A090388 -id:A090388 - cp's OEIS frontend

A385256 Decimal expansion of the volume of a gyroelongated triangular bicupola with unit edge.

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

Offset: 1

Paolo Xausa, Jun 24 2025

The gyroelongated triangular bicupola is Johnson solid J_44.

			4.6945643929158936762142216512961490819695690656940...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Gyroelongated triangular bicupola.

Cf. A385257 (surface area).

Cf. A002193, A002193, A384143.

First[RealDigits[Sqrt[2]*(5/3 + Sqrt[1 + Sqrt[3]]), 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J44", "Volume"], 10, 100]]

Equals sqrt(2)*(5/3 + sqrt(1 + sqrt(3))) = A002193*(5/3 + sqrt(A090388)).

Equals the largest real root of 6561*x^8 - 198288*x^6 + 1506600*x^4 - 7125696*x^2 + 2704.

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

A377796 Decimal expansion of the surface area of a truncated icosidodecahedron (great rhombicosidodecahedron) with unit edge length.

Original entry on oeis.org

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

Offset: 3

Paolo Xausa, Nov 07 2024

			174.292030342323920882932107526283465728485221920...

Eric Weisstein's World of Mathematics, Great Rhombicosidodecahedron.
Wikipedia, Truncated icosidodecahedron.

Cf. A377797 (volume), A377798 (circumradius), A377799 (midradius).

Cf. A090388, A019970.

First[RealDigits[30*(1 + Sqrt[3] + Sqrt[5 + Sqrt[20]]), 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TruncatedIcosidodecahedron", "SurfaceArea"], 10, 100]]

Equals 30*(1 + sqrt(3) + sqrt(5 + 2*sqrt(5))) = 30*(A090388 + A019970).

A378131 Decimal expansion of sqrt(1 + sqrt(3))L/(Pi12^(1/8)), where L is the lemniscate constant (A062539).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Nov 18 2024

			1.011204695537690090572855988569625803283536658...

Wikipedia, Lemniscate constant (includes this constant).
Index to sequences related to the Gamma function.

Cf. A000796, A062539, A068465, A090388.

Cf. A377999, A378128, A378129, A378130, A378132.

First[RealDigits[Sqrt[(1 + Sqrt[3])*Pi]/(2^(3/4)*3^(1/8)*Gamma[3/4]^2), 10, 100]] (* or *)
First[RealDigits[Hypergeometric2F1[1/3, 2/3, 1, (3*Sqrt[3] - 5)/4], 10, 100]]

Equals sqrt((1 + sqrt(3))*Pi)/(2^(3/4)*3^(1/8)*Gamma(3/4)^2) = sqrt(A090388*A000796)/(2^(3/4)*3^(1/8)*A068465^2).
Equals Sum_{j,k integers} exp(-2*Pi*(j^2 + j*k + k^2)).
Equals 2F1(1/3, 2/3, 1, (3*sqrt(3) - 5)/4), where 2F1 is the ordinary hypergeometric function.

A144982 Decimal expansion of cos(Pi/24) = cos(7.5 degrees).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Sep 28 2008

Octic number of denominator 2 and minimal polynomial 256x^8 - 512x^6 + 320x^4 - 64x^2 + 1. - Charles R Greathouse IV, May 13 2019

			Equals 0.9914448613738104111445575269285628712777382744...

Index entries for algebraic numbers, degree 8

evalf( sqrt(2*sqrt(2)+sqrt(3)+1)/2^(5/4)) ;

RealDigits[ Sqrt[2 + Sqrt[2 + Sqrt[3]]]/2, 10, 105] // First (* Jean-François Alcover, Feb 20 2013 *)
RealDigits[Cos[Pi/24],10,120][[1]] (* Harvey P. Dale, Feb 11 2024 *)

cos(Pi/24) \\ Charles R Greathouse IV, May 13 2019

sqrt(2*sqrt(2)+sqrt(3)+1)/2^(5/4) =sqrt(A010466+A090388)/A011027.
Equals 2F1(9/16,7/16;1/2;3/4) / 2 . - R. J. Mathar, Oct 27 2008
4*this^3 -3*this = A144981. - R. J. Mathar, Aug 29 2025
Equals 2F1(-1/16,1/16;1/2;3/4) = 2F1(-1/12,1/12;1/2;1/2). - R. J. Mathar, Aug 31 2025

A224837 Surface area of Johnson square pyramid (rounded down) with all the edge-lengths equal to n.

