A360927 Expansion of the g.f. x*(1 + 3*x + 4*x^2 + 4*x^3)/((1 - x)^2*(1 + x)).

0, 1, 4, 9, 16, 21, 28, 33, 40, 45, 52, 57, 64, 69, 76, 81, 88, 93, 100, 105, 112, 117, 124, 129, 136, 141, 148, 153, 160, 165, 172, 177, 184, 189, 196, 201, 208, 213, 220, 225, 232, 237, 244, 249, 256, 261, 268, 273, 280, 285, 292, 297, 304, 309, 316, 321, 328

Offset

0,3

Comments

The sequence gives the number of "ON" cells in the cellular automaton on a quadrant of a square grid after the n-th stage, where the "ON" cells lie only on the perimeter and the two diagonals of the square.

Links

Table of n, a(n) for n=0..56.

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

Formula

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

a(0) = 0, a(1) = 1, a(n) = 6*n - 8 for n even, and a(n) = 6*n - 9 for n odd.

E.g.f.: 4*(x + 2) + (6*x - 8)*cosh(x) + (6*x - 9)*sinh(x).

a(2*n) = A017569(n-1) = 4*A016777(n-1).

a(2*n+1) = A017629(n-1).

Example

Illustrations for n = 1..8:
      o          o o          o o o
                 o o          o o o
                              o o o
  a(1) = 1    a(2) = 4      a(3) = 9
   o o o o    o o o o o    o o o o o o
   o o o o    o o   o o    o o     o o
   o o o o    o   o   o    o   o o   o
   o o o o    o o   o o    o   o o   o
              o o o o o    o o     o o
                           o o o o o o
  a(4) = 16   a(5) = 21     a(6) = 28
   o o o o o o o       o o o o o o o o
   o o       o o       o o         o o
   o   o   o   o       o   o     o   o
   o     o     o       o     o o     o
   o   o   o   o       o     o o     o
   o o       o o       o   o     o   o
   o o o o o o o       o o         o o
                       o o o o o o o o
     a(7) = 33            a(8) = 40

Mathematica

LinearRecurrence[{1, 1, -1}, {0, 1, 4, 9, 16}, 57]

Crossrefs

Cf. A194274, A016777, A017569, A017629.

Sequence in context: A313341 A313342 A140868 * A313343 A261849 A246336

Adjacent sequences: A360924 A360925 A360926 * A360928 A360929 A360930

Keyword

nonn,easy

Author

Stefano Spezia, Feb 25 2023

Status

approved