A073010 -id:A073010 - cp's OEIS frontend

A381155 Decimal expansion of the isoperimetric quotient of a regular 10-gon.

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

Offset: 0

Paolo Xausa, Feb 15 2025

For the definition of isoperimetric quotient, see A381152.

			0.96688279904640254032818321918275298846986824108440...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Isoperimetric Quotient.
Wikipedia, Isoperimetric inequality.

Cf. A019916, A178816.

Cf. isoperimetric quotient of other regular polygons: A073010 (triangle), A003881 (square), A381152 (pentagon), A093766 (hexagon), A381153 (heptagon), A196522 (octagon), A381154 (9-gon), A381156 (11-gon), A381157 (12-gon).

First[RealDigits[Pi/(10*Tan[Pi/10]), 10, 100]]

Equals Pi/(10*tan(Pi/10)) = Pi/(10*A019916).

Equals (1/25)*Pi*A178816.

A381156 Decimal expansion of the isoperimetric quotient of a regular 11-gon.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Feb 15 2025

For the definition of isoperimetric quotient, see A381152.

			0.97266200091990681953828897938526763171296541114234...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Isoperimetric Quotient.
Wikipedia, Isoperimetric inequality.

Cf. A256854.

Cf. isoperimetric quotient of other regular polygons: A073010 (triangle), A003881 (square), A381152 (pentagon), A093766 (hexagon), A381153 (heptagon), A196522 (octagon), A381154 (9-gon), A381155 (10-gon), A381157 (12-gon).

First[RealDigits[Pi/(11*Tan[Pi/11]), 10, 100]]

Equals Pi/(11*tan(Pi/11)).

Equals (4/121)*Pi*A256854.

A381157 Decimal expansion of the isoperimetric quotient of a regular 12-gon.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Feb 15 2025

For the definition of isoperimetric quotient, see A381152.

			0.97704861665685333572562679495712274710387812858570...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Isoperimetric Quotient.
Wikipedia, Isoperimetric inequality.

Cf. A019913, A178809.

Cf. isoperimetric quotient of other regular polygons: A073010 (triangle), A003881 (square), A381152 (pentagon), A093766 (hexagon), A381153 (heptagon), A196522 (octagon), A381154 (9-gon), A381155 (10-gon), A381156 (11-gon).

First[RealDigits[Pi/(12*Tan[Pi/12]), 10, 100]]

Equals Pi/(12*tan(Pi/12)) = Pi/(12*A019913).

Equals (1/36)*Pi*A178809.

A381671 Decimal expansion of the isoperimetric quotient of a regular tetrahedron.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Mar 03 2025

Polya (1954) defines the isoperimetric quotient of a solid as 36*Pi*V^2/(S^3), where V and S are the volume and surface area of the solid, respectively.

The isoperimetric quotient of a sphere is 1.

			0.30229989403903630843234637627369262204734437468212...

George Polya, Mathematics and Plausible Reasoning, Vol. 1: Induction and Analogy in Mathematics, Princeton University Press, Princeton, New Jersey, 1954. See pp. 188-189, exercise 43.

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A273633 (sphericity).
Cf. isoperimetric quotient of other Platonic solids: A019673 (cube), A073010 (octahedron), A374772 (dodecahedron), A381672 (icosahedron).
Cf. A002194, A019673.

First[RealDigits[Pi/(6*Sqrt[3]), 10, 100]]

Equals Pi/(6*sqrt(3)) = A019673/A002194.

A086464 Decimal expansion of 17/36*zeta(4).

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Jul 21 2003

			0.51109708258581525710477952336666262075474350507273...

Renzo Sprugnoli, Sums of reciprocals of the central binomial coefficients, El. J. Combin. Numb. Th. 6 (2006) # A27
Eric Weisstein's World of Mathematics, Central Binomial Coefficient.

Cf. A073010, A073016, A086463.

RealDigits[17*Zeta[4]/36, 10, 120][[1]] (* Amiram Eldar, May 25 2023 *)

zeta(4)*17/36 \\ Michel Marcus, Jul 31 2015

Equals Sum_{n>=1} 1/(n^4 * binomial(2*n,n)).

A210453 Decimal expansion of Sum_{n>=1} 1/(nbinomial(3n,n)).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jan 21 2013

			0.37121697526024703447477166607535880558762946905197...

George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press, 2006, p. 60.

Necdet Batir, On the series Sum_{k=1..oo} binomial(3k,k)^{-1} k^{-n} x^k, Proc. Indian Acad. Sci. (Math. Sci.), Vol. 115, No. 4 (2005), pp. 371-381; arXiv preprint, arXiv:math/0512310 [math.AC], 2005.
Courtney Moen, Infinite series with binomial coefficients, Math. Mag. 64 (1) (1991) 53-55.

