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A014820 a(n) = (1/3)*(n^2 + 2*n + 3)*(n+1)^2.

%I A014820 #87 Feb 16 2025 08:32:33
%S A014820 1,8,33,96,225,456,833,1408,2241,3400,4961,7008,9633,12936,17025,
%T A014820 22016,28033,35208,43681,53600,65121,78408,93633,110976,130625,152776,
%U A014820 177633,205408,236321,270600,308481
%N A014820 a(n) = (1/3)*(n^2 + 2*n + 3)*(n+1)^2.
%C A014820 a(n) is the number of 4 X 4 pandiagonal magic squares with sum 2n. - Sharon Sela (sharonsela(AT)hotmail.com), May 10 2002
%C A014820 Figurate numbers based on the 4-dimensional regular convex polytope called the 16-cell, hexadecachoron, 4-cross polytope or 4-hyperoctahedron with Schlaefli symbol {3,3,4}. a(n)=(n^2*(n^2+2))/3 if the offset were 1. - Michael J. Welch (mjw1(AT)ntlworld.com), Apr 01 2004, _R. J. Mathar_, Jul 18 2009
%C A014820 If X is an n-set and Y_i (i=1,2,3) mutually disjoint 2-subsets of X then a(n-6) is equal to the number of 7-subsets of X intersecting each Y_i (i=1,2,3). - _Milan Janjic_, Aug 26 2007
%C A014820 Equals binomial transform of [1, 7, 18, 20, 8, 0, 0, 0, ...], where (1, 7, 18, 20, 8) = row 4 of the Chebyshev triangle A081277. Also = row 4 of the array in A142978. - _Gary W. Adamson_, Jul 19 2008
%D A014820 T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.
%H A014820 Vincenzo Librandi, <a href="/A014820/b014820.txt">Table of n, a(n) for n = 0..1000</a>
%H A014820 M. Ahmed, J. De Loera and R. Hemmecke, <a href="http://arxiv.org/abs/math/0201108">Polyhedral Cones of Magic Cubes and Squares</a>, arXiv:math/0201108 [math.CO], 2002.
%H A014820 Maya Ahmed, Jesús De Loera and Raymond Hemmecke, <a href="http://dx.doi.org/10.1007/978-3-642-55566-4_2">Polyhedral cones of magic cubes and squares</a>, in Discrete and Computational Geometry, Springer, Berlin, 2003, pp. 25-41.
%H A014820 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>
%H A014820 Milan Janjić, <a href="https://arxiv.org/abs/1905.04465">On Restricted Ternary Words and Insets</a>, arXiv:1905.04465 [math.CO], 2019.
%H A014820 M. Janjic and B. Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv 1301.4550 [math.CO], 2013.
%H A014820 Hyun Kwang Kim, <a href="http://dx.doi.org/10.1090/S0002-9939-02-06710-2">On Regular Polytope Numbers</a>, Proc. Amer. Math. Soc., 131 (2002), 65-75.
%H A014820 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/16-Cell.html">16-Cell</a>
%H A014820 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1)
%F A014820 Or, a(n-1) = n^2*(n^2+2)/3. - Corrected by _R. J. Mathar_, Jul 18 2009
%F A014820 From _Vladeta Jovovic_, Apr 03 2002: (Start)
%F A014820 G.f.: (1+x)^3/(1-x)^5.
%F A014820 Recurrence: a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). (End)
%F A014820 a(n-1) = C(n+3,4) + 3 C(n+2,4) + 3 C(n+1,4) + C(n,4).
%F A014820 Sum_{n>=0} 1/((1/3*(n^2 + 2*n + 3))*(n+1)^2) = (1/4)*Pi^2 - 3*sqrt(2)*Pi*coth(Pi*sqrt(2))*(1/8) + 3/8 = 1.1758589... - _Stephen Crowley_, Jul 14 2009
%F A014820 a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5), with n > 4, a(0)=1, a(1)=8, a(2)=33, a(3)=96, a(4)=225. - _Yosu Yurramendi_, Sep 03 2013
%F A014820 From _Bruce J. Nicholson_, Jan 23 2019: (Start)
%F A014820 Sum_{i=0..n} a(i) = A061927(n+1).
%F A014820 a(n) = 4*A002415(n+1) + A000290(n+1) = A039623(n+1) + A002415(n+1). (End)
%F A014820 E.g.f.: (3 + 21*x + 27*x^2 + 10*x^3 + x^4)*exp(x)/3. - _G. C. Greubel_, Feb 10 2019
%F A014820 Sum_{n >= 0} (-1)^n/(a(n)*a(n+1)) = 17/3 - 8*log(2) = 1/(8 + 2/(8 + 6/(8 + ... + n*(n-1)/(8 + ...)))). See A142983. - _Peter Bala_, Mar 06 2024
%p A014820 al:=proc(s,n) binomial(n+s-1,s); end; be:=proc(d,n) local r; add( (-1)^r*binomial(d-1,r)*2^(d-1-r)*al(d-r,n),r=0..d-1); end; [seq(be(4,n),n=0..100)];
%t A014820 LinearRecurrence[{5, -10, 10, -5, 1}, {1, 8, 33, 96, 225}, 31] (* _Jean-François Alcover_, Jan 17 2018 *)
%o A014820 (Magma) [(1/3)*(n^2+2*n+3)*(n+1)^2: n in [0..40]]; // _Vincenzo Librandi_, May 22 2011
%o A014820 (PARI) a(n)=(n+1)^2*(n^2+2*n+3)/3 \\ _Charles R Greathouse IV_, Apr 17 2012
%o A014820 (R)
%o A014820 a <- c(1, 8, 33, 96,225)
%o A014820 for(n in (length(a)+1):30) a[n] <- 5*a[n-1]-10*a[n-2]+10*a[n-3]-5*a[n-4]+a[n-5]
%o A014820 a # _Yosu Yurramendi_, Sep 03 2013
%o A014820 (Sage) [((n+1)^2+2)*(n+1)^2/3 for n in range(40)] # _G. C. Greubel_, Feb 10 2019
%o A014820 (GAP) List([0..40], n -> (n+1)^2*((n+1)^2 +2)/3); # _G. C. Greubel_, Feb 10 2019
%Y A014820 Cf. A000332, A000583, A005900, A069038, A069039, A070212, A081277, A092181, A092182, A092183, A099175, A099193, A099195, A099196, A099197, A142978.
%Y A014820 Cf. A061927, A002415, A000290, A039623.
%K A014820 nonn,easy
%O A014820 0,2
%A A014820 _N. J. A. Sloane_
%E A014820 Formula index corrected by _R. J. Mathar_, Jul 18 2009