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%I #82 Aug 31 2022 08:16:30
%S 1,16,100,400,1225,3136,7056,14400,27225,48400,81796,132496,207025,
%T 313600,462400,665856,938961,1299600,1768900,2371600,3136441,4096576,
%U 5290000,6760000,8555625,10732176,13351716,16483600,20205025,24601600,29767936,35808256
%N Squares of tetrahedral numbers: a(n) = binomial(n+3,n)^2.
%C Total area of all square and rectangular regions from an n+1 X n+1 grid. E.g., n = 2, there are 9 individual squares, 4 2 X 2's and 1 3 X 3, total area 9 + 16 + 9 = 34. The rectangular regions include 6 2 X 1's, 6 1 X 2's, 3 3 X 1's, 3 1 X 3's, 2 3 X 2's and 2 2 X 3's, total area 12 + 12 + 9 + 9 + 12 + 12 = 66, hence a(2) = 34 + 66 = 100. - _Jon Perry_, Jul 29 2003 [Index/grid size adjusted by _Rick L. Shepherd_, Jun 27 2017]
%C Number of 3 X 3 submatrices of an n+3 X n+3 matrix. - _Rick L. Shepherd_, Jun 27 2017
%C The inverse binomial transform gives row n=2 of A087107. - _R. J. Mathar_, Aug 31 2022
%H T. D. Noe, <a href="/A001249/b001249.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F From _R. J. Mathar_, Aug 19 2008: (Start)
%F a(n) = (A000292(n+1))^2.
%F O.g.f.: (1+x)(x^2+8x+1)/(1-x)^7. (End)
%F a(n) = C(n+4, 3)*C(n+4, 4)/(n+4) + A001303(n) = C(n+4, 3)*C(n+3, 3)/4 + A001303(n) = C(n+4, 6) + 3*C(n+5, 6) + C(n+6,6) + A001303(n). - _Gary Detlefs_, Aug 07 2013
%F -n^2*a(n) + (n+3)^2*a(n-1) = 0. - _R. J. Mathar_, Aug 15 2013
%F a(n) = 9*A040977(n-1) + A000579(n+6) + A000579(n+3). - _R. J. Mathar_, Aug 15 2013
%F a(n) = (n+3)*C(n+2, 2)*C(n+3, 3)/3. - _Gary Detlefs_, Jan 06 2014
%F a(n) = A000290(n+1)*A000290(n+2)*A000290(n+3)/36. - _Bruno Berselli_, Nov 12 2014
%F G.f. 2F1(4,4;1;x). - _R. J. Mathar_, Aug 09 2015
%F E.g.f.: exp(x)*(1 + 15*x + 69*x^2/2! + 147*x^3/3! + 162*x^4/4! + 90*x^5/5! + 20*x^6/6!). Computed from the o.g.f with the formulas (23) - (25) of the W. Lang link given in A060187. - _Wolfdieter Lang_, Jul 27 2017
%F From _Amiram Eldar_, Jan 24 2022: (Start)
%F Sum_{n>=0} 1/a(n) = 9*Pi^2 - 351/4.
%F Sum_{n>=0} (-1)^n/a(n) = 63/4 - 3*Pi^2/2. (End)
%F a(n) = 7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7). - _Wesley Ivan Hurt_, Aug 29 2022
%F a(n) = a(n-1)+A000217(n+1)*A000330(n+1). - _J. M. Bergot_, Aug 29 2022
%p A001249 := proc(n) binomial(n+3,n)^2 end proc: seq(A001249(n),n=0..10) ; # _Zerinvary Lajos_, May 17 2006
%t Table[Binomial[n + 3, 3]^2, {n, 0, 100}] (* _T. D. Noe_, Jun 26 2012 *)
%o (PARI) a(n)=binomial(n+3,3)^2 \\ _Charles R Greathouse IV_, Sep 24 2015
%Y Cf. A000290, A000292, A006542, A033455, A108674 (first diffs.), A086020 (partial sums).
%Y Third column of triangle A008459.
%Y Cf. A000579, A001303, A040977.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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