Cf. A005809, A073010, A075778, A210462, A210463.

A075778neg := proc()
        1/3-root[3](25/2-3*sqrt(69)/2)/3 -root[3](25/2+3*sqrt(69)/2)/3;
end proc:
A210462 := proc()
        local a075778 ;
        a075778 := A075778neg() ;
        (1+1/a075778/(a075778-1))/2 ;
end proc:
A210463 := proc()
        local a075778,a210462 ;
        a075778 := A075778neg() ;
        a210462 := A210462() ;
        -1/a075778-a210462^2 ;
        sqrt(%) ;
end proc:
A210453 := proc()
        local v,x;
        v := 0.0 ;
        for x in [ A075778neg(), A210462()+I*A210463(), A210462()-I*A210463() ] do
                v := v+ x*log(1-1/x)/(3*x-2) ;
        end do:
        evalf(v) ;
end proc:
A210453() ;

RealDigits[ HypergeometricPFQ[{1, 1, 3/2}, {4/3, 5/3}, 4/27]/3, 10, 105] // First (* Jean-François Alcover, Feb 11 2013 *)

Equals Sum_{n>=1} 1/(n*A005809(n)).
Equals Integral_{x=0..1} x^2/(1-x^2+x^3) dx.
Equals Sum_(R) R*log(1-1/R)/(3*R-2) where R is summed over the set of the three constants -A075778, A210462-i*A210463 and A210462-i*A210463, i=sqrt(-1), that is, over the set of the three roots of x^3-x^2+1.
Equals (1/sqrt(23)) * (arctan(sqrt(3)/(2*phi-1)) * 18*phi/(phi^2-phi+1) - log((phi^3+1)/(phi+1)^3) * (3*sqrt(3)*phi*(1-phi))/(phi^3+1)), where phi = ((25+3*sqrt(69))/2)^(1/3) (Batir, 2005, p. 378, eq. (3.2)). - Amiram Eldar, Dec 07 2024

A273984 Decimal expansion of the odd Bessel moment s(5,1) (see the referenced paper about the elliptic integral evaluations of Bessel moments).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 06 2016

			1.07128505542180765851871197803081716076317977716705621702469365995...

David H. Bailey, Jonathan M. Borwein, David Broadhurst and M. L. Glasser, Elliptic integral evaluations of Bessel moments, arXiv:0801.0891 [hep-th], 2008, page 21.

Cf. A073010 (s(3,1)), A121839 (1+s(3,3)), A222068 (s(4,1)), A244854 (2s(4,3)), A273959, A273985 (s(5,3)), A273986 (s(5,5)).

s[5, 1] = NIntegrate[x*BesselI[0, x]*BesselK[0, x]^4, {x, 0, Infinity}, WorkingPrecision -> 105];
RealDigits[s[5, 1]][[1]]

intnumosc(x=0,x*besseli(0,x)*besselk(0,x)^4,Pi) \\ Charles R Greathouse IV, Oct 23 2023

s(5,1) = Integral_{0..inf} x*BesselI_0(x)*BesselK_0(x)^4 dx.

Equals Pi^2 C (conjectural, where C is A273959).

Offset corrected by Rick L. Shepherd, Jun 07 2016

A273985 Decimal expansion of the odd Bessel moment s(5,3) (see the referenced paper about the elliptic integral evaluations of Bessel moments).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 06 2016

			0.0859372906917684524238417457876469580337873779130649806431684669637579...

David H. Bailey, Jonathan M. Borwein, David Broadhurst and M. L. Glasser, Elliptic integral evaluations of Bessel moments, arXiv:0801.0891, page 21.

Cf. A073010 (s(3,1)), A121839 (1+s(3,3)), A222068 (s(4,1)), A244854 (2s(4,3)), A273959, A273984 (s(5,1)), A273986 (s(5,5)).

s[5, 3] = NIntegrate[x^3*BesselI[0, x]*BesselK[0, x]^4, {x, 0, Infinity}, WorkingPrecision -> 103];
Join[{0}, RealDigits[s[5, 3]][[1]]]

s(5,3) = Integral_{0..inf} x^3*BesselI_0(x)*BesselK_0(x)^4 dx.

Equals Pi^2 (2/15)^2 (13 C - 1/(10 C)) (conjectural, where C is A273959).

A273986 Decimal expansion of the odd Bessel moment s(5,5) (see the referenced paper about the elliptic integral evaluations of Bessel moments).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 06 2016

			0.054514253132761880363033919802009596877614349544575913649940264634...

