Search: a346585 -id:a346585
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A093766
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Decimal expansion of Pi/(2*sqrt(3)).
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+10
33
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9, 0, 6, 8, 9, 9, 6, 8, 2, 1, 1, 7, 1, 0, 8, 9, 2, 5, 2, 9, 7, 0, 3, 9, 1, 2, 8, 8, 2, 1, 0, 7, 7, 8, 6, 6, 1, 4, 2, 0, 3, 3, 1, 2, 4, 0, 4, 6, 3, 7, 0, 2, 8, 7, 7, 8, 4, 9, 4, 2, 4, 6, 7, 6, 9, 4, 0, 6, 1, 5, 9, 0, 5, 6, 3, 1, 7, 6, 9, 4, 1, 8, 4, 2, 0, 6, 2, 4, 9, 4, 1, 0, 6, 0, 3, 0, 0, 8, 4, 4, 2, 8
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OFFSET
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0,1
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COMMENTS
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Density of densest packing of equal circles in two dimensions (achieved for example by the A2 lattice).
The number gives the areal coverage (90.68... percent) of the close hexagonal (densest) packing of circles in the plane. The hexagonal unit cell is a rhombus of side length 1 and height sqrt(3)/2; the area of the unit cell is sqrt(3)/2 and the four parts of circles add to an area of one circle of radius 1/2, which is Pi/4. - R. J. Mathar, Nov 22 2011
Ratio of surface area of a sphere to the regular octahedron whose edge equals the diameter of the sphere. - Omar E. Pol, Dec 09 2013
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REFERENCES
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J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer, 3rd. ed., 1998. See p. xix.
L. B. W. Jolley, Summation of Series, Dover (1961), Eq. (84) on page 16.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, p. 30.
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LINKS
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FORMULA
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Equals (5/6)*(7/6)*(11/12)*(13/12)*(17/18)*(19/18)*(23/24)*(29/30)*(31/30)*..., where the numerators are primes > 3 and the denominators are the nearest multiples of 6.
Equals Sum_{n>=1} 1/A134667(n). [Jolley]
Equals Sum_{n>=0} (-1)^n/A124647(n). [Jolley eq. 273]
Continued fraction expansion: 1 - 2/(18 + 12*3^2/(24 + 12*5^2/(32 + ... + 12*(2*n - 1)^2/((8*n + 8) + ... )))). See A254381 for a sketch proof. - Peter Bala, Feb 04 2015
Equals 4*Sum_{n >= 0} 1/((6*n + 1)*(6*n + 5)).
Continued fraction: 1/(1 + 1^2/(4 + 5^2/(2 + 7^2/(4 + 11^2/(2 + ... + (6*n + 1)^2/(4 + (6*n + 5)^2/(2 + ... ))))))). (End)
The inverse is (2*sqrt(3))/Pi = Product_{n >= 1} 1 + (1 - 1/(4*n))/(4*n*(9*n^2 - 9*n + 2)) = (35/32) * (1287/1280) * (8075/8064) * (5635/5632) * (72819/72800) * ... = 1.102657790843585... - Dimitris Valianatos, Aug 31 2019
Equals Integral_{x=0..oo} 1/(x^2 + 3) dx.
Equals Integral_{x=0..oo} 1/(3*x^2 + 1) dx. (End)
Equals 1 + Sum_{k>=1} ( 1/(6*k+1) - 1/(6*k-1) ). - Sean A. Irvine, Jul 24 2021
For positive integer k, Pi/(2*sqrt(3)) = Sum_{n >= 0} (6*k + 4)/((6*n + 1)*(6*n + 6*k + 5)) - Sum_{n = 0..k-1} 1/(6*n + 5). - Peter Bala, Jul 10 2024
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EXAMPLE
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0.906899682117108925297039128821077866142033124046370287784942...
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MATHEMATICA
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PROG
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CROSSREFS
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Related constants: A020769, A020789, A093825, A222066, A222067, A222068, A222069, A222070, A222071, A222072, A260646, (1/2)*A093602, A346585, A346584, A346583.
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A122373
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Expansion of (c(q)^3 + c(q^2)^3) / 27 in powers of q where c() is a cubic AGM theta function.
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+10
3
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1, 4, 9, 16, 24, 36, 50, 64, 81, 96, 120, 144, 170, 200, 216, 256, 288, 324, 362, 384, 450, 480, 528, 576, 601, 680, 729, 800, 840, 864, 962, 1024, 1080, 1152, 1200, 1296, 1370, 1448, 1530, 1536, 1680, 1800, 1850, 1920, 1944, 2112, 2208, 2304, 2451, 2404
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OFFSET
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1,2
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COMMENTS
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a(n) = n^2 and n > 0 if and only if n = 2^i * 3^j with i, j >=0 (numbers in A003586). - Michael Somos, Jun 08 2012
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LINKS
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FORMULA
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Expansion of eta(q^2)^5 * eta(q^3)^4 * eta(q^6) / eta(q)^4 in powers of q.
a(n) is multiplicative with a(2^e) = 4^e, a(3^e) = 9^e, a(p^e) = (p^(2*e + 2) - f^(e+1)) / (p^2 - f) where f = 1 if p == 1 (mod 6), f = -1 if p == 5 (mod 6).
