A010527 -id:A010527

A325732

First term of n-th difference sequence of (floor(k*r)), r = sqrt(3/4), k >= 0.

+10
2

0, 1, -1, 1, -1, 1, -1, 0, 7, -35, 119, -329, 791, -1715, 3430, -6419, 11319, -18767, 28763, -38759, 38759, 1, -149228, 572057, -1615429, 3979001, -9014851, 19251001, -39309301, 77558760, -149239771, 282712561, -532577025, 1008032953, -1934671809, 3787949521

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,9

LINKS

Clark Kimberling, Table of n, a(n) for n = 1..200

MATHEMATICA

Table[First[Differences[Table[Floor[Sqrt[3/4]*n], {n, 0, 50}], n]], {n, 1, 50}]

CROSSREFS

Cf. A325664. Inverse binomial Transform of A171970 and A172475.

Cf. A010527 (sqrt(3/4)).

KEYWORD

easy,sign

AUTHOR

Clark Kimberling, May 20 2019

STATUS

approved

A343055

Decimal expansion of the imaginary part of i^(1/16), or sin(Pi/32).

+10
2

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

An algebraic number of degree 16 and denominator 2. - Charles R Greathouse IV, Jan 09 2022

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..10000

Michael Penn, The exact value of sin 5⅝°??, YouTube video, 2021.

Wikipedia, Trigonometric constants expressed in real radicals

Index entries for algebraic numbers, degree 16

FORMULA

Equals (1/2) * sqrt(2-sqrt(2+sqrt(2+sqrt(2)))).

EXAMPLE

0.09801714032956060199419...

MATHEMATICA

RealDigits[Sin[Pi/32], 10, 100, -1][[1]] (* Amiram Eldar, Apr 27 2021 *)

PROG

(PARI) imag(I^(1/16))

(PARI) sin(Pi/32)

(PARI) sqrt(2-sqrt(2+sqrt(2+sqrt(2))))/2

(Sage) numerical_approx(sin(pi/32), digits=123) # G. C. Greubel, Sep 30 2022

CROSSREFS

sin(Pi/m): A010527 (m=3), A010503 (m=4), A019845 (m=5), A323601 (m=7), A182168 (m=8), A019829 (m=9), A019827 (m=10), A019824 (m=12), A232736 (m=14), A019821 (m=15), A232738 (m=16), A241243 (m=17), A019819 (m=18), A019818 (m=20), A343054 (m=24), A019815 (m=30), this sequence (m=32), A019814 (m=36).

KEYWORD

nonn,cons,easy

AUTHOR

Seiichi Manyama, Apr 04 2021

STATUS

approved

A343056

Decimal expansion of the real part of i^(1/16), or cos(Pi/32).

+10
2

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..10000

Leon D. Fairbanks, Powers of Cosine and Sine, arXiv:2308.04437 [math.GM], 2023. See p. 3.

Wikipedia, Trigonometric constants expressed in real radicals.

FORMULA

Equals (1/2) * sqrt(2+sqrt(2+sqrt(2+sqrt(2)))).

EXAMPLE

0.9951847266721968862448369...

MATHEMATICA

RealDigits[Cos[Pi/32], 10, 100][[1]] (* Amiram Eldar, Apr 27 2021 *)

PROG

(PARI) real(I^(1/16))

(PARI) cos(Pi/32)

(PARI) sqrt(2+sqrt(2+sqrt(2+sqrt(2))))/2

(Magma) R:= RealField(127); Cos(Pi(R)/32) // G. C. Greubel, Sep 30 2022

(SageMath) numerical_approx(cos(pi/32), digits=122) # G. C. Greubel, Sep 30 2022

CROSSREFS

cos(Pi/m): A010503 (m=4), A019863 (m=5), A010527 (m=6), A073052 (m=7), A144981 (m=8), A019879 (m=9), A019881 (m=10), A019884 (m=12), A232735 (m=14), A019887 (m=15), A232737 (m=16), A210649 (m=17), A019889 (m=18), A019890 (m=20), A144982 (m=24), A019893 (m=30). this sequence (m=32), A019894 (m=36).

KEYWORD

nonn,cons,easy

AUTHOR

Seiichi Manyama, Apr 04 2021

STATUS

approved

A093822

Decimal expansion of -109/121 - 82/(121*sqrt(3)) + (2*sqrt(-35139 + 28634*sqrt(3)))/121 - Pi/3 + arccos((-1 + sqrt(3))/2).