David H. Bailey, Jonathan M. Borwein, David Broadhurst and M. L. Glasser, Elliptic integral evaluations of Bessel moments, arXiv:0801.0891, page 21.

Cf. A073010 (s(3,1)), A121839 (1+s(3,3)), A222068 (s(4,1)), A244854 (2s(4,3)), A273959, A273984 (s(5,1)), A273985 (s(5,3)).

s[5, 5] = NIntegrate[x^5*BesselI[0, x]*BesselK[0, x]^4, {x, 0, Infinity}, WorkingPrecision -> 103];
Join[{0}, RealDigits[s[5, 5]][[1]]]

s(5,5) = Integral_{0..inf} x^5*BesselI_0(x)*BesselK_0(x)^4 dx.

Equals Pi^2 (4/15)^3 (43 C - 19/(40 C)) (conjectural, where C is A273959).

A113448 Expansion of (eta(q^2)^2 * eta(q^9) * eta(q^18)) / (eta(q) * eta(q^6)) in powers of q.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 2, 1, 0, 0, 0, 0, 2, 2, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 1, 2, 0, 2, 0, 0, 2, 1, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 3, 1, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 2, 2, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 2, 3, 0, 1, 0, 0, 2, 2, 0

Offset: 1

Michael Somos, Nov 02 2005

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

			G.f. = x + x^2 + x^4 + 2*x^7 + x^8 + 2*x^13 + 2*x^14 + x^16 + 2*x^19 + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A004016, A005882, A005928, A033687, A073010, A097195.

a[ n_] := If[ n < 1, 0, If[ Mod[n, 3] == 0, 0, DivisorSum[ n, KroneckerSymbol[ -12, #] &]]]; (* Michael Somos, Jul 30 2015 *)
a[ n_] := SeriesCoefficient[ x QPochhammer[ x^9]^3 / QPochhammer[ x^3] + x^2 QPochhammer[ x^18]^3 / QPochhammer[ x^6], {x, 0, n}]; (* Michael Somos, Jul 30 2015 *)

{a(n) = if( n<1, 0, if( n%3, sumdiv(n,d, kronecker(-12, d))))};

{a(n) = if( n<1, 0, direuler(p=2, n, if( p==3, 1, 1 / ((1 - X) * (1 - kronecker(-12, p)*X))))[n])}

{a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 1, p==3, 0, p%6==1, e+1, !(e%2))))};

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^9 + A) * eta(x^18 + A) / (eta(x + A) * eta(x^6 + A)), n))};

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^9 + A)^3 / eta(x^3 + A) + x * eta(x^18 + A)^3 / eta(x^6 + A), n))};

Euler transform of period 18 sequence [ 1, -1, 1, -1, 1, 0, 1, -1, 0, -1, 1, 0, 1, -1, 1, -1, 1, -2, ...].
Moebius transform is period 18 sequence [ 1, 0, -1, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 1, 0, -1, 0, ...].
a(n) is multiplicative with a(2^e) = 1, a(3^e) = 0^e, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1 + (-1)^e)/2 if p == 5 (mod 6).
a(3*n) = 0, a(2*n) = a(n).
G.f.: Sum_{k>0} x^(6*k - 5) / (1 - x^(6*k - 5)) - x^(6*k - 1) / (1 - x^(6*k - 1)) - x^(18*k - 15) / (1 - x^(18*k - 15)) + x^(18*k - 6) / (1 - x^(18*k - 6)).
G.f.: Sum_{k>0} x^k * (1 - x^(2*k)) * (1 - x^(4*k)) * (1-x^(10*k)) / (1 - x^(18*k)).
Expansion of (c(q) + c(q^2))/3 in powers of q^(1/3) where c(q) is a cubic AGM theta function.
a(3*n + 1) = A033687(n). a(6*n + 1) = A097195(n). - Michael Somos, Jul 30 2015
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(3*sqrt(3)) = 0.604599... (A073010). - Amiram Eldar, Oct 15 2022

cp's OEIS Frontend

A381155 Decimal expansion of the isoperimetric quotient of a regular 10-gon.

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A381156 Decimal expansion of the isoperimetric quotient of a regular 11-gon.

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A381157 Decimal expansion of the isoperimetric quotient of a regular 12-gon.

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A381671 Decimal expansion of the isoperimetric quotient of a regular tetrahedron.

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A086464 Decimal expansion of 17/36*zeta(4).

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A210453 Decimal expansion of Sum_{n>=1} 1/(n*binomial(3*n,n)).

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A273984 Decimal expansion of the odd Bessel moment s(5,1) (see the referenced paper about the elliptic integral evaluations of Bessel moments).

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A210453 Decimal expansion of Sum_{n>=1} 1/(nbinomial(3n,n)).