A333637 The number of cells which contain multiple squares of a Genealodron formed from 2^n - 1 equal-sized squares (when viewed from above).

0, 0, 0, 2, 7, 15, 27, 41, 57, 75, 95, 117, 141, 167, 195, 225, 257, 291, 327, 365, 405, 447, 491, 537, 585, 635, 687, 741, 797, 855, 915, 977, 1041, 1107, 1175, 1245, 1317, 1391, 1467, 1545, 1625, 1707, 1791, 1877, 1965, 2055, 2147, 2241, 2337, 2435, 2535, 2637, 2741, 2847, 2955, 3065, 3177, 3291, 3407, 3525

Offset: 1

Author

Andrew Smith, Mar 30 2020

Keywords

• nonn
• easy

Comments

See A179178 for the definition of a Genealodron. In this variation, a Genealodron is a rooted binary tree constructed from squares. One edge of each square is attached to its parent and the two adjacent edges to its child trees.

The first Genealodron consists of one square.

The second Genealodron is formed by joining another equal-sized square to the left edge and to the right edge of the first so that the second Genealodron is made up of three squares.

The third Genealodron is formed by joining squares to the upper and lower edges of both the second and third square of the second Genealodron so that the third Genealodron is made up of seven squares.

This continues, with the edges to which the new squares are attached alternating between left/right and upper/lower.

From the fourth generation onwards, some squares will overlap. a(n) is the number of cells which contain overlapping squares.

Links

• Andrew Smith, Illustration of initial terms

Crossrefs

Cf. A179178, A179316, A297103.

Formula

Conjecture: for n>=6, a(n) = n^2 - n - 15. - Vaclav Kotesovec, Apr 07 2020

Conjectures from Colin Barker, Apr 07 2020: (Start)

G.f.: x^4*(1 + x^2)*(2 + x - 2*x^2) / (1 - x)^3.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>8.

(End)