A179316 -id:A179316 - cp's OEIS frontend

A179178 The number of equal-sized equilateral triangles visible (when viewed from above) in successive Genealodrons formed from 2^n -1 same size equilateral triangles.

1, 3, 7, 14, 25, 40, 57, 77, 100, 126, 155, 187, 222, 260, 301, 345, 392, 442, 495, 551, 610, 672, 737, 805, 876, 950, 1027, 1107, 1190, 1276, 1365, 1457, 1552, 1650, 1751, 1855, 1962, 2072, 2185, 2301, 2420, 2542, 2667, 2795, 2926, 3060, 3197, 3337, 3480

Offset: 1

Elizabeth Hignell (elizabethhignell(AT)hotmail.com), Jun 30 2010

A Genealodron represents a rooted binary tree and is composed of equal sized equilateral triangles. One edge of each triangle is attached to its parent and the other two to its child trees. The first triangle, which is the root of the tree, has a designated unattached parent edge. Triangles may overlap as needed.
The first Genealodron consists of one equilateral triangle.
The second Genealodron is formed by joining another same size equilateral triangle to the left edge and to the right edge of the first so that the second Genealodron is made up of three triangles.
The third Genealodron is formed by joining same size equilateral triangles to the left and right edges of both the second and third triangle of the second Genealodron so that the third Genealodron is made up of seven triangles.
The fourth Genealodron is formed by joining same size equilateral triangles to the left and right edges of the fourth, fifth, sixth and seventh triangles of the third Genealodron so that the fourth Genealodron has fifteen triangles. The fourth Genealodron has the first overlap so although it contains 15 triangles only 14 are seen when it is viewed from above.
The fifth Genealodron is formed by adding 16 more triangles to the edges of last eight triangles added to the fourth Genealodron so the fifth Genealodron has 31 triangles, only 25 of which are seen when it is viewed from above because of the increasing number of overlaps.
The sixth Genealodron has 63 triangles only 40 of which are visible.
Gradually within the Genealodron spirals (which are hexagonal in cross-section) are building counterclockwise on the left hand edge of every triangle and clockwise on the right hand edge of every triangle. Because of the way the triangles form into hexagonal stacks although the total number of triangles in successive Genealodrons is 2^n - 1 the rate at which the number of visible triangles increases becomes arithmetic with a common difference of 3.
A Genealodron formed from an infinite number of same size equilateral triangles creates a hyperbolic plane.
Also, the crystal ball sequence for the honeycomb point lattice with a single edge removed at the origin. Without this removal the sequence would be A005448. The sixth Genealodron when viewed from above has the shape of a hexagon (see illustration of initial terms). All subsequent generations will retain this shape and so the sequence becomes A005449, the second pentagonal numbers. - Andrew Howroyd, Mar 24 2016

Colin Barker, Table of n, a(n) for n = 1..1000
Kival Ngaokrajang, Illustration of initial terms
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A179316.

CoefficientList[Series[x (1 + x^2 + x^3 + x^4 - 2 x^6 + x^7)/(1 - x)^3, {x, 0, 49}], x] (* Michael De Vlieger, Mar 26 2016 *)

Vec(x*(1+x^2+x^3+x^4-2*x^6+x^7)/(1-x)^3 + O(x^50)) \\ Colin Barker, Mar 26 2016

a(n) = A005449(n-1) for n>=6. - Andrew Howroyd, Mar 24 2016
From Colin Barker, Mar 26 2016: (Start)
a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n>5.
a(n) = (2-5*n+3*n^2)/2 for n>5.
G.f.: x*(1+x^2+x^3+x^4-2*x^6+x^7) / (1-x)^3.
(End)

a(12)-a(40) from Andrew Howroyd, Mar 24 2016

Definition improved by Andrew Howroyd, Apr 16 2020

A297103 The number of equal-sized squares in the highest stack of squares contained in successive Genealodrons formed from 2^n - 1 equal-sized squares.

Original entry on oeis.org

1, 1, 1, 2, 5, 7, 10, 20, 41, 67, 110, 220, 441, 767, 1335, 2670, 5341, 9587, 17211, 34422, 68845, 126011, 230655, 461310, 922621, 1711595, 3175311, 6350622, 12701245, 23796515, 44584536, 89169072, 178338145

Offset: 1

Andrew Smith, Dec 25 2017

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.
The fourth Genealodron is formed by joining squares to the left and right edges of the fourth, fifth, sixth and seventh squares of the third Genealodron so that the fourth Genealodron has fifteen squares. The fourth Genealodron has the first overlaps, so although it contains 15 squares only 13 are seen when it is viewed from above.
The fifth Genealodron is formed by adding 16 more squares to the upper and lower edges of the last eight squares added to the fourth Genealodron so the fifth Genealodron has 31 squares, only 21 of which are seen when it is viewed from above because of the increasing number of overlaps.
The sixth Genealodron is formed by adding 32 more squares to the left and right edge of the last 16 squares added to the fifth Genealodron. So the sixth Genealodron has 63 squares only 31 of which are visible.
This continues, and the edges on which the new squares are added keep alternating between left and right and then upper and lower.
Gradually within the Genealodron, spirals are building counterclockwise and clockwise. The sequence that the Genealodron built with squares generates is different from the one built with equilateral triangles, because when a square is added, the spiral then turns through 90 degrees rather than just 60 degrees.

Andrew Smith, Illustration of initial terms

Cf. A179178, A179316.

%I solved the problem by representing each Genealodron as a matrix
n=input('how many terms?');
%preallocation of length of output (length n)
vec=zeros(1,n);
%below I initialize the first 3 terms which are easily done with pen and paper
vec(1)=1;
vec(2)=1;
vec(3)=1;
%imat is the intermediate matrix to go from 3rd to 4th matrix.
imat=[1,0,1;0,0,0;1,0,1];
%mat is the 3rd matrix
mat=[1,0,1;1,1,1;1,0,1];
%loop
for i=4:n
    %when i is even
    if mod(i,2)==0
        imat2=[zeros(i-1,2),imat];
        imat3=[imat,zeros(i-1,2)];
        %superposing two variations of previous intermediate matrix to get next one
        imat=imat2+imat3;
        %making mat same size as imat
        mat=[zeros(i-1,1),mat,zeros(i-1,1)];
        %calculating new matrix (=old matrix+intermediate)
        mat=mat+imat;
    %similarly when i is odd
    else
       imat2=[zeros(2,i);imat];
       imat3=[imat;zeros(2,i)];
       imat=imat2+imat3;
       mat=[zeros(1,i);mat;zeros(1,i)];
       mat=mat+imat;
    end
    %working out the maximum value of new matrix and allocating it to a position in the output vector
    vec(i)=max(max(mat));
end
format long g
disp(vec)

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

Original entry on oeis.org

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

Andrew Smith, Mar 30 2020

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.

Andrew Smith, Illustration of initial terms

Cf. A179178, A179316, A297103.

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)

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A179178 The number of equal-sized equilateral triangles visible (when viewed from above) in successive Genealodrons formed from 2^n -1 same size equilateral triangles.

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A297103 The number of equal-sized squares in the highest stack of squares contained in successive Genealodrons formed from 2^n - 1 equal-sized squares.

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MATLAB

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

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Author

Keywords

Comments

Links

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Formula