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A134507 Number of rectangles in a pyramid built with squares. The squares counted in A092498 are excluded.

%I A134507 #33 Apr 09 2024 12:57:06
%S A134507 0,4,19,57,134,269,486,813,1281,1926,2788,3910,5340,7130,9335,12015,
%T A134507 15234,19059,23562,28819,34909,41916,49928,59036,69336,80928,93915,
%U A134507 108405,124510,142345,162030,183689,207449,233442,261804,292674,326196
%N A134507 Number of rectangles in a pyramid built with squares. The squares counted in A092498 are excluded.
%C A134507 At the first step, the pyramid contains only one unitary square. At each step of rank n we add a row of 2*n-1 squares below the previous pyramid. The sequence is the number of rectangles of any size which can be seen in this pyramid of height n.
%C A134507 .__..........___.
%C A134507 |..|.........|..|
%C A134507 |__|......___|__|__
%C A134507 ..........|..|..|..|
%C A134507 ..0.......|__|__|__| 3 rectangles 2X1, 1 rectangle 3X1
%H A134507 Vincenzo Librandi, <a href="/A134507/b134507.txt">Table of n, a(n) for n = 1..1000</a>
%H A134507 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,5,-5,6,-4,1).
%F A134507 For n == 0 mod 3, a(n) = n*(3*n^3+5*n^2-3*n-3)/18; for n == 1 mod 3, a(n) = (n-1)*(3*n^3+8*n^2+5*n+2)/18; for n == 2 mod 3, a(n) = (3*n^4+5*n^3-3*n^2-3*n+2)/18. [corrected and edited by _Michel Marcus_, Apr 09 2024]
%F A134507 G.f.: -x^2*(5*x^2+3*x+4)/((x-1)^5*(x^2+x+1)). [_Colin Barker_, Nov 16 2012]
%F A134507 a(n) = (3*n^4+5*n^3-3*n^2-5*n+6*floor((n+1)/3))/18. - _Luce ETIENNE_, Jul 31 2015
%e A134507 G.f. = 4*x^2 + 19*x^3 + 57*x^4 + 134*x^5 + 269*x^6 + 486*x^7 + 813*x^8 + ...
%t A134507 a[ n_] := SeriesCoefficient[ x^2 (4 + 3 x + 5 x^2) / ((1 - x)^5 (1 + x + x^2)), {x, 0, n}]; (* _Michael Somos_, Feb 25 2014 *)
%t A134507 a[ n_] := Quotient[ 3 n^4 + 5 n^3 - 3 n^2 - 3 n + 2, 18]; (* _Michael Somos_, Feb 25 2014 *)
%t A134507 CoefficientList[Series[-x (5 x^2 + 3 x + 4)/((x - 1)^5 (x^2 + x + 1)), {x, 0, 40}], x] (* _Vincenzo Librandi Mar 01 2014 *)
%o A134507 (PARI) {a(n) = (3*n^4 + 5*n^3 - 3*n^2 - 3*n + 2) \ 18}; /* _Michael Somos_, Feb 17 2008 */
%o A134507 (Magma) I:=[0,4,19,57,134,269,486]; [n le 7 select I[n] else 4*Self(n-1)-6*Self(n-2)+5*Self(n-3)-5*Self(n-4)+6*Self(n-5)-4*Self(n-6)+Self(n-7): n in [1..50]]; // _Vincenzo Librandi_, Mar 01 2014
%Y A134507 Cf. A092498.
%K A134507 easy,nonn
%O A134507 1,2
%A A134507 Philippe Lallouet (philip.lallouet(AT)orange.fr), Jan 19 2008