A165663 -id:A165663 - cp's OEIS frontend

A240735 a(n) = floor(6^n/(3+sqrt(3))^n).

1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 13, 17, 21, 27, 35, 44, 56, 71, 90, 115, 146, 185, 235, 298, 378, 479, 607, 770, 977, 1238, 1570, 1991, 2525, 3202, 4060, 5148, 6527, 8276, 10494, 13306, 16872, 21393, 27125, 34393, 43609, 55294, 70111, 88897, 112717, 142919

Offset: 0

Kival Ngaokrajang, Apr 11 2014

a(n) is the perimeter (rounded down) of a dodecaflake after n iterations, let a(0) = 1. The total number of sides is 12*A000400(n). The total number of holes is A240846. 3 + sqrt(3) = A165663.

Kival Ngaokrajang, Illustration of dodecaflake for n = 0..3
Wikipedia, n-flake

Cf. A000400, A240846, A165663, A240523 (pentaflake), A240671 (heptaflake), A240572 (octaflake), A240733 (nonaflake), A240734 (decaflake), A240735 (dodecaflake).

A240735:=n->floor(6^n/(3+sqrt(3))^n); seq(A240735(n), n=0..50); # Wesley Ivan Hurt, Apr 12 2014

Table[Floor[6^n/(3 + Sqrt[3])^n], {n, 0, 50}] (* Wesley Ivan Hurt, Apr 12 2014 *)

{a(n)=floor(6^n/(3+sqrt(3))^n)}
       for (n=0, 100, print1(a(n), ", "))

A094433 a(n) is the left term in M^n * [1 0 0], M = the 3 X 3 matrix [1 -1 0 / -1 3 -2 / 0 -2 2].

Original entry on oeis.org

1, 1, 2, 6, 24, 108, 504, 2376, 11232, 53136, 251424, 1189728, 5629824, 26640576, 126064512, 596543616, 2822874624, 13357986048, 63210668544, 299116094976, 1415432558592, 6697898781696, 31694797338624, 149981391341568, 709719564017664, 3358429036056576

Offset: 0

Gary W. Adamson, May 02 2004

Right term of M^n * [1 0 0] = A094434(n).
a(n)/a(n-1) tends to 3 + sqrt(3) = 4.732050807... (A165663).
A094434(n)/a(n) tends to 1 + sqrt(3) = 2.732050807... (A090388).
M is a "stiffness matrix" with k1 = 1, k2 = 2; in K = [k1 -k1 0 / -k1 (k1 + k2) -k2 / 0 -k2 k2], where K relates to Hooke's Law governing the force on nodes of springs resulting from stretching or compressing the springs (see A094431).
The eigenvalues of M are 3+sqrt(3), 3-sqrt(3) and 0. - Tamas Kalmar-Nagy (integers(AT)kalmarnagy.com), Mar 23 2008
a(n) is the number of permutations of length n+1 avoiding the partially ordered pattern (POP) {1>2, 1>3, 1>4, 5>2, 5>3, 5>4} of length 5. That is, the number of length n+1 permutations having no subsequences of length 5 in which the elements in positions 1 and 5 are larger than the elements in positions 2, 3 and 4. - Sergey Kitaev, Dec 11 2020

			a(4) = 24 since M^4 * [1 0 0] = [24 -84 60].
G.f. = 1 + x + 2*x^2 + 6*x^3 + 24*x^4 + 108*x^5 + 504*x^6 + 2376*x^7 + ...

Carl D. Meyer, "Matrix Analysis and Applied Linear Algebra", SIAM, 2000, p. 86-87.

Michael De Vlieger, Table of n, a(n) for n = 0..1483
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
Sergey Kitaev and Artem Pyatkin, On permutations avoiding partially ordered patterns defined by bipartite graphs, arXiv:2204.08936 [math.CO], 2022.
Takao Komatsu, Asymmetric Circular Graph with Hosoya Index and Negative Continued Fractions, arXiv:2105.08277 [math.CO], 2021.
Index entries for linear recurrences with constant coefficients, signature (6,-6).