Original entry on oeis.org

2, 10, 24, 43, 68, 98, 133, 174, 221, 273, 330, 393, 461, 535, 614, 699, 789, 885, 986, 1092, 1204, 1322, 1445, 1573, 1707, 1846, 1991, 2141, 2297, 2458, 2625, 2797, 2975, 3158, 3346, 3540, 3740, 3945, 4155, 4371, 4592, 4819, 5051, 5289, 5532, 5781, 6035, 6294

Offset: 1

K. D. Bajpai, Sep 18 2013

nonn

Johnson square pyramid: a square base with four equilateral triangular-faces. All the edge-lengths are equal.

			a(3) = 24: Surface area = (1+sqrt(3))*3^2 = 24.588... and floor(24.588...) = 24.

K. D. Bajpai, Table of n, a(n) for n = 1..1000 [a(613), a(864) corrected by _Georg Fischer_]
Wikipedia, Square pyramid

Cf. A090388, A228189.

a:= n-> floor((1+sqrt(3))*n^2):
seq(a(n), n=1..48);

Mathematica
```
Table[Floor[(1+Sqrt[3])*k^2], {k, 500}]
```
PARI
```
vector(500, k, floor((1+sqrt(3))*k^2))
```

a(n)=n^2+sqrtint(3*n^4) \\ Charles R Greathouse IV, Sep 18 2013

a(n) = floor((1+sqrt(3))*n^2).

A256965 Decimal expansion of sqrt(2) + sqrt(3/2).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Apr 16 2015

Conjectured minimal length of fence ensuring privacy of a square garden.

			2.6389584337646840979003307615626437745526456157052831373930...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.11, p. 517.

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Bernd Kawohl, The opaque square and the opaque circle, in: General Inequalities VII, Int. Ser. Numer. Math. 123 (1997), pp. 339-346.
Bernd Kawohl, Some nonconvex shape optimization problems, in: Optimal Shape Design, Eds. A.Cellina u. A. Ornelas, Springer Lecture Notes in Math.1740 (2000), pp. 7-46.
Ian Stewart, Pursuing Polygonal Privacy, Mathematical Recreations Column, Scientific American, 284 (No. 2, 2001), 88-89.

Cf. A090388.

SetDefaultRealField(RealField(100)); Sqrt(2) + Sqrt(3/2); // G. C. Greubel, Aug 19 2018

RealDigits[Sqrt[2] + Sqrt[3/2], 10, 110][[1]] (* Vincenzo Librandi, Aug 21 2016 *)

sqrt(2)+sqrt(3/2) \\ Michel Marcus, Dec 20 2015

A317502 Triangle read by rows: T(0,0) = 1; T(n,k) = 3 T(n-1,k) - 2 * T(n-3,k-1) for k = 0..floor(n/3); T(n,k)=0 for n or k < 0.

