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A051472 a(n) = A028317(n)/2.

%I A051472 #14 Jul 02 2025 16:01:58
%S A051472 3,3,6,4,4,19,5,18,18,5,23,65,23,6,6,102,189,231,189,102,7,41,291,420,
%T A051472 420,291,41,7,48,711,840,711,48,8,605,1551,1551,605,8,281,3102,281,9,
%U A051472 72,2574,4433,4433,2574,72,9,81,1456,7007,11583,7007,1456,81,10,10,588
%N A051472 a(n) = A028317(n)/2.
%H A051472 G. C. Greubel, <a href="/A051472/b051472.txt">Table of n, a(n) for n = 0..1000</a>
%e A051472 Even elements of (1/2)*A028317 as an irregular triangle:
%e A051472    3,  3;
%e A051472    6;
%e A051472    4,  4;
%e A051472   19;
%e A051472    5, 18, 18, 5;
%e A051472   23, 65, 23;
%e A051472    6,  6;
%e A051472   ...
%t A051472 A028313[n_, k_]:= If[n<2, 1, Binomial[n,k] +3*Binomial[n-2,k-1]];
%t A051472 f= Table[A028313[n,k], {n,0,100}, {k,0,n}]//Flatten;
%t A051472 b[n_]:= DeleteCases[{f[[n+1]]}, _?OddQ]/2;
%t A051472 Table[b[n], {n,0,200}]//Flatten (* _G. C. Greubel_, Jan 06 2024 *)
%o A051472 (Magma)
%o A051472 A028313:= func< n, k | n le 1 select 1 else Binomial(n, k) +3*Binomial(n-2, k-1) >;
%o A051472 a:=[A028313(n, k): k in [0..n], n in [0..100]];
%o A051472 [a[n]/2: n in [1..200] | (a[n] mod 2) eq 0]; // _G. C. Greubel_, Jan 06 2024
%o A051472 (SageMath)
%o A051472 def A028313(n, k): return 1 if n<2 else binomial(n, k) + 3*binomial(n-2, k-1)
%o A051472 a=flatten([[A028313(n, k) for k in range(n+1)] for n in range(101)])
%o A051472 [a[n]/2 for n in (0..200) if a[n]%2==0] # _G. C. Greubel_, Jan 06 2024
%Y A051472 Cf. A028313, A028317.
%K A051472 nonn,tabf
%O A051472 0,1
%A A051472 _James Sellers_