A179178 -id:A179178 - cp's OEIS frontend

A179316 The number of equal-sized equilateral triangles in the highest stack of triangles contained in successive Genealodrons formed from 2^n - 1 same size equilateral triangles.

1, 1, 1, 2, 2, 3, 5, 7, 12, 22, 37, 66, 118, 228, 413, 762, 1441, 2718, 5147, 9804, 18594, 35420, 67729, 129976, 249176, 479112, 921625, 1777474, 3429822, 6632522, 12826031, 24850926, 48165224, 93507918, 181566683, 353075420, 686711066, 1337373564, 2604998105, 5080131368

Offset: 1

Elizabeth Hignell (elizabethhignell(AT)hotmail.com), Jul 10 2010

nonn

As explained in the comments to A179178 the n-th Genealodron can be formed by adding 2^(n-1) same size equilateral triangles to the left and right edges of the last 2^(n-2) triangles added to the (n-1)th Genealodron. It is easier however to imagine the n-th Genealodron formed by taking a new same size equilateral triangle and joining the bottom edge of the first triangle of a (n-1)th Genealodron to its left edge and similarly the bottom edge of the first triangle of another (n-1)th Genealodron to its right edge.
The shape formed is the same. Expressed in genealogical terms, instead of adding a round of equivalent many-times great-grandparent triangles to the structure, a child triangle has been put in forcing each triangle to move up a generation on both the father and mother's side.
The overlaps become increasingly complex as the equilateral triangles stack into spirals within the structure and as n gets larger the child triangle method becomes the only feasible way of generating successive Genealodrons.
For n>=18, with reference to the illustration of the initial terms of A179178, the location of the highest stack of triangles will stabilize at the cell labeled 11 for even n and at the cells labeled 5 and 6 for odd n. The sequence can be computed as the number of walks in the honeycomb lattice of length less than or equal to n that don't double back on themselves and that start at the origin and finish at the location with the greatest number of such walks. Also when making the first step only two of the three cells adjacent to the origin must be considered. - Andrew Howroyd, Mar 26 2016

			From _Andrew Howroyd_, Mar 26 2016: (Start)
See illustration of initial terms of A179178 for cell numbering.
a(4) = 2 because there are 2 permissible walks of length <= 4 ending on the cell labeled 11: {1,3,6,11} and {1,2,5,11}.
a(6) = 3 because there are 3 permissible walks of length <= 6 ending on the cell labeled 10: {1,2,5,10}, {1,3,6,11,5,10} and {1,2,4,9,18,10}.
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..250
Kival Ngaokrajang, Illustration of initial terms, A179178.

A179178 is a related Genealodron sequence.

a(13)-a(40) from Andrew Howroyd, Mar 25 2016

A253547 Total number of star-shaped dodecagons appearing in a variant of hexagon expansion after n iterations.

Original entry on oeis.org

0, 0, 0, 1, 3, 9, 16, 23, 33, 43, 56, 69, 85, 101, 120, 139, 161, 183, 208, 233, 261, 289, 320, 351, 385, 419, 456, 493, 533, 573, 616, 659, 705, 751, 800, 849, 901, 953, 1008, 1063, 1121, 1179, 1240, 1301, 1365, 1429, 1496, 1563, 1633, 1703, 1776, 1849, 1925, 2001, 2080

Offset: 1

Kival Ngaokrajang, Jan 03 2015

Total number of hexagons after n iterations is A179178. See illustration.

Kival Ngaokrajang, Illustration of initial terms
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A179178.

LinearRecurrence[{2,0,-2,1},{0,0,0,1,3,9,16,23,33},60] (* Harvey P. Dale, Oct 30 2015 *)

{
a=1;d1=0;print1("0, 0, 0, 1",", ");
for(n=4,100,
   if(n<5,d1=2,
     if(n<6,d1=6,
       if(n<7,d1=7,
         if(Mod(n,2)==0,d1=d1+3
         )
       )
     )
   );
   a=a+d1;
   print1(a,", ")
)
}

Conjectures from Colin Barker, Jan 03 2015: (Start)
a(n) = (27 - 3*(-1)^n - 28*n + 6*n^2)/8 for n>5.
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n>9.
G.f.: -x^4*(2*x^5 - 4*x^4 + 3*x^2 + x + 1) / ((x-1)^3*(x+1)). (End)

A253895 Total number of octagons in two variants of an octagon expansion after n iterations: either "side-to-side" or "vertex-to-vertex", respectively.

