A255870 -id:A255870 - cp's OEIS frontend

A256429 a(n) is the total number of pentagrams in a variant of pentagram fractal after n iterations.

1, 6, 11, 16, 41, 106, 211, 416, 941, 2106, 4411, 9316, 20341, 44106, 94111, 201716, 435741, 938606, 2014311, 4330116, 9324641, 20060606, 43122511, 92747016, 199552041, 429222606, 923076211, 1985467416, 4270895441, 9186237106, 19758020411, 42498043816, 91411232341

Offset: 0

Kival Ngaokrajang, Mar 28 2015

nonn

Inspired by A255870. But at the higher iterations, the perimeter would be a pentagram instead of a pentagon. See illustration in the links.

Kival Ngaokrajang, Illustration of initial terms, Illustration of n = 6

Cf. A255870.

Conjectures from Colin Barker, Mar 29 2015: (Start)
a(n) = 3*a(n-1)-4*a(n-2)+7*a(n-3)-5*a(n-4) for n>3.
G.f.: -(3*x^2-3*x-1) / ((x-1)*(5*x^3-2*x^2+2*x-1)).
(End)

A256569 a(n) is the total number of pentagrams in a variant of pentagram fractal after n iterations.

Original entry on oeis.org

1, 6, 21, 61, 171, 461, 1181, 2951

Offset: 1

Kival Ngaokrajang, Apr 02 2015

Inspired by A255870. But at the higher iterations, the perimeter would be an irregular polygon instead of a pentagon. The construction rule is vertex to vertex expansion. See illustration in the links.

Kival Ngaokrajang, Illustration of initial terms

Cf. A255870, A256429, A256571.

A256571 a(n) is the total number of pentagrams in a variant of pentagram fractal after n iterations.

Original entry on oeis.org

1, 6, 21, 56, 131, 281, 556, 1046, 1896, 3346, 5796, 9916, 16806, 28316, 47506, 79476, 132706

Offset: 1

Kival Ngaokrajang, Apr 02 2015

Inspired by A255870. But at the higher iterations, the perimeter is an irregular polygon instead of a pentagon. The construction rule is sides to vertices expansion. See illustration in the links.

Table of n, a(n) for n = 1..17
Kival Ngaokrajang, Illustration of initial terms

Cf. A255870, A256429, A256569.

a(9) corrected and a(10)-a(15) added by Udo Vieth, Jan 16 2023

a(16)-a(17) added by Udo Vieth, Aug 19 2024

A256582 a(n) is the total number of pentagrams in a variant of pentagram fractal after n iterations.

Original entry on oeis.org

1, 11, 46, 146, 416, 1136, 2776

Offset: 1

Kival Ngaokrajang, Apr 04 2015

Inspired by A255870. But the scale factor is tan(3*Pi/10)/(1+phi) instead of 1/phi.

See illustration in the links.

Kival Ngaokrajang, Illustration of initial terms

Cf. A255870, A256429, A256569, A256571, A256583, A256597.

A256583 a(n) is the total number of pentagrams in a variant of pentagram fractal after n iterations.

Original entry on oeis.org

1, 11, 66, 336, 1666, 8186

Offset: 1

Kival Ngaokrajang, Apr 04 2015

Inspired by A255870. But the scale factor is (1/phi)^2 instead of 1/phi.

See illustration in the links.

Kival Ngaokrajang, Illustration of initial terms

Cf. A255870, A256429, A256569, A256571, A256582, A256597.

A256597 a(n) is the total number of pentagrams in a variant of pentagram fractal after n iterations.

Original entry on oeis.org

1, 11, 76, 436, 2441

Offset: 1

Kival Ngaokrajang, Apr 04 2015

Inspired by A255870. But the scale factor is tan(Pi/10) instead of 1/phi.

See illustration in the links.

Kival Ngaokrajang, Illustration of initial terms

Cf. A255870, A256429, A256569, A256571, A256582, A256583.

A263418 a(n) is the total number of pentagrams in a variant of pentagram fractal after n iterations.

Original entry on oeis.org

1, 6, 21, 51, 106, 201, 361, 626, 1061, 1771, 2926, 4801, 7841, 12766, 20741, 33651, 54546, 88361, 143081, 231626, 374901, 606731, 981846, 1588801, 2570881, 4159926, 6731061, 10891251, 17622586, 28514121, 46137001, 74651426, 120788741, 195440491, 316229566

Offset: 0

Kival Ngaokrajang, Oct 17 2015

Inspired by A255870.

Colin Barker, Table of n, a(n) for n = 0..1000
Kival Ngaokrajang, Illustration of initial terms
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Cf. A001911, A255870, A256429, A256569, A256571, A256582, A256583, A263419.