Original entry on oeis.org

1, 3, 9, 27, -2, 81, -12, 243, -54, 729, -216, 4, 2187, -810, 36, 6561, -2916, 216, 19683, -10206, 1080, -8, 59049, -34992, 4860, -96, 177147, -118098, 20412, -720, 531441, -393660, 81648, -4320, 16, 1594323, -1299078, 314928, -22680, 240, 4782969, -4251528, 1180980, -108864, 2160

Offset: 0

Shara Lalo, Aug 02 2018

The numbers in rows of the triangle are along "second layer" skew diagonals pointing top-right in center-justified triangle given in A303901 ((3-2*x)^n) and along "second layer" skew diagonals pointing top-left in center-justified triangle given in A317498 ((-2+3x)^n), see links. (Note: First layer skew diagonals in center-justified triangles of coefficients in expansions of (3-2*x)^n and (-2+3x)^n are given in A303941 and A302747 respectively.) The coefficients in the expansion of 1/(1-3x+2x^3) are given by the sequence generated by the row sums. The row sums give A077846. If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 2.7320508075688772... (A090388: 1+sqrt(3)), when n approaches infinity.

			Triangle begins:
        1;
        3;
        9;
        27,        -2;
        81,       -12;
       243,       -54;
       729,      -216,        4;
      2187,      -810,       36;
      6561,     -2916,      216;
     19683,    -10206,     1080,       -8;
     59049,    -34992,     4860,      -96;
    177147,   -118098,    20412,     -720;
    531441,   -393660,    81648,    -4320,    16;
   1594323,  -1299078,   314928,   -22680,   240;
   4782969,  -4251528,  1180980,  -108864,  2160;
  14348907, -13817466,  4330260,  -489888, 15120,  -32;
  43046721, -44641044, 15588936, -2099520, 90720, -576;

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 136, 396, 397.

Row sums give A077846.
Cf. A303901, A317498.
Cf. A090388.
Cf. A303941, A302747.

t[n_, k_] := t[n, k] = 3^(n - 3k) * (-2)^k/((n - 3 k)! k!) * (n - 2 k)!; Table[t[n, k], {n, 0, 15}, {k, 0, Floor[n/3]} ]  // Flatten
t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, 3 * t[n - 1, k] - 2 * t[n - 3, k - 1]]; Table[t[n, k], {n, 0, 15}, {k, 0, Floor[n/3]}] // Flatten

T(n,k) = 3^(n - 3k) * (-2)^k / ((n - 3k)! k!) * (n - 2k)! where n is a nonnegative integer and k = 0..floor(n/3).

A317503 Triangle read by rows: T(0,0) = 1; T(n,k) = -2 T(n-1,k) + 3 * T(n-3,k-1) for k = 0..floor(n/3); T(n,k)=0 for n or k < 0.

Original entry on oeis.org

1, -2, 4, -8, 3, 16, -12, -32, 36, 64, -96, 9, -128, 240, -54, 256, -576, 216, -512, 1344, -720, 27, 1024, -3072, 2160, -216, -2048, 6912, -6048, 1080, 4096, -15360, 16128, -4320, 81, -8192, 33792, -41472, 15120, -810, 16384, -73728, 103680, -48384, 4860, -32768, 159744, -253440, 145152, -22680, 243

Offset: 0

Shara Lalo, Aug 02 2018

The numbers in rows of the triangle are along "second layer" skew diagonals pointing top-left in center-justified triangle given in A303901 ((3-2*x)^n) and along "second layer" skew diagonals pointing top-right in center-justified triangle given in A317498 ((-2+3x)^n), see links. (Note: First layer skew diagonals in center-justified triangles of coefficients in expansions of (3-2*x)^n and (-2+3x)^n are given in A303941 and A302747 respectively.) The coefficients in the expansion of 1/(1 + 2x - 3x^3) are given by the sequence generated by the row sums. The row sums give A317499.

			Triangle begins:
       1;
      -2;
       4;
      -8,       3;
      16,     -12;
     -32,      36;
      64,     -96,       9;
    -128,     240,     -54;
     256,    -576,     216;
    -512,    1344,    -720,      27;
    1024,   -3072,    2160,    -216;
   -2048,    6912,   -6048,    1080;
    4096,  -15360,   16128,   -4320,     81;
   -8192,   33792,  -41472,   15120,   -810;
   16384,  -73728,  103680,  -48384,   4860;
  -32768,  159744, -253440,  145152, -22680,   243;
   65536, -344064,  608256, -414720,  90720, -2916;

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 136, 396, 397.