+10
1

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

Area of lamina found by Sprague in the Lebesgue minimal problem.

LINKS

Table of n, a(n) for n=0..101.

Eric Weisstein's World of Mathematics, Lebesgue Minimal Problem

EXAMPLE

0.844137123...

MATHEMATICA

RealDigits[-109/121-82/(121Sqrt[3])+(2Sqrt[-35139+28634Sqrt[3]])/121-Pi/3+ ArcCos[(-1+Sqrt[3])/2], 10, 120][[1]] (* Harvey P. Dale, Sep 22 2020 *)

CROSSREFS

Cf. A010527, A093821.

KEYWORD

nonn,cons,easy,changed

AUTHOR

Eric W. Weisstein, Apr 16 2004

STATUS

approved

A179639

Decimal expansion of the volume of gyroelongated pentagonal pyramid with edge length 1.

+10
1

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Gyroelongated pentagonal pyramid: 11 vertices,25 edges,and 16 faces.

LINKS

Table of n, a(n) for n=1..105.

Wolfram Alpha, Johnson solid 11

FORMULA

Digits of (25+9*sqrt(5))/24.

EXAMPLE

1.88019215822908780282010679244089525495689855152098881326825313369561...

MATHEMATICA

RealDigits[N[(25+9*Sqrt[5])/24, 200]]

CROSSREFS

Cf. A001622, A010527, A102208, A179290, A179292, A179294, A179449, A179450, A179451, A179452, A179552, A179553, A179554, A179587, A179588, A179589, A179590, A179591, A179592, A179593, A179637, A179638.

KEYWORD

cons,nonn,easy

AUTHOR

Vladimir Joseph Stephan Orlovsky, Jul 21 2010

STATUS

approved

A179640

Decimal expansion of the surface area of gyroelongated pentagonal pyramid with edge length 1.

+10
1

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Gyroelongated pentagonal pyramid: 11 vertices, 25 edges, and 16 faces.

LINKS

Table of n, a(n) for n=1..105.

Wolfram Alpha, Johnson solid 11

FORMULA

Digits of sqrt(5/2*(70+sqrt(5)+3*sqrt(75+30*sqrt(5))))/2.

EXAMPLE

8.21566792897225677348693575803563097544289387179912568441637087996861...

MATHEMATICA

RealDigits[N[Sqrt[5/2*(70+Sqrt[5]+3*Sqrt[75+30*Sqrt[5]])]/2, 200]]

CROSSREFS

Cf. A001622, A010527, A102208, A179290, A179292, A179294, A179449, A179450, A179451, A179452, A179552, A179553, A179554, A179587, A179588, A179589, A179590, A179591, A179592, A179593, A179637, A179638, A179639.

KEYWORD

cons,nonn,easy

AUTHOR

Vladimir Joseph Stephan Orlovsky, Jul 21 2010

STATUS

approved

A194082

Sum{floor(sqrt(3)*k/2) : 1<=k<=n}

+10
1

0, 1, 3, 6, 10, 15, 21, 27, 34, 42, 51, 61, 72, 84, 96, 109, 123, 138, 154, 171, 189, 208, 227, 247, 268, 290, 313, 337, 362, 387, 413, 440, 468, 497, 527, 558, 590, 622, 655, 689, 724, 760, 797, 835, 873, 912, 952, 993, 1035, 1078, 1122, 1167, 1212

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

Partial sums of A171970.

Comment from R. J. Mathar, Dec 02 2012 (Start):

a(n-1) is the number of unit squares regularly packed into the isosceles triangle of edge length n.

The triangle may be aligned with the Cartesian axes by putting its bottom edge on the horizontal axis, so its vertices are at (x,y) = (0,0), (n,0) and (n/2,sqrt(3)*n/2), see A010527.

The area inside the triangle is sqrt(3)*n^2/4 = A120011*n^2. There is an obvious upper limit of floor(sqrt(3)*n^2/4) = A171971(n) to the count of non-overlapping unit squares inside this triangle.

Regular packing: We place the first row of unit squares so they touch the bottom edge of the triangle. Their number is limited by the length of the horizontal section of the line y=1 inside the triangle, n-2*y/sqrt(3), which touches all of these first-row squares at their top.