Cf. A094431, A094432, A094434, A165663, A090388.

a:= n-> (<<1|-1|0>, <-1|3|-2>, <0|-2|2>>^n)[1$2]:
seq(a(n), n=0..28);  # Alois P. Heinz, Dec 11 2020

Table[(MatrixPower[{{1, -1, 0}, {-1, 3, -2}, {0, -2, 2}}, n].{1, 0, 0})[[1]], {n, 24}] (* Robert G. Wilson v *)
Table[(3 + Sqrt[3])^n + (3 - Sqrt[3])^n, {n, 0, 20}] // Simplify (* Tamas Kalmar-Nagy (integers(AT)kalmarnagy.com), Mar 23 2008 *)
Rest@ CoefficientList[Series[x (1 - 4 x)/(1 - 6 x + 6 x^2), {x, 0, 23}], x] (* Michael De Vlieger, May 01 2019 *)

[lucas_number2(n,6,6)for n in range(-1,23)] # Zerinvary Lajos, Jul 08 2008

a(n) = (3+sqrt(3))^(n-2) + (3-sqrt(3))^(n-2). - Tamas Kalmar-Nagy (integers(AT)kalmarnagy.com), Mar 23 2008 [Corrected by R. J. Mathar, Mar 28 2010, Jun 02 2010]

G.f.: 1 + x*(1-4*x)/(1-6*x+6*x^2). - R. J. Mathar, Mar 28 2010

More terms from Robert G. Wilson v, May 08 2004

a(0)=1 prepended by Alois P. Heinz, Dec 11 2020

A383852 Decimal expansion of the volume of an elongated triangular pyramid with unit edge.

Original entry on oeis.org

5, 5, 0, 8, 6, 3, 8, 3, 2, 0, 8, 9, 9, 7, 7, 2, 4, 4, 1, 1, 5, 3, 3, 5, 6, 4, 5, 7, 2, 7, 2, 7, 6, 2, 6, 4, 9, 4, 9, 8, 4, 0, 6, 3, 6, 4, 0, 0, 6, 7, 4, 1, 6, 3, 1, 1, 2, 0, 0, 8, 3, 8, 9, 6, 9, 5, 5, 4, 4, 2, 9, 4, 0, 9, 9, 0, 4, 2, 2, 6, 2, 5, 0, 7, 8, 1, 8, 8, 4, 1

Offset: 0

Paolo Xausa, May 19 2025

The elongated triangular pyramid is Johnson solid J_7.

			0.55086383208997724411533564572727626494984063640067...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Wikipedia, Elongated triangular pyramid.

Cf. A165663 (surface area).

Cf. A002193, A010482.

First[RealDigits[(Sqrt[2] + Sqrt[27])/12, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J7", "Volume"], 10, 100]]

Equals (sqrt(2) + 3*sqrt(3))/12 = (A002193 + A010482)/12.

Minimal polynomial: 20736*x^4 - 8352*x^2 + 625. - Stefano Spezia, May 19 2025

A384139 Decimal expansion of the volume of an elongated triangular bipyramid with unit edges.

Original entry on oeis.org

6, 6, 8, 7, 1, 4, 9, 6, 2, 2, 8, 7, 7, 3, 5, 1, 6, 4, 8, 4, 8, 8, 0, 9, 7, 0, 6, 0, 7, 8, 0, 8, 4, 4, 3, 8, 1, 6, 3, 9, 7, 9, 9, 5, 9, 3, 4, 8, 7, 5, 3, 1, 6, 9, 2, 1, 0, 0, 6, 5, 0, 3, 4, 5, 2, 8, 1, 0, 5, 3, 3, 3, 9, 7, 0, 8, 8, 4, 5, 1, 5, 7, 4, 5, 3, 5, 1, 1, 3, 5

Offset: 0

Paolo Xausa, May 20 2025

The elongated triangular bipyramid is Johnson solid J_14.

Also the volume of an augmented triangular prism (Johnson solid J_49) with unit edges. - Paolo Xausa, Aug 04 2025

			0.66871496228773516484880970607808443816397995934875...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Wikipedia, Augmented triangular prism.
Wikipedia, Elongated triangular bipyramid.

Cf. A165663 (surface area - 2).

Cf. A010482, A010482, A383852, A384138, A384140.

First[RealDigits[(Sqrt[8] + Sqrt[27])/12, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J14", "Volume"], 10, 100]]

Equals (2*sqrt(2) + 3*sqrt(3))/12 = (A010466 + A010482)/12.

