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A134507 Number of rectangles in a pyramid built with squares. The squares counted in A092498 are excluded. 2
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 (list; graph; refs; listen; history; text; internal format)
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, c(n) = n*((3*n^3)+(5*n^2)-(3^n)-3)/18; for n == 1 mod 3, c(n) = (n-1)*(3*(n^3]+8*(n^2)+5*n+2)/18; for n == 2 mod 3, c(n) = (3*(n^4)+5*(n^3)-3*(n^2)-3*n+2)/18

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: A020496 A108484 A283333 * 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

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Last modified February 16 21:46 EST 2020. Contains 331975 sequences. (Running on oeis4.)