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

1,1

Links

Table of n, a(n) for n=1..49.

DefElement, Serendipity

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

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

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 *)

Prog

(Python) print([4 if n == 1 else n  * (n + 3) // 2 + 3 for n in range(1, 50)])
(PARI) a(n) = if (n==1, 4, n*(n+3)/2 + 3); \\ Michel Marcus, Jan 04 2021

Crossrefs

Sequence in context: A311552 A311553 A311554 * A194274 A098573 A092753

Adjacent sequences: A340263 A340264 A340265 * A340267 A340268 A340269

Keyword

nonn,easy

Author

Matthew Scroggs, Jan 02 2021

Status

approved