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A321237 Start with a square of dimension 1 X 1, and repeatedly append along the squares of the previous step squares with half their side length that do not overlap with any prior square; a(n) gives the number of squares appended at n-th step. 2
1, 8, 28, 68, 148, 308, 628, 1268, 2548, 5108, 10228, 20468, 40948, 81908, 163828, 327668, 655348, 1310708, 2621428, 5242868, 10485748, 20971508, 41943028, 83886068, 167772148, 335544308, 671088628, 1342177268, 2684354548, 5368709108, 10737418228, 21474836468 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The following diagram depicts the first three steps of the construction:

                       +----+----+----+----+

                       |  3 |  3 |  3 |  3 |

                  +----+----+----+----+----+----+

                  |  3 |         |         |  3 |

             +----+----+    2    |    2    +----+----+

             |  3 |  3 |         |         |  3 |  3 |

        +----+----+----+---------+---------+----+----+----+

        |  3 |         |                   |         |  3 |

        +----+    2    |                   |    2    +----+

        |  3 |         |                   |         |  3 |

        +----+---------+         1         +---------+----+

        |  3 |         |                   |         |  3 |

        +----+    2    |                   |    2    +----+

        |  3 |         |                   |         |  3 |

        +----+----+----+---------+---------+----+----+----+

             |  3 |  3 |         |         |  3 |  3 |

             +----+----+    2    |    2    +----+----+

                  |  3 |         |         |  3 |

                  +----+----+----+----+----+----+

                       |  3 |  3 |  3 |  3 |

                       +----+----+----+----+

A square of step n+1 touches one or two squares of step n.

The limiting construction is an octagon (truncated square); its area is 7 times the area of the initial square.

See A321257 for a similar sequence.

LINKS

Colin Barker, Table of n, a(n) for n = 1..1000

Rémy Sigrist, Illustration of the construction after 7 steps

Index entries for linear recurrences with constant coefficients, signature (3,-2).

FORMULA

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

a(n) = 4 * A154117(n-1) for any n > 1.

Sum_{n > 0} a(n) / 4^(n-1) = 7.

From Colin Barker, Nov 02 2018: (Start)

G.f.: x*(1 + 2*x)*(1 + 3*x) / ((1 - x)*(1 - 2*x)).

a(n) = 5*2^n - 12 for n>1.

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

(End)

PROG

(PARI) a(n) = if (n==1, return (1), return (4*( 2^(n-1) + 3 * floor( (2^(n-2)-1) ) )))

(PARI) Vec(x*(1 + 2*x)*(1 + 3*x) / ((1 - x)*(1 - 2*x)) + O(x^40)) \\ Colin Barker, Nov 02 2018

CROSSREFS

Cf. A154117, A321257.

Sequence in context: A083013 A028553 A100182 * A328535 A119515 A304343

Adjacent sequences:  A321234 A321235 A321236 * A321238 A321239 A321240

KEYWORD

nonn,easy

AUTHOR

Rémy Sigrist, Nov 01 2018

STATUS

approved

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Last modified February 21 23:41 EST 2020. Contains 332113 sequences. (Running on oeis4.)