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.

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

Offset: 1

Author

Rémy Sigrist, Nov 01 2018

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A154117, A321257.

Programs

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

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)