Row sums give A317499.
Cf. A303901, A317498.
Cf. A090388.
Cf. A303941, A302747.

t[n_, k_] := t[n, k] = (-2)^(n - 3k) * 3^k/((n - 3 k)! k!) * (n - 2 k)!; Table[t[n, k], {n, 0, 15}, {k, 0, Floor[n/3]} ]  // Flatten
t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, -2 * t[n - 1, k] + 3 * t[n - 3, k - 1]]; Table[t[n, k], {n, 0, 15}, {k, 0, Floor[n/3]}] // Flatten

T(n,k) = (-2)^(n - 3k) * 3^k / ((n - 3k)! k!) * (n - 2k)! where n is a nonnegative integer and k = 0..floor(n/3).

A337402 Decimal expansion of the length of third shortest diagonal in a regular 12-gon with unit edge length.

Original entry on oeis.org

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

Offset: 1

Mohammed Yaseen, Aug 26 2020

The distinct diagonal lengths in a regular 12-gon ABC...JKL with unit edge length are
AC = sqrt(2 +   sqrt(3)) =     sqrt(2)/(-1+sqrt(3)) = A188887
AD = sqrt(4 + 2*sqrt(3)) =          2 /(-1+sqrt(3)) = A090388
AE = sqrt(6 + 3*sqrt(3)) =     sqrt(6)/(-1+sqrt(3))
AF = sqrt(7 + 4*sqrt(3)) = (1+sqrt(3))/(-1+sqrt(3)) = A019973
AG = sqrt(8 + 4*sqrt(3)) =   2*sqrt(2)/(-1+sqrt(3)) = A214726

			3.34606521495123162230117512366749281383748155339375...

Paolo Xausa, Table of n, a(n) for n = 1..10000
N. J. A. Sloane and Gavin A. Theobald, On Dissecting Polygons into Rectangles, arXiv:2309.14866 [math.CO], 2023. See Eq. (2.3).
I. J. Zucker, G. S. Joyce, Special values of the hypergeometric series II, Math. Proc. Cam. Phil. Soc. 131 (2001) 309 eq (8.9)

Cf. A337301, A188887, A090388, A019973, A214726, A145439.

First[RealDigits[Sqrt[6+3Sqrt[3]],10,100]] (* Paolo Xausa, Oct 19 2023 *)

sqrt(6 + 3*sqrt(3)) \\ Michel Marcus, Aug 26 2020

Equals sin(Pi/3)/sin(Pi/12) = sqrt(2) + 2*cos(Pi/12) = sqrt(3*cot(Pi/12)).
Equals sqrt(6 + 3*sqrt(3)) = sqrt(6)/(-1+sqrt(3)) = (3+sqrt(3))/sqrt(2).
Equals 3*A145439.
Equals Gamma(1/24)*Gamma(11/24)/(Gamma(5/24)*Gamma(7/24)) [Zucker] - R. J. Mathar, Jun 24 2024

cp's OEIS Frontend

A385256 Decimal expansion of the volume of a gyroelongated triangular bicupola with unit edge.

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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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A377796 Decimal expansion of the surface area of a truncated icosidodecahedron (great rhombicosidodecahedron) with unit edge length.

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A378131 Decimal expansion of sqrt(1 + sqrt(3))*L/(Pi*12^(1/8)), where L is the lemniscate constant (A062539).

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A144982 Decimal expansion of cos(Pi/24) = cos(7.5 degrees).

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A224837 Surface area of Johnson square pyramid (rounded down) with all the edge-lengths equal to n.

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A256965 Decimal expansion of sqrt(2) + sqrt(3/2).

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A378131 Decimal expansion of sqrt(1 + sqrt(3))L/(Pi12^(1/8)), where L is the lemniscate constant (A062539).