Original entry on oeis.org

1, 3, 7, 14, 25, 41, 63, 90, 120, 154, 192, 233, 278, 328, 382, 439, 500, 566, 636, 709, 786, 868, 954, 1043, 1136, 1234, 1336, 1441, 1550, 1664, 1782, 1903, 2028, 2158, 2292, 2429, 2570, 2716, 2866, 3019, 3176, 3338, 3504, 3673, 3846, 4024, 4206, 4391, 4580, 4774, 4972

Offset: 1

Kival Ngaokrajang, Jan 17 2015

nonn

Inspired by A061777 and A179178 which are "vertex-to-vertex" and "side-to-side" versions of equilateral triangle expansion respectively.
In these octagon expansions there is allowed an expansion obeying "two sides separated by one side" or one obeying "two vertices separated by one vertex" for "side-to-side" or "vertex-to-vertex" versions respectively.
Two star shaped hexadecagons (16-gons) and a 4-star appear for n = 8 in the "side-to-side" version, and in the "vertex-to-vertex" version there appear two irregular star shaped icositetragons (24-gons). There are also rare type of polygons appearing for n > 8. See illustrations.

Kival Ngaokrajang, Illustration of initial terms, Rare type polygons

Cf. A253896, A061777 (Triangle expansion, vertex-to-vertex, 3 vertices), A179178 (Triangle expansion, side-to-side, 2 sides), A253687 (Pentagon expansion, side-to-side, 2 consecutive sides and 1 isolated side), A253688 (Pentagon expansion, vertex-to-vertex, 2 consecutive vertices and 1 isolated vertex), A253547 (Hexagon expansion, vertex-to-vertex, 2 vertices separated by 1 vertex).

{
a=1;d1=0;p=a;print1(a,", ");\\8s2a, total oct.
for(n=2,100,
   if(n<=7,d1=n-1,
     if(n<9,d1=5,
       if(n<10,d1=3,
         if(n<11,d1=4,
           if(Mod(n,4)==0,d1=3,
             if(Mod(n,4)==1,d1=4,
               if(Mod(n,4)==2,d1=5,d1=4
               )
             )
           )
         )
       )
     )
   );
   a=a+d1;p=p+a;
   print1(p,", ")
)
}

Conjectures from Colin Barker, Jan 17 2015: (Start)
a(n) = (-4-i*(-i)^n+i*i^n-18*n+8*n^2)/4 for n>8, where i=sqrt(-1).
G.f.: -x*(x^12-2*x^10-x^8+2*x^6+2*x^5+2*x^4+x^3+2*x^2+1) / ((x-1)^3*(x^2+1)).
(End)

A253896 Total number of either concave decagons or concave hexadecagons in two variants of an octagon expansion after n iterations: either "side-to-side" or "vertex-to-vertex", respectively.

Original entry on oeis.org

0, 0, 0, 1, 3, 7, 13, 22, 34, 48, 62, 81, 99, 121, 143, 170, 196, 226, 256, 291, 325, 363, 401, 444, 486, 532, 578, 629, 679, 733, 787, 846, 904, 966, 1028, 1095, 1161, 1231, 1301, 1376, 1450, 1528, 1606, 1689, 1771, 1857, 1943, 2034, 2124, 2218, 2312, 2411, 2509, 2611

Offset: 1

Kival Ngaokrajang, Jan 17 2015

nonn

Inspired by A061777 and A179178 which are "vertex-to-vertex" and "side-to-side" versions of equilateral triangle expansion, respectively.
In these octagon expansions, there is allowed only an expansion obeying "two sides separated by one side" or one by obeying "two vertices separated by one vertex" for the "side-to-side" or "vertex-to-vertex" versions, respectively.
Two star-shaped hexadecagons (16-gons) and a 4-star appear when n = 8 for the "side-to-side" version, and in the "vertex-to-vertex" version there appears an irregular star-shaped icositetragons (24-gons). Rare type of polygons also appear for  n > 8. See illustrations.

