%I #40 Oct 24 2021 16:24:07
%S 4,8,12,17,23,30,38,47,57,68,80,93,107,122,138,155,173,192,212,233,
%T 255,278,302,327,353,380,408,437,467,498,530,563,597,632,668,705,743,
%U 782,822,863,905,948,992,1037,1083,1130,1178,1227,1277
%N The number of degrees of freedom in a quadrilateral cell for a serendipity finite element space of order n.
%H DefElement, <a href="https://defelement.com/elements/serendipity">Serendipity</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(1) = 4, a(n) = n*(n+3)/2 + 3 (if n > 1).
%F From _Stefano Spezia_, Jan 02 2021: (Start)
%F G.f.: x*(4 - 4*x + x^3)/(1 - x)^3.
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 4. (End)
%F a(n) = (A111802(n+2)+1)/2 + 2. - _Hugo Pfoertner_, Jan 02 2021
%t A340266[n_] := Module[{a}, a[1] = 4; a[i_] := a[i] = i*(i + 3)/2 + 3; a[n]]; Table[A340266[n], {n, 1, 49}] (* _Robert P. P. McKone_, Jan 29 2021 *)
%t LinearRecurrence[{3,-3,1},{4,8,12,17},50] (* _Harvey P. Dale_, Oct 24 2021 *)
%o (Python) print([4 if n == 1 else n * (n + 3) // 2 + 3 for n in range(1, 50)])
%o (PARI) a(n) = if (n==1, 4, n*(n+3)/2 + 3); \\ _Michel Marcus_, Jan 04 2021
%K nonn,easy
%O 1,1
%A _Matthew Scroggs_, Jan 02 2021
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