Euler transform of period 6 sequence [4, -1, 0, -1, 4, -6, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 3^(1/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is g.f. for A132000.
G.f.: Sum_{k>0} k^2 * x^k / (1 + x^k + x^(2*k)) * (1 + (1+(-1)^k)/8).
G.f.: Product_{k>0} (1 - x^k) * (1 + x^(3*k)) * (1 + x^k)^5 * (1 - x^(3*k))^5.
Expansion of psi(q)^2 * psi(q^3)^2 * phi(-q^3)^3 / phi(-q) in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Jun 23 2012
Expansion of c(q) * c(q^2) * b(q^2)^2 / (9 * b(q)) in powers of q where b(), c() are cubic AGM theta functions. - Michael Somos, Jun 23 2012
G.f.: Sum_{k>0} k^2 * x^k * (1 + x^(2*k)) / (1 + x^(2*k) + x^(4*k)). - Michael Somos, Jul 05 2020
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = Pi^3/(18*sqrt(3)) = 0.994526... (A346585). - Amiram Eldar, Dec 22 2023
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EXAMPLE
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G.f. = q + 4*q^2 + 9*q^3 + 16*q^4 + 24*q^5 + 36*q^6 + 50*q^7 + 64*q^8 + 81*q^9 + 96*q^10 + ...
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MATHEMATICA
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terms = 50; QP = QPochhammer; s = QP[q^2]^5*QP[q^3]^4*(QP[q^6]/QP[q]^4) + O[q]^terms; CoefficientList[s, q] (* Jean-François Alcover, Jul 04 2017, from first formula *)
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PROG
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(PARI) {a(n) = my(A, p, e, f); if( n<0, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p<5, p^(2*e), f =- (-1)^(p%3); (p^(2*e + 2) - f^(e+1)) / (p^2 - f))))};
(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^3 + A)^4 * eta(x^6 + A) / eta(x + A)^4, n))};
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CROSSREFS
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KEYWORD
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nonn,easy,mult
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AUTHOR
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STATUS
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approved
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A346583
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Decimal expansion of 301 * Pi^7 / (524880 * sqrt(3)).
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+10
3
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9, 9, 9, 9, 8, 8, 3, 7, 7, 4, 0, 9, 4, 0, 5, 7, 8, 7, 8, 6, 2, 1, 9, 1, 3, 6, 8, 3, 6, 1, 8, 6, 3, 1, 4, 4, 4, 1, 0, 6, 6, 1, 8, 2, 2, 4, 7, 5, 5, 4, 0, 7, 2, 1, 9, 8, 4, 6, 1, 3, 5, 7, 5, 9, 1, 1, 0, 2, 3, 3, 0, 8, 6, 5, 2, 2, 7, 4, 6, 3, 0, 1, 0, 1, 2, 3, 1
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OFFSET
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0,1
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REFERENCES
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L. B. W. Jolley, Summation of Series, Dover, 1961, Eq. (314).
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LINKS
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FORMULA
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Equals 7 * 43 * Pi^7 / (6! * 3^6 * sqrt(3)).
Equals 1 + Sum_{k>=1} ( 1/(6*k+1)^7 - 1/(6*k-1)^7 ).
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EXAMPLE
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0.999988377409405787862191368361863144410...
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MATHEMATICA
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First[RealDigits[N[7 * 43 * Pi^7 / (6! * 3^6 * Sqrt[3]), 88]]] (* Stefano Spezia, Jul 24 2021 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A346584
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Decimal expansion of 11 * Pi^5 / (1944 * sqrt(3)).
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+10
3
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9, 9, 9, 7, 3, 5, 6, 0, 7, 6, 4, 8, 7, 5, 1, 7, 3, 8, 9, 9, 5, 0, 2, 8, 6, 9, 4, 7, 8, 8, 5, 0, 6, 9, 0, 7, 7, 8, 1, 3, 6, 7, 4, 4, 8, 3, 1, 2, 2, 6, 5, 8, 9, 1, 3, 6, 6, 3, 9, 3, 7, 4, 5, 5, 8, 0, 6, 5, 8, 0, 7, 4, 6, 4, 6, 4, 1, 6, 4, 0, 3, 3, 2, 2, 2, 3, 2
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,1
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REFERENCES
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L. B. W. Jolley, Summation of Series, Dover, 1961, Eq. (314).
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LINKS
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FORMULA
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Equals 11 * Pi^5 / (4! * 3^4 * sqrt(3)).
Equals 1 + Sum_{k>=1} ( 1/(6*k+1)^5 - 1/(6*k-1)^5 ).
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EXAMPLE
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0.99973560764875173899502869478850690...
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MATHEMATICA
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First[RealDigits[N[11 * Pi^5 / (4! * 3^4 * Sqrt[3]), 88]]] (* Stefano Spezia, Jul 24 2021 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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