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

0, 4, 19, 57, 134, 269, 486, 813, 1281, 1926, 2788, 3910, 5340, 7130, 9335, 12015, 15234, 19059, 23562, 28819, 34909, 41916, 49928, 59036, 69336, 80928, 93915, 108405, 124510, 142345, 162030, 183689, 207449, 233442, 261804, 292674, 326196

Offset

1,2

Comments

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.

.__..........___.

|..|.........|..|

|__|......___|__|__

..........|..|..|..|

..0.......|__|__|__| 3 rectangles 2X1, 1 rectangle 3X1

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (4,-6,5,-5,6,-4,1).

Formula

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]

G.f.: -x^2*(5*x^2+3*x+4)/((x-1)^5*(x^2+x+1)). [Colin Barker, Nov 16 2012]

a(n) = (3*n^4+5*n^3-3*n^2-5*n+6*floor((n+1)/3))/18. - Luce ETIENNE, Jul 31 2015

Example

G.f. = 4*x^2 + 19*x^3 + 57*x^4 + 134*x^5 + 269*x^6 + 486*x^7 + 813*x^8 + ...

Mathematica

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 *)
a[ n_] := Quotient[ 3 n^4 + 5 n^3 - 3 n^2 - 3 n + 2, 18]; (* Michael Somos, Feb 25 2014 *)
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 *)

Prog

(PARI) {a(n) = (3*n^4 + 5*n^3 - 3*n^2 - 3*n + 2) \ 18}; /* Michael Somos, Feb 17 2008 */
(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

Crossrefs

Cf. A092498.

Sequence in context: A108484 A283333 A332697 * A098813 A212039 A055485

Adjacent sequences: A134504 A134505 A134506 * A134508 A134509 A134510

Keyword

easy,nonn

Author

Philippe Lallouet (philip.lallouet(AT)orange.fr), Jan 19 2008

Status

approved