Equals the largest root of 20736*x^4 - 10080*x^2 + 361.

A156309 Decimal expansion of the absolute value of the larger solution of (n^2+n)/2 = -1/12. (Real root q of 6n^2 + 6n + 1, the other root being p = -1-q.)

Original entry on oeis.org

2, 1, 1, 3, 2, 4, 8, 6, 5, 4, 0, 5, 1, 8, 7, 1, 1, 7, 7, 4, 5, 4, 2, 5, 6, 0, 9, 7, 4, 9, 0, 2, 1, 2, 7, 2, 1, 7, 6, 1, 9, 9, 1, 2, 4, 3, 6, 4, 9, 3, 6, 5, 6, 1, 9, 9, 0, 6, 9, 8, 8, 3, 6, 7, 5, 8, 0, 1, 1, 1, 6, 3, 8, 4, 8, 5, 3, 3, 3, 2, 7, 1, 5, 3, 1, 4, 2, 3, 0, 2, 2, 0, 7, 1, 2, 5, 2, 3, 7, 3, 8, 7, 3, 9

Offset: 0

Daniele P. Morelli, Feb 07 2009

The formula returning the n-th triangular number (A000217) is (n^2+n)/2. On the other hand, Ramanujan's identity claims that the value of the infinite sum 1+2+3+.... is -1/12. This irrational number is the solution of the equation (n^2+n)/2 = -1/12, that is, the "limit" triangular number.
Equals the Knuth's random generators constant, that is, the ratio c/m in congruence random number generators of the type X_(n+1) = (aX_n +c) mod (m) which minimizes the correlation between successive values. - Stanislav Sykora, Nov 13 2013
It is also the fraction of the full solid angle cut out by a cone having the magic angle (A195696) as its polar angle. - Stanislav Sykora, Nov 13 2013

			The two roots of 6n^2 + 6n + 1 = 0 are -0.21132... and -0.78867513... (Cf. A020769.)

B. Candelpergher, Ramanujan summation of divergent series. Lectures notes in mathematics 2185, Springer 2017.
D. E. Knuth, The Art of Computer Programming, Vol. 2, Addison-Wesley, 1969, Chapter 3.3.3.

P. J. Cameron and V. Yildiz, Counting false entries in truth tables of bracketed formulas connected by implication, arXiv:1106.4443 [math.CO], 2011.
Michael Penn, What is the radius of 🔴 ?, YouTube video, 2021.

Cf. A000217, A020760, A020769, A165663, A195696, A334843.

First[RealDigits[(3 - Sqrt[3])/6, 10, 100]] (* Paolo Xausa, Jun 25 2024 *)

abs(solve(n=-1/2, 0, 6*n^2+6*n+1)) \\ Michel Marcus, Oct 05 2013

(1 - 1/sqrt(3))/2 = (1 - A020760)/2 = 1/2 - A020769. - R. J. Mathar, Feb 10 2009
Equals - HurwitzZeta(-1, (9 - sqrt(3))/6). - Peter Luschny, Jul 05 2020
Equals (3 - sqrt(3))/6. - Michel Marcus, Jun 10 2021
Equals 1/A165663 = A334843/3. - Hugo Pfoertner, Jun 25 2024

Flipped sign of definition, corrected offset, simplified formula R. J. Mathar, Feb 10 2009

A321120 Decimal expansion of (3 + sqrt(3))/12.

Original entry on oeis.org

3, 9, 4, 3, 3, 7, 5, 6, 7, 2, 9, 7, 4, 0, 6, 4, 4, 1, 1, 2, 7, 2, 8, 7, 1, 9, 5, 1, 2, 5, 4, 8, 9, 3, 6, 3, 9, 1, 1, 9, 0, 0, 4, 3, 7, 8, 1, 7, 5, 3, 1, 7, 1, 9, 0, 0, 4, 6, 5, 0, 5, 8, 1, 6, 2, 0, 9, 9, 4, 4, 1, 8, 0, 7, 5, 7, 3, 3, 3, 3, 6, 4, 2, 3, 4, 2, 8

Offset: 0

Franck Maminirina Ramaharo, Nov 09 2018

The smallest weight in Holladay-Sard's quadrature formula for semi-infinite integrals.

