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 A006331 a(n) = n*(n+1)*(2*n+1)/3. (Formerly M1963) 44
 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, {a3, 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 From Bruno Berselli, May 17 2018: (Start) a(n) = n*A002378(n) - Sum_{k=0..n-1} A002378(k) for n>0, a(0)=0. Also: A163102(n) = n*a(n) - Sum_{k=0..n-1} a(k) for n>0, A163102(0)=0. (End) a(n) = A005900(n) - A000290(n) = A096000(n) - A000578(n+1) = A000578(n+1) - A084980(n+1) = A000578(n+1) - A077415(n)-1 = A112524(n) + 1 = A188475(n) - 1 = A061317(n) - A100178(n) = A035597(n+1) - A006331(n+1). - Bruce J. Nicholson, Jun 24 2018 E.g.f.: (1/3)*exp(x)*x*(6 + 9*x + 2*x^2). - Stefano Spezia, Jan 05 2020 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 = 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. Cf. A002378, A048147, A098077, A163102. Cf. A005900, A084980, A077415, A112524, A188475, A100178, A035597, A096000. Sequence in context: A060515 A109723 A053594 * A252591 A296849 A296380 Adjacent sequences:  A006328 A006329 A006330 * A006332 A006333 A006334 KEYWORD nonn,easy,nice AUTHOR STATUS approved

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Last modified January 27 19:46 EST 2021. Contains 340479 sequences. (Running on oeis4.)