{a=1; print1(a, ", "); for(n=1,100, b=fibonacci(n+3)-2; a=a+5*b; print1 (a, ", "))}

Vec(-(x^3+5*x^2+3*x+1)/((x-1)^2*(x^2+x-1)) + O(x^50)) \\ Colin Barker, Oct 18 2015

a(0) = 1, for n > 0, a(n) = a(n-1) + 5*(fibonacci(n+3)-2) or a(n) = a(n-1) + 5*A001911(n).
From Colin Barker, Oct 18 2015: (Start)
a(n) = 3*a(n-1)-2*a(n-2)-a(n-3)+a(n-4) for n>3.
G.f.: -(x^3+5*x^2+3*x+1) / ((x-1)^2*(x^2+x-1)).
(End)
a(n) = -14 + 2^(-1-n)*((25-11*sqrt(5))*(1-sqrt(5))^n + (1+sqrt(5))^n*(25+11*sqrt(5))) - 10*(1+n). - Colin Barker, Mar 12 2017

A263419 a(n) is the total number of pentagrams in a variant of pentagram fractal after n iterations.

Original entry on oeis.org

1, 6, 11, 26, 51, 106, 201, 396, 751, 1446, 2741

Offset: 0

Kival Ngaokrajang, Oct 17 2015

Inspired by A255870, also same as A263418 but overlaps are prohibited.

Kival Ngaokrajang, Illustration of initial terms

Cf. A255870, A256429, A256569, A256571, A256582, A256583, A263418.

A256107 Irregular triangle read by rows, T(n,k) is the number of pentagrams on the k layers at n iterations of a pentagram fractal (see comment).

Original entry on oeis.org

1, 2, 4, 1, 7, 2, 2, 12, 4, 5, 2, 1, 20, 7, 10, 4, 2, 2, 33, 12, 18, 8, 4, 4, 2, 1, 54, 20, 31, 14, 7, 8, 4, 2, 2, 88, 33, 52, 24, 12, 14, 8, 4, 4, 2, 1, 143, 54, 86, 40, 20, 24, 14, 7, 8, 4, 2, 2, 232, 88, 141, 66, 33, 40, 24, 12, 14, 8, 4, 4, 2, 1, 376, 143, 230, 108, 54, 66

Offset: 0

Kival Ngaokrajang, Mar 14 2015

Refer to A255870, the number of pentagrams on one side of the outer layer (including pentagrams on two vertices) at n iterations would be T(n,0), the next layers k >= 1 T(n,k) are the number of pentagrams toward the center. For k >= 2, the row length is A032766. The first differences of A255870 = 5*(rows sum - 1). T(n,k) = A000071 with a shift for k = 0 or k mod 3 = 1. T(n,2) = A006327 with a shift. For k >= 3, T(n,k) = 2*A000071 with a shift for k mod 3 = 0 or 2. See illustration in the links.

			Irreuglar triangle begins:
n/k  0  1  2  3  4  5  6  7  8 ...
0   1
1   2
2   4  1
3   7  2  2
4  12  4  5  2  1
5  20  7 10  4  2  2
6  33 12 18  8  4  4  2  1
7  54 20 31 14  7  8  4  2  2
8  88 33 52 24 12 14  8  4  4
...

Kival Ngaokrajang, Illustration of initial terms, T(n,k) for n = 0..10, k = 0..13

Cf. A255870, A032766, A000071, A006327.

{for(n=0, 20, if(n<2, lk=0, lk=floor(3*(n-2)/2)+1); for (k=0, lk, if(k<>0, if(k<>2, if(Mod(k,3)==1, t=fibonacci(n+1-2*(k-1)/3)-1, t=2*(fibonacci(n+2-ceil((2*k+1)/3))-1)), t=fibonacci(n+2)-3), t=fibonacci(n+3-2*k/3)-1); print1(t, ", ")))}

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A256429 a(n) is the total number of pentagrams in a variant of pentagram fractal after n iterations.

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A256569 a(n) is the total number of pentagrams in a variant of pentagram fractal after n iterations.

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A256571 a(n) is the total number of pentagrams in a variant of pentagram fractal after n iterations.

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A256582 a(n) is the total number of pentagrams in a variant of pentagram fractal after n iterations.

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A256583 a(n) is the total number of pentagrams in a variant of pentagram fractal after n iterations.

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A256597 a(n) is the total number of pentagrams in a variant of pentagram fractal after n iterations.

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A263418 a(n) is the total number of pentagrams in a variant of pentagram fractal after n iterations.

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A263419 a(n) is the total number of pentagrams in a variant of pentagram fractal after n iterations.

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A256107 Irregular triangle read by rows, T(n,k) is the number of pentagrams on the k layers at n iterations of a pentagram fractal (see comment).

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