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

%I A371532 #26 Aug 23 2025 10:02:27
%S A371532 1,19,93,263,569,1051,1749,2703,3953,5539,7501,9879,12713,16043,19909,
%T A371532 24351,29409,35123,41533,48679,56601,65339,74933,85423,96849,109251,
%U A371532 122669,137143,152713,169419,187301,206399,226753,248403,271389,295751,321529,348763
%N A371532 Centered cuboctahedral numbers: the number of integer triples (x,y,z) such that max(|x|,|y|,|z|) <= n and |x|+|y|+|z| <= 2n.
%H A371532 Paolo Xausa, <a href="/A371532/b371532.txt">Table of n, a(n) for n = 0..10000</a>
%H A371532 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Cuboctahedron.html">Cuboctahedron</a>.
%H A371532 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FigurateNumber.html">Figurate Number</a>.
%H A371532 Wikipedia, <a href="https://en.wikipedia.org/wiki/Ehrhart_polynomial">Ehrhart polynomial</a>.
%H A371532 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A371532 a(n) = (20*n^3 + 24*n^2 + 10*n + 3)/3.
%F A371532 a(n) =  A016755(n) - A130809(n-2).
%F A371532 G.f.: (x^3 + 23*x^2 + 15*x + 1)/(x-1)^4. - _Paolo Xausa_, Apr 02 2024
%F A371532 From _Elmo R. Oliveira_, Aug 22 2025: (Start)
%F A371532 E.g.f.: exp(x)*(3 + 54*x + 84*x^2 + 20*x^3)/3.
%F A371532 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)
%e A371532 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)).
%t A371532 Array[(20*#^3 + 24*#^2 + 10*# + 3)/3 &, 50, 0] (* or *)
%t A371532 LinearRecurrence[{4, -6, 4, -1}, {1, 19, 93, 263}, 50] (* _Paolo Xausa_, Apr 02 2024 *)
%o A371532 (Python)
%o A371532 def A371532(n): return n*(n*(5*n+6<<2)+10)//3+1 # _Chai Wah Wu_, Apr 02 2024
%Y A371532 Cf. A016755, A130809, A371515.
%K A371532 nonn,easy,changed
%O A371532 0,2
%A A371532 _Peter Kagey_, Mar 26 2024