A340266 The number of degrees of freedom in a quadrilateral cell for a serendipity finite element space of order n.

4, 8, 12, 17, 23, 30, 38, 47, 57, 68, 80, 93, 107, 122, 138, 155, 173, 192, 212, 233, 255, 278, 302, 327, 353, 380, 408, 437, 467, 498, 530, 563, 597, 632, 668, 705, 743, 782, 822, 863, 905, 948, 992, 1037, 1083, 1130, 1178, 1227, 1277

Offset: 1

Author

Matthew Scroggs, Jan 02 2021

Keywords

• nonn
• easy

Links

• DefElement, Serendipity
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Programs

Mathematica

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 *)
LinearRecurrence[{3,-3,1},{4,8,12,17},50] (* Harvey P. Dale, Oct 24 2021 *)

PARI

a(n) = if (n==1, 4, n*(n+3)/2 + 3); \\ Michel Marcus, Jan 04 2021

Python

print([4 if n == 1 else n  * (n + 3) // 2 + 3 for n in range(1, 50)])

Formula

a(1) = 4, a(n) = n*(n+3)/2 + 3 (if n > 1).

From Stefano Spezia, Jan 02 2021: (Start)

G.f.: x*(4 - 4*x + x^3)/(1 - x)^3.

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

a(n) = (A111802(n+2)+1)/2 + 2. - Hugo Pfoertner, Jan 02 2021