			0.3943375672974064411272871951...

Harold J. Ahlberg, Edwin N. Nilson and Joseph L. Walsh, The Theory of Splines and Their Applications, Academic Press, 1967.

John C. Holladay, A smoothest curve approximation, Math. Comp. Vol. 11 (1957), 233-243.
Leroy F. Meyers and Arthur Sard, Best approximate integration formulas, J. Math. Phys. Vol. 29 (1950), 118-123.
Arthur Sard, Best approximate integration formulas; best approximation formulas, American Journal of Mathematics Vol. 71 (1949), 80-91.
Frans Schurer, On natural cubic splines, with an application to numerical integration formulae, EUT report. WSK, Dept. of Mathematics and Computing Science Vol. 70-WSK-04 (1970), 1-32.

Cf. A020805, A165663.

Cf. A321118, A321119.

Digits := 1000; evalf((3 + sqrt(3))/12);

RealDigits[(3 + Sqrt[3])/12, 10, 100][[1]]

PARI
```
(3 + sqrt(3))/12
```

Equals lim_{n->infinity} A321118(0,n)/A321119(n).
Irrational number represented by the periodic continued fraction [0, 2, 1, 1; [6, 2]].
Largest real root of 1 - 12*x + 24*x^2.

A297701 Decimal expansion of 1 + sqrt(2) + sqrt(3).

Original entry on oeis.org

4, 1, 4, 6, 2, 6, 4, 3, 6, 9, 9, 4, 1, 9, 7, 2, 3, 4, 2, 3, 2, 9, 1, 3, 5, 0, 6, 5, 7, 1, 5, 5, 7, 0, 4, 4, 5, 5, 1, 2, 4, 7, 7, 1, 2, 9, 1, 8, 7, 3, 2, 8, 7, 0, 1, 2, 3, 2, 4, 8, 6, 7, 1, 7, 4, 4, 2, 6, 6, 5, 4, 9, 5, 3, 7, 0, 9, 0, 7, 0, 7, 5, 9, 3, 1, 5, 3, 3, 7, 2, 1, 0, 8, 4, 8, 9, 0, 1, 4, 8, 4, 1, 0, 6, 3, 9, 9, 8, 7, 6, 4, 6, 3, 1, 9, 0, 0, 0, 0, 5

Offset: 1

Alonso del Arte, Jan 03 2018

This is an algebraic integer of degree 4, with minimal polynomial x^4 - 4*x^3 - 4*x^2 + 16*x - 8.

			  1.0000000000000000000000000000...
+ 1.4142135623730950488016887242...
+ 1.7320508075688772935274463415...
= 4.1462643699419723423291350657...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for algebraic numbers, degree 4

Essentially the same as A135611. Cf. A002193, A002194, A014176, A165663, A188582.

SetDefaultRealField(RealField(100)); 1 + Sqrt(2) + Sqrt(3); // G. C. Greubel, Nov 20 2018

RealDigits[1 + Sqrt[2] + Sqrt[3], 10, 100][[1]]

1+sqrt(2)+sqrt(3) \\ Felix Fröhlich, Jan 06 2018

numerical_approx(1+sqrt(2)+sqrt(3), digits=100) # G. C. Greubel, Nov 20 2018

1 + sqrt(2) + sqrt(3) = 1 + sqrt(5 + 2 sqrt(6)).

1 + A002193 + A002194 = A014176 + A002194 = A165663 + A188582. - Felix Fröhlich, Jan 06 2018

Terms a(52) onward corrected by G. C. Greubel, Nov 20 2018

A276265 Expansion of (1 + 2x)/(1 - 6x + 6*x^2).

