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A321257
Start with an equilateral triangle, and repeatedly append along the triangles of the previous step equilateral triangles with half their side length that do not overlap with any prior triangle; a(n) gives the number of triangles appended at n-th step.
3
1, 6, 21, 60, 147, 330, 705, 1464, 2991, 6054, 12189, 24468, 49035, 98178, 196473, 393072, 786279, 1572702, 3145557, 6291276, 12582723, 25165626, 50331441, 100663080, 201326367, 402652950, 805306125, 1610612484, 3221225211, 6442450674, 12884901609, 25769803488
OFFSET
1,2
COMMENTS
The following diagram depicts the first three steps of the construction:
- - - -
/ \ / \ / \ / \
/ 3 \ / 3 \ / 3 \ / 3 \
-------------------------
/ \ / \ / \
/ 3 \ 2 / \ 2 / 3 \
------- / \ -------
/ \ / \ / \ / \ / \
/ 3 \ / 3 \ / \ / 3 \ / 3 \
------------- -------------
/ \ / \ / \
/ 3 \ 2 / 1 \ 2 / 3 \
------- / \ -------
/ \ / \ / \
/ 3 \ / \ / 3 \
-------------------------------------
/ \ / \ / \
/ 3 \ 2 / 3 \ 2 / 3 \
------- ------- -------
/ \ / \ / \ / \
/ 3 \ / 3 \ / 3 \ / 3 \
-------------------------
A triangle of step n+1 touches one or two triangles of step n.
The construction presents holes from the 3rd step onwards; these will be gradually filled in the subsequent steps.
The limiting construction is a hexagon; its area is 6 times the area of the initial triangle.
See A321237 for a similar sequence.
FORMULA
a(n) = 3*(2^(n-1) + 3*(2^(n-1)-n)) for any n > 1.
Sum_{n > 0} a(n) / 4^(n-1) = 6.
G.f.: x*(1 + 2*x + 2*x^2 + 4*x^3)/((1-2*x)*(1-x)^2). - Vincenzo Librandi, Nov 02 2018
a(n) - 4*a(n-1) + 5*a(n-2) - 2*a(n-3) = 0, with n>1. - Vincenzo Librandi, Nov 02 2018
MAPLE
1, seq(3*(2^(n-1)+3*(2^(n-1)-n)), n=2..35); # Muniru A Asiru, Nov 02 2018
MATHEMATICA
CoefficientList[Series[(1 + 2 x + 2 x^2 + 4 x^3) / ((1 - 2 x) (1 - x)^2), {x, 0, 40}], x] (* Vincenzo Librandi, Nov 02 2018 *)
PROG
(PARI) a(n) = if (n==1, 1, 3*(2^(n-1) + 3*(2^(n-1)-n)))
(Magma) [1] cat [3*(2^(n-1) + 3*(2^(n-1)-n)): n in [2..35]]; // Vincenzo Librandi, Nov 02 2018
(GAP) Concatenation([1], List([2..35], n->3*(2^(n-1)+3*(2^(n-1)-n)))); # Muniru A Asiru, Nov 02 2018
CROSSREFS
Cf. A321237.
Sequence in context: A341221 A143115 A258142 * A305120 A066524 A113070
KEYWORD
nonn
AUTHOR
Rémy Sigrist, Nov 01 2018
STATUS
approved