n.

Maximum accumulated number of electrons at energy level n. - Scott A. Brown (scottbrown(AT)neo.rr.com), Feb 28 2000

Let M_n denote the n X n matrix M_n(i,j)=i^2+j^2; then the characteristic polynomial of M_n is x^n - a(n)x^(n-1) - .... - Michael Somos, Nov 14 2002

Convolution of odds (A005408) and evens (A005843). - Graeme McRae, Jun 06 2006

10*a(n) = A016755(n) - A001845(n); since A016755 are odd cubes and A001845 centered octahedral numbers, 10*a(n) are the "odd cubes without their octahedral contents." - Damien Pras, Mar 19 2011

a(n) is the number of non-monotonic functions with domain {0,1,2} and codomain {0,1,...,n}. - Dennis P. Walsh, Apr 25 2011

For any odd number 2n+1, find sum a*b, {a<b and a+b=2n+1}. This sum is equal to the n-th nonzero term of this sequence. Thus for 13=2*n+1, n=6; there are six products 1*12+2*11+3*10+4*9+5*8+6*7=182, which is also twice the sum of the squares for n=6. - J. M. Bergot, Jul 16 2011

a(n) gives the number of (n+1) X (n+1) symmetric (0,1)-matrices containing three ones (see [Cameron]). - L. Edson Jeffery, Feb 18 2012

a(n) is the number of 4-tuples (w,x,y,z) with all terms in {0,...,n} and |w-x|<y. - Clark Kimberling, Jun 02 2012

Partial sums of A001105. - Omar E. Pol, Jan 12 2013

Total number of square diagonals (of any size) in an n X n square grid. - Wesley Ivan Hurt, Mar 24 2015

Number of diagonal attacks of two queens on (n+1) X (n+1) chessboard. - Antal Pinter, Sep 20 2015

REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

J. L. Bailey, Jr., A table to facilitate the fitting of certain logistic curves, Annals Math. Stat., 2 (1931), 355-359.

J. L. Bailey, A table to facilitate the fitting of certain logistic curves, Annals Math. Stat., 2 (1931), 355-359. [Annotated scanned copy]

P. Cameron, T. Prellberg and D. Stark, Asymptotics for incidence matrix classes, Electron. J. Combin. 13 (2006), #R85, p. 11.

G. Kreweras, Sur une classe de problèmes de denombrement liés au treillis des partitions des entiers, Cahiers du Bureau Universitaire de Recherche Opérationnelle}, Institut de Statistique, Université de Paris, 6 (1965), circa p. 82.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Author?, Basic atomic information [broken link ?]

Dennis Walsh, Notes on finite monotonic and non-monotonic functions

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

FORMULA

G.f.: 2*x*(1+x)/(1-x)^4. - Simon Plouffe (in his 1992 dissertation)

a(n) = 2*binomial(n+1,3) + 2*binomial(n+2,3).

a(n) = 2*A000330(n) = A002492(n)/2.

From the formula for the sum of squares of positive integers 1^2 + 2^2 + 3^2 + ... + n^2 = n(n+1)(2*n+1)/6, if we multiply both sides by 2 we get Sum_{k=0..n} 2*k^2 = n(n+1)(2*n+1)/3, which is an alternative formula for this sequence. - Mike Warburton (mikewarb(AT)gmail.com), Sep 08 2007

a(n) = sum(a*b), where the summing is over all unordered partitions 2*n+1=a+b. - Vladimir Shevelev, May 11 2012

a(n) = binomial(2n+2, 3)/2. - Ronan Flatley, Dec 13 2012

a(n) = A000292(n) + A002411(n). - Omar E. Pol, Jan 11 2013

a(0)=0, a(1)=2, a(2)=10, a(3)=28, a(n)=4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Harvey P. Dale, Apr 12 2013

a(n) = A208532(n+1,2). - Philippe Deléham, Dec 05 2013

Sum_{n>0} 1/a(n) = 9 - 12*log(2). - Enrique Pérez Herrero, Dec 03 2014

a(n) = A000292(n-1) + (n+1)*A000217(n). - J. M. Bergot, Sep 02 2015

a(n) = 2*(A000332(n+3) - A000332(n+1)). - Antal Pinter, Sep 20 2015

EXAMPLE

For n=2, a(2)=10 since there are 10 non-monotonic functions f from {0,1,2} to {0,1,2}, namely, functions f = <f(1),f(2),f(3)> given by <0,1,0>, <0,2,0>, <0,2,1>, <1,0,1>, <1,0,2>, <1,2,0>, <1,2,1>, <2,0,1>, <2,0,2>, and <2,1,2>. - Dennis P. Walsh, Apr 25 2011

Let n=4, 2*n+1 = 9. Since 9 = 1+8 = 3+6 = 5+4 = 7+2, a(4) = 1*8 + 3*6 + 5*4 + 7*2 = 60. - Vladimir Shevelev, May 11 2012

MAPLE

A006331 := proc(n)

    n*(n+1)*(2*n+1)/3 ;

end proc:

seq(A006331(n), n=0..80) ; # R. J. Mathar, Sep 27 2013

MATHEMATICA

Table[n(n+1)(2n+1)/3, {n, 0, 40}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {0, 2, 10, 28}, 50] (* Harvey P. Dale, Apr 12 2013 *)

PROG

(PARI) a(n)=if(n<0, 0, n*(n+1)*(2*n+1)/3)

(MAGMA) [n*(n+1)*(2*n+1)/3: n in [0..40]]; // Vincenzo Librandi, Aug 15 2011

(Haskell)

a006331 n = sum $ zipWith (*) [2*n-1, 2*n-3 .. 1] [2, 4 ..]

-- Reinhard Zumkeller, Feb 11 2012

CROSSREFS

A row of A132339.

a(n) = Sum_{i=0..n} T(i,n-i), array T as in A048147.

Sequence in context: A060515 A109723 A053594 * A252591 A104657 A000900

Adjacent sequences:  A006328 A006329 A006330 * A006332 A006333 A006334

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified May 28 21:38 EDT 2017. Contains 287241 sequences.