Kival Ngaokrajang, Illustration of initial terms, Rare type polygons

Cf. A253895, A061777 (Triangle expansion, vertex-to-vertex, 3 vertices), A179178 (Triangle expansion, side-to-side, 2 sides), A253687 (Pentagon expansion, side-to-side, 2 consecutive sides and 1 isolated side), A253688 (Pentagon expansion, vertex-to-vertex, 2 consecutive vertices and 1 isolated vertex), A253547 (Hexagon expansion, vertex-to-vertex, 2 vertices separated by 1 vertex).

{
a=0;d1=0;p=1;print1("0, 0, 0, ",p,", ");\\8s2a1
for(n=2,100,
   if(n<5,d1=2,
     if(n<7,d1=3,
       if(n<8,d1=2,
         if(Mod(n,4)==0,d1=0,
           if(Mod(n,4)==1,d1=5,
             if(Mod(n,4)==2,d1=-1,d1=4
             )
           )
         )
       )
     )
   );
   a=a+d1;p=p+a;
   print1(p,", ")
)
}

Empirical g.f.: -x^4*(2*x^10 -4*x^9 +2*x^8 -2*x^7 +2*x^5 +2*x^4 +2*x^3 +2*x^2 +x +1) / ((x -1)^3*(x +1)*(x^2 +1)). - Colin Barker, Jan 17 2015

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)

A254835 Total number of nonagons in a variant of a nonagon expansion ("side-to-side", two consecutive sides) after n iterations.

Original entry on oeis.org

2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, 133, 136, 144, 153, 161, 170, 180, 187, 197, 206, 216, 225, 233, 242, 248, 259, 269, 278, 286, 295, 305, 314, 322, 331, 341, 350, 358, 367, 377, 386, 394, 403, 413, 422, 430, 439, 449, 458, 466, 475, 485, 494, 502

Offset: 1

Kival Ngaokrajang, Feb 08 2015

nonn

Two irregular star-shaped 18-gons appear for n = 17.

There are also rare types of polygons appearing for n >= 16. See illustrations.

Kival Ngaokrajang, Illustration of initial terms, Rare type polygons

Cf. A061777 (Triangle expansion, vertex-to-vertex, 3 vertices), A179178 (Triangle expansion, side-to-side, 2 sides), A253687 (Pentagon expansion, side-to-side, 2 consecutive sides and 1 isolated side), A253688 (Pentagon expansion, vertex-to-vertex, 2 consecutive vertices and 1 isolated vertex), A253547 (Hexagon expansion, vertex-to-vertex, 2 vertices separated by 1 vertex), A253895 and A253896 (Octagon expansion).

{a=259;print1("2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, 133, 136, 144, 153, 161, 170, 180, 187, 197, 206, 216, 225, 233, 242, 248, ",a,", "); for(n=32,100,if(Mod(n,4)==0,d=10,if(Mod(n,4)==1,d=9,if(Mod(n,4)==2, d=8, d=9)));a=a+d;print1(a,", "))}

Conjectures from Colin Barker, Feb 08 2015: (Start)
a(n) = 2*a(n-1)-2*a(n-2)+2*a(n-3)-a(n-4) for n>21.
G.f.: -x*(2*x^33 -4*x^32 +4*x^31 -6*x^30 +4*x^29 +2*x^26 -4*x^25 +4*x^24 -4*x^23 +2*x^22 -2*x^20 -4*x^19 +8*x^18 -2*x^17 +8*x^16 +2*x^15 -2*x^14 -2*x^13 -2*x^12 -2*x^11 -2*x^10 -2*x^9 -2*x^8 -2*x^7 -2*x^6 -2*x^5 -2*x^4 -x^3 -3*x^2 -2) / ((x -1)^2*(x^2 +1)).
(End)

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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A179316 The number of equal-sized equilateral triangles in the highest stack of triangles contained in successive Genealodrons formed from 2^n - 1 same size equilateral triangles.

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A253547 Total number of star-shaped dodecagons appearing in a variant of hexagon expansion after n iterations.

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A253895 Total number of octagons in two variants of an octagon expansion after n iterations: either "side-to-side" or "vertex-to-vertex", respectively.

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A253896 Total number of either concave decagons or concave hexadecagons in two variants of an octagon expansion after n iterations: either "side-to-side" or "vertex-to-vertex", respectively.

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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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A254835 Total number of nonagons in a variant of a nonagon expansion ("side-to-side", two consecutive sides) after n iterations.

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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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