The number of unit squares in the next row, between y=1 and y=2, is limited by the length of the horizontal section of the line y=2 inside the triangle, n-2*y/sqrt(3). Continuing, in row y=1, 2, ... we insert floor(n-2*y/sqrt(3)) unit squares, all with the same orientation.

The total number of squares is sum_{ y=1, 2, ..., floor(n*sqrt(3)/2) } floor( n-2*y/sqrt(3) ), and resummation yields, up to an index shift, this sequence here.

(End)

LINKS

Table of n, a(n) for n=1..53.

MATHEMATICA

r = Sqrt[3]/2;

c[k_] := Sum[Floor[j*r], {j, 1, k}];

Table[c[k], {k, 1, 90}]

PROG

(PARI) a(n)=sum(k=1, n, sqrtint(3*k^2\4)) \\ Charles R Greathouse IV, Jan 06 2013

CROSSREFS

Cf. A171970.

KEYWORD

nonn

AUTHOR

Clark Kimberling, Aug 17 2011

STATUS

approved

A333322

Decimal expansion of (3/8) * sqrt(3).

+10
1

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

This is the area of the regular hexagon of diameter 1.

From Bernard Schott, Apr 09 2022 and Oct 01 2022: (Start)

For any triangle ABC, where (A,B,C) are the angles:

sin(A) * sin(B) * sin(C) <= (3/8) * sqrt(3) [Bottema reference],

cos(A/2) * cos(B/2) * cos(C/2) <= (3/8) * sqrt(3) [Mitrinovic reference],

and if (ha,hb,hc) are the altitude lengths and (a,b,c) the side lengths of this triangle [Scott Brown link]:

(ha+hb) * (hb+hc) * (hc+ha) / (a+b) * (b+c) * (c+a) <= (3/8) * sqrt(3).

The equalities are obtained only when triangle ABC is equilateral. (End)

REFERENCES

O. Bottema et al., Geometric Inequalities, Groningen, 1969, item 2.7, page 19.

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.15, p. 526.

D. S. Mitrinovic, E. S. Barnes, D. C. B. Marsh, J. R. M. Radok, Elementary Inequalities, Tutorial Text 1 (1964), P. Noordhoff LTD, Groningen, problem 6.2.2, page 111.

LINKS

Table of n, a(n) for n=0..119.

Scott Brown, Problem 3453, Crux Mathematicorum, Vol. 36, No. 5 (2010), pp. 342 and 343.

FORMULA

Equals A104954/2 or A104956/4.

EXAMPLE

0.649519052838328985...

MATHEMATICA

RealDigits[(3/8) * Sqrt[3], 10, 120][[1]]

PROG

(PARI) sqrt(27)/8 \\ Charles R Greathouse IV, Apr 09 2022

CROSSREFS

Cf. A002194 (sqrt(3)), A104954.

Cf. A010527, A020821, A104956, A152623 (other geometric inequalities).

KEYWORD

cons,nonn

AUTHOR

Kritsada Moomuang, Mar 15 2020

STATUS

approved

A179453

Decimal expansion of the inradius of an icosidodecahedron with edge length 1.

+10
0

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Icosidodecahedron: 32 faces, 30 vertices, and 60 edges.

LINKS

Table of n, a(n) for n=1..105.

Icosidodecahedron

FORMULA

Digits of (5+3*sqrt(5))/8.

EXAMPLE

1.46352549156242113615344012577422858829023188485432214660158646702894...

MATHEMATICA

RealDigits[N[(5+3*Sqrt[5])/8, 200]]

PROG

(PARI) (sqrt(45)+5)/8 \\ Charles R Greathouse IV, Apr 25 2016

CROSSREFS

Cf. A001622, A010527, A102208, A179290, A179292, A179294, A179449, A179450, A179451, A179452.

KEYWORD

nonn,cons,easy

AUTHOR

Vladimir Joseph Stephan Orlovsky, Jul 14 2010

STATUS

approved

A179641

Decimal expansion of the volume of pentagonal dipyramid with edge length 1.

+10
0

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

Pentagonal dipyramid: 7 vertices, 15 edges, and 10 faces.

LINKS

Table of n, a(n) for n=0..104.

Wolfram Alpha, Johnson solid 13

FORMULA

Digits of (5+sqrt(5))/12.

EXAMPLE

0.60300566479164914136743113906093968628671819663429381035590810378421...