A210375 Number of 2 X 2 matrices with all terms in {0,1,...,n} and (sum of terms) = n + 3.

0, 1, 16, 44, 80, 125, 180, 246, 324, 415, 520, 640, 776, 929, 1100, 1290, 1500, 1731, 1984, 2260, 2560, 2885, 3236, 3614, 4020, 4455, 4920, 5416, 5944, 6505, 7100, 7730, 8396, 9099, 9840, 10620, 11440, 12301, 13204, 14150, 15140, 16175, 17256, 18384, 19560

Offset: 0

Author

Clark Kimberling, Mar 20 2012

Keywords

• nonn
• easy

Comments

A210375 is also the number of 2 X 2 matrices with all terms in {0,1,...,n} and (sum of terms) = 3*n - 3.

See A210000 for a guide to related sequences.

Links

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

Crossrefs

Cf. A210000.

Programs

Mathematica

a = 0; b = n; z1 = 45;
t[n_] := t[n] = Flatten[Table[w + x + y + z, {w, a, b}, {x, a, b}, {y, a, b}, {z, a, b}]]
c[n_, k_] := c[n, k] = Count[t[n], k]
Table[c[n, n + 3], {n, 0, z1}]   (* A210375 *)
Table[c[n, 3 n - 3], {n, 0, z1}] (* A210375 *)

Formula

From Colin Barker, Dec 07 2017: (Start)

G.f.: x*(1 + 12*x - 14*x^2 - 4*x^3 + 6*x^4) / (1 - x)^4.

a(n) = (-120 + 74*n + 15*n^2 + n^3) / 6 for n > 1.

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 5. (End)