A253688 -id:A253688 - cp's OEIS frontend

A255870 a(n) is the total number of pentagrams in a pentagram fractal after n iterations.

1, 6, 26, 76, 191, 411, 816, 1521, 2726, 4741, 8081, 13566, 22536, 37146, 60896, 99436, 161921, 263151, 427086, 692481, 1122056, 1817281, 2942351, 4762926, 7708866, 12475686, 20188766, 32668996, 52862651, 85536891, 138405156, 223948041, 362359586, 586314421

Offset: 0

Kival Ngaokrajang, Mar 08 2015

Inspired by the "calice" (see detail in the CNRS link).

The pentagrams appearing in the calice are scaled down by a factor of 1/phi = 0.61803398... from the pentagrams whose vertex-to-vertex length = d. See illustration of the nested pentagrams in the links.

Colin Barker, Table of n, a(n) for n = 0..1000
Shalom Eliahou, Pavages, symétrie d'ordre 5 et Fibonacci: un amateur passionné, Images des Mathématiques, CNRS, 2015 (in French).
Frédéric Mansuy, Supersymétrie d'ordre cinq périodique (in French).
Kival Ngaokrajang, Illustration of initial terms, Excel calculation sheet
Index entries for linear recurrences with constant coefficients, signature (3,-1,-4,3,1,-1).

Cf. A000071, A006327, A249852, A253688.

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I:=[1,6,26,76,191,411,816]; [n le 7 select I[n] else 3*Self(n-1)-Self(n-2)-4*Self(n-3)+3*Self(n-4)+Self(n-5)-Self(n-6): n in [1..40]]; // Vincenzo Librandi, Mar 18 2015

CoefficientList[Series[(5 x^6 + x^5 - 10 x^4 - 8 x^3 - 9 x^2 - 3 x - 1)/((1-x)^3 (x+1) (x^2+x-1)), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 18 2015 *)

Vec(-(5*x^6+x^5-10*x^4-8*x^3-9*x^2-3*x-1)/((x-1)^3*(x+1)*(x^2+x-1)) + O(x^100)) \\ Colin Barker, Mar 12 2015

a(n) = 3*a(n-1)-a(n-2)-4*a(n-3)+3*a(n-4)+a(n-5)-a(n-6) for n>6. - Colin Barker, Mar 12 2015
G.f.: -(5*x^6+x^5-10*x^4-8*x^3-9*x^2-3*x-1) / ((x-1)^3*(x+1)*(x^2+x-1)). - Colin Barker, Mar 12 2015
a(n) = (-307 + 5*(-1)^n - 3*2^(2-n)*s*((11-5*s)*(1-s)^n - (1+s)^n*(11+5*s)) - 150*(1+n) - 50*(1+n)*(2+n)) / 8 for n>0 where s=sqrt(5). - Colin Barker, Mar 12 2017
E.g.f.: 3*exp(x/2)*(25*cosh(sqrt(5)*x/2) + 11*sqrt(5)*sinh(sqrt(5)*x/2)) - (12 + 5*x)*(23 + 5*x)*cosh(x)/4 - (281 + 25*x*(7 + x))*sinh(x)/4 - 5. - Stefano Spezia, Dec 07 2022

A253687 The total number of pentagons in a variant of pentagon expansion (side-to-side, two consecutive sides and one isolated side) after n iterations.

Original entry on oeis.org

1, 4, 10, 21, 39, 64, 94, 129, 167, 212, 262, 317, 375, 440, 510, 585, 663, 748, 838, 933, 1031, 1136, 1246, 1361, 1479, 1604, 1734, 1869, 2007, 2152, 2302, 2457, 2615, 2780, 2950, 3125, 3303, 3488, 3678, 3873, 4071, 4276, 4486, 4701, 4919, 5144, 5374, 5609, 5847, 6092

Offset: 1

Kival Ngaokrajang, Jan 09 2015

nonn

Two decagons appearing at n >= 6. See illustration.

Kival Ngaokrajang, Illustration of initial terms

Cf. A253688 (vertex-to-vertex).

{
a=1;d1=0;p=a;print1(a,", ");\\5s3b
for(n=2,100,
   if(n<3,d1=2,
     if(n<4,d1=3,
       if(n<5,d1=5,
         if(n<6,d1=7,
           if(Mod(n,4)==0,d1=5,
             if(Mod(n,4)==1,d1=3,
               if(Mod(n,4)==2,d1=7,d1=5
               )
             )
           )
         )
       )
     )
   );
   a=a+d1;p=p+a;
   print1(p,", ")
)
}

Conjecture: a(n) = 2*a(n-1)-a(n-2)+a(n-4)-2*a(n-5)+a(n-6) for n>7. - Colin Barker, Jan 09 2015

Empirical g.f.: x*(4*x^8-2*x^6-5*x^5-6*x^4-5*x^3-3*x^2-2*x-1) / ((x-1)^3*(x+1)*(x^2+1)). - Colin Barker, Jan 09 2015

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

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)

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A255870 a(n) is the total number of pentagrams in a pentagram fractal after n iterations.

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A253687 The total number of pentagons in a variant of pentagon expansion (side-to-side, two consecutive sides and one isolated side) 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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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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