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A360927
Expansion of the g.f. x*(1 + 3*x + 4*x^2 + 4*x^3)/((1 - x)^2*(1 + x)).
0
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.
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:
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a(1) = 1 a(2) = 4 a(3) = 9
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a(4) = 16 a(5) = 21 a(6) = 28
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a(7) = 33 a(8) = 40
MATHEMATICA
LinearRecurrence[{1, 1, -1}, {0, 1, 4, 9, 16}, 57]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Stefano Spezia, Feb 25 2023
STATUS
approved