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A006331
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n*(n+1)*(2n+1)/3.
(Formerly M1963)
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21
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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; internal format)
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OFFSET
| 0,2
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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 denotes 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 (g_m(AT)mcraefamily.com), 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}. [From Dennis P. Walsh, April 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 nth 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
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REFERENCES
| G. Kreweras, Sur une classe de problemes de denombrement lies au treillis des partitions des entiers, Cahiers du Bureau Universitaire de Recherche Op\'{e}rationnelle, Institut de Statistique, Universit\'{e} de Paris, 6 (1965), circa p. 82.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..10000
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Author?, Basic atomic information
Index to sequences with linear recurrences with constant coefficients, signature (4,-6,4,-1).
Dennis Walsh, Notes on finite monotonic and non-monotonic functions
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FORMULA
| G.f.: x*(2+2*x)/(1-x)^4. a(n)=2*C(n+1, 3)+2*C(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(2*k^2,k=0..n) = n(n+1)(2*n+1)/3, which is an alternative formula for this sequence. - Mike Warburton (mikewarb(AT)gmail.com), Sep 08 2007
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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>. [From Dennis P. Walsh, April 25 2011]
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MAPLE
| A006331:=2*(1+z)/(z-1)**4; [S. Plouffe in his 1992 dissertation, without the initial 0.]
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MATHEMATICA
| f[n_]:=4*n+2; s1=s2=0; lst={}; Do[a=f[n]; s1+=a; s2+=s1; AppendTo[lst, s2], {n, 0, 6!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Jun 25 2009]
s = 0; lst = {s}; Do[s += 2*n^2; AppendTo[lst, s], {n, 1, 40, 1}]; lst [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 15 2009]
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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
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CROSSREFS
| A row of A132339.
a(n)=Sum{T(i, n-i): i=0, 1, ..., n}, array T as in A048147.
Sequence in context: A060515 A109723 A053594 * A104657 A000900 A124023
Adjacent sequences: A006328 A006329 A006330 * A006332 A006333 A006334
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KEYWORD
| nonn,easy,nice,changed
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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