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A001249 Squares of tetrahedral numbers: binomial(n+3,n)^2. 19
1, 16, 100, 400, 1225, 3136, 7056, 14400, 27225, 48400, 81796, 132496, 207025, 313600, 462400, 665856, 938961, 1299600, 1768900, 2371600, 3136441, 4096576, 5290000, 6760000, 8555625, 10732176, 13351716, 16483600, 20205025, 24601600, 29767936, 35808256 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

Number of 3 X 3 submatrices of an n+3 X n+3 matrix. - Rick L. Shepherd, Jun 27 2017

LINKS

T. D. Noe, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).

FORMULA

From R. J. Mathar, Aug 19 2008: (Start)

a(n) = (A000292(n+1))^2.

O.g.f.: (1+x)(x^2+8x+1)/(1-x)^7. (End)

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

-n^2*a(n) + (n+3)^2*a(n-1) = 0. - R. J. Mathar, Aug 15 2013

a(n) = 9*A040977(n-1) + A000579(n+6) + A000579(n+3). - R. J. Mathar, Aug 15 2013

a(n) = (n+3)*C(n+2, 2)*C(n+3, 3)/3. - Gary Detlefs, Jan 06 2014

a(n) = A000290(n+1)*A000290(n+2)*A000290(n+3)/36. - Bruno Berselli, Nov 12 2014

G.f. 2F1(4,4;1;x). - R. J. Mathar, Aug 09 2015

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

MAPLE

[seq(binomial(n+3, n)^2, n=0..50)]; # Zerinvary Lajos, May 17 2006

a:=n->sum(sum(binomial(j, 2)*binomial(k, 2), j=0..n), k=0..n): seq(a(n), n=2..28); # Zerinvary Lajos, Jun 02 2007

MATHEMATICA

Table[Binomial[n + 3, 3]^2, {n, 0, 100}] (* T. D. Noe, Jun 26 2012 *)

PROG

(PARI) a(n)=binomial(n+3, 3)^2 \\ Charles R Greathouse IV, Sep 24 2015

CROSSREFS

Cf. A000290, A000292, A006542, A033455.

Third column of triangle A008459.

Sequence in context: A045784 A016958 A108677 * A014796 A052206 A169721

Adjacent sequences:  A001246 A001247 A001248 * A001250 A001251 A001252

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

Index/grid size adjusted in first comment by Rick L. Shepherd, Jun 27 2017

STATUS

approved

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Last modified February 20 01:06 EST 2018. Contains 299357 sequences. (Running on oeis4.)