Original entry on oeis.org

1, 8, 42, 204, 972, 4608, 21816, 103248, 488592, 2312064, 10940832, 51772608, 244990656, 1159308288, 5485905792, 25959585024, 122842075392, 581294942208, 2750717200896, 13016533552128, 61594898107392, 291470187331584, 1379251735345152, 6526689288081408, 30884625316417536

Offset: 0

Ilya Gutkovskiy, Aug 26 2016

Satisfies recurrence relations system a(n) = 4*a(n-1) + 2*b(n-1), b(n) = 2*b(n-1) + a(n-1), a(0)=1, b(0)=2.
More generally, for the recurrence relations system a(n) = 4*a(n-1) + 2*b(n-1), b(n) = 2*b(n-1) + a(n-1), a(0)=k, b(0)=m solution is a(n) = (((sqrt(3) - 1)*k - 2*m)*(3 - sqrt(3))^n + (sqrt(3)*k + k + 2*m)*(3 + sqrt(3))^n)/(2*sqrt(3)), b(n) = ((-k + sqrt(3)*m + m)*(3 - sqrt(3))^n + (k + (sqrt(3) - 1)*m)*(3 + sqrt(3))^n)/(2*sqrt(3)).
Convolution of A030192 and {1, 2, 0, 0, 0, 0, 0, ...}.

Index entries for linear recurrences with constant coefficients, signature (6,-6)

Cf. A030192, A165663.

a:=series((1+2*x)/(1-6*x+6*x^2),x=0,25): seq(coeff(a,x,n),n=0..24); # Paolo P. Lava, Mar 27 2019

LinearRecurrence[{6, -6}, {1, 8}, 25]
CoefficientList[Series[(1 + 2 x)/(1 - 6 x + 6 x^2), {x, 0, 24}], x] (* Michael De Vlieger, Aug 26 2016 *)

Vec((1+2*x)/(1-6*x+6*x^2) + O(x^99)) \\ Altug Alkan, Aug 26 2016

O.g.f.: (1 + 2*x)/(1 - 6*x + 6*x^2).
E.g.f.: (5*sqrt(3)*sinh(sqrt(3)*x) + 3*cosh(sqrt(3)*x))*exp(3*x)/3.
a(n) = 6*a(n-1) - 6*a(n-2).
a(n) = ((-5 + sqrt(3))*(3 - sqrt(3))^n + (5 + sqrt(3))*(3 + sqrt(3))^n)/(2*sqrt(3)).
Lim_{n->infinity} a(n+1)/a(n) = 3 + sqrt(3) = A165663.
a(n) = A030192(n)+2*A030192(n-1). - R. J. Mathar, Jan 25 2023

A344111 Decimal expansion of 4 + sqrt(3).

Original entry on oeis.org

5, 7, 3, 2, 0, 5, 0, 8, 0, 7, 5, 6, 8, 8, 7, 7, 2, 9, 3, 5, 2, 7, 4, 4, 6, 3, 4, 1, 5, 0, 5, 8, 7, 2, 3, 6, 6, 9, 4, 2, 8, 0, 5, 2, 5, 3, 8, 1, 0, 3, 8, 0, 6, 2, 8, 0, 5, 5, 8, 0, 6, 9, 7, 9, 4, 5, 1, 9, 3, 3, 0, 1, 6, 9, 0, 8, 8, 0, 0, 0, 3, 7, 0, 8, 1, 1, 4, 6, 1, 8, 6, 7, 5

Offset: 1

Wesley Ivan Hurt, May 10 2021

Decimal expansion of the surface area of a gyrobifastigium with regular faces and unit edge length.

Essentially the same sequence of digits as A176102, A165663, A160390, A090388, A019973 and A002194. - R. J. Mathar, May 16 2021

			5.73205080756887729...

Wikipedia, Gyrobifastigium

Cf. A002194, A019973, A090388, A160390, A165663.

RealDigits[4 + Sqrt[3], 10, 100] // Flatten

cp's OEIS Frontend

A240735 a(n) = floor(6^n/(3+sqrt(3))^n).

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A094433 a(n) is the left term in M^n * [1 0 0], M = the 3 X 3 matrix [1 -1 0 / -1 3 -2 / 0 -2 2].

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A383852 Decimal expansion of the volume of an elongated triangular pyramid with unit edge.

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A384139 Decimal expansion of the volume of an elongated triangular bipyramid with unit edges.

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A156309 Decimal expansion of the absolute value of the larger solution of (n^2+n)/2 = -1/12. (Real root q of 6n^2 + 6n + 1, the other root being p = -1-q.)

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A321120 Decimal expansion of (3 + sqrt(3))/12.

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A297701 Decimal expansion of 1 + sqrt(2) + sqrt(3).

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A276265 Expansion of (1 + 2x)/(1 - 6x + 6*x^2).