A371532 Centered cuboctahedral numbers: the number of integer triples (x,y,z) such that max(|x|,|y|,|z|) <= n and |x|+|y|+|z| <= 2n.

1, 19, 93, 263, 569, 1051, 1749, 2703, 3953, 5539, 7501, 9879, 12713, 16043, 19909, 24351, 29409, 35123, 41533, 48679, 56601, 65339, 74933, 85423, 96849, 109251, 122669, 137143, 152713, 169419, 187301, 206399, 226753, 248403, 271389, 295751, 321529, 348763

Offset: 0

Author

Peter Kagey, Mar 26 2024

Keywords

• nonn
• easy
• changed

Examples

			The a(1) = 19 lattice points are all permutations of the points (0,0,0), (0,0,1), and (0,1,1), where any number of the coordinates can also be made negative (e.g., (1,-1,0)).
		

Links

• Paolo Xausa, Table of n, a(n) for n = 0..10000
• Eric Weisstein's World of Mathematics, Cuboctahedron.
• Eric Weisstein's World of Mathematics, Figurate Number.
• Wikipedia, Ehrhart polynomial.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A016755, A130809, A371515.

Programs

Mathematica

Array[(20*#^3 + 24*#^2 + 10*# + 3)/3 &, 50, 0] (* or *)
LinearRecurrence[{4, -6, 4, -1}, {1, 19, 93, 263}, 50] (* Paolo Xausa, Apr 02 2024 *)

Python

def A371532(n): return n*(n*(5*n+6<<2)+10)//3+1 # Chai Wah Wu, Apr 02 2024

Formula

a(n) = (20*n^3 + 24*n^2 + 10*n + 3)/3.

a(n) = A016755(n) - A130809(n-2).

G.f.: (x^3 + 23*x^2 + 15*x + 1)/(x-1)^4. - Paolo Xausa, Apr 02 2024

From Elmo R. Oliveira, Aug 22 2025: (Start)

E.g.f.: exp(x)*(3 + 54*x + 84*x^2 + 20*x^3)/3.

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