A001249 Squares of tetrahedral numbers: a(n) = binomial(n+3,n)^2.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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 [Index/grid size adjusted by Rick L. Shepherd, Jun 27 2017]

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

The inverse binomial transform gives row n=2 of A087107. - R. J. Mathar, Aug 31 2022

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).

Crossrefs

Cf. A000290, A000292, A006542, A033455, A108674 (first diffs.), A086020 (partial sums).

Third column of triangle A008459.

Cf. A000579, A001303, A040977.

Programs

Maple

A001249 := proc(n) binomial(n+3,n)^2 end proc: seq(A001249(n),n=0..10) ; # Zerinvary Lajos, May 17 2006

Mathematica

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

PARI

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

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

From Amiram Eldar, Jan 24 2022: (Start)

Sum_{n>=0} 1/a(n) = 9*Pi^2 - 351/4.

Sum_{n>=0} (-1)^n/a(n) = 63/4 - 3*Pi^2/2. (End)

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

a(n) = a(n-1)+A000217(n+1)*A000330(n+1). - J. M. Bergot, Aug 29 2022

a(n) = A002415(n+2)^2 - 20*A006857(n-1). - Yasser Arath Chavez Reyes, Nov 08 2024