A059817 -id:A059817

Search: a059817 -id:a059817

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A059818

Let s_n be the simplex packing n-width for the manifold torus X square; sequence gives denominator of s_n/Pi.

+10
7

1, 1, 3, 7, 2, 2, 2, 2, 7, 37, 5

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

OFFSET

1,3

REFERENCES

F. Miller Maley et al., Symplectic packings in cotangent bundles of tori, Experimental Mathematics, 9 (No. 3, 2000), 435-455.

LINKS

Table of n, a(n) for n=1..11.

EXAMPLE

1, 1, 2/3, 4/7, 1/2, 1/2, 1/2, 1/2, 3/7, 15/37, 2/5, ...

CROSSREFS

Cf. A059812 - A059817.

KEYWORD

nonn,frac

AUTHOR

N. J. A. Sloane, Feb 24 2001

EXTENSIONS

It is conjectured that the sequence continues 3/8, 4/11, 7/20, 15/44, 46/137, 1/3, 1/3, ...

Edited by N. J. A. Sloane, May 23 2014

STATUS

approved

A059812

Let g_n be the ball packing n-width for the manifold torus X interval; sequence gives denominator of (g_n/Pi)^2.

+10
5

1, 4, 4, 4, 25, 25, 64, 289, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64

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

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..64.

F. Miller Maley et al., Symplectic packings in cotangent bundles of tori, Experimental Mathematics, 9 (No. 3, 2000), 435-455.

Index entries for linear recurrences with constant coefficients, signature (2,-1).

FORMULA

From Colin Barker, Nov 06 2019: (Start)

G.f.: x*(1 + 2*x - 3*x^2 + 21*x^4 - 21*x^5 + 39*x^6 + 186*x^7 - 505*x^8 + 281*x^9) / (1 - x)^2.

a(n) = 2*a(n-1) - a(n-2) for n>10.

a(n) = n for n>8.

(End)

EXAMPLE

1, 1/4, 1/4, 1/4, 4/25, 4/25, 9/64, 36/289, 1/9, 1/10, ...

CROSSREFS

Cf. A059811, A059813, A059814, A059815, A059816, A059817, A059818.

KEYWORD

nonn,frac,easy

AUTHOR

N. J. A. Sloane, Feb 24 2001

EXTENSIONS

Edited by N. J. A. Sloane, May 23 2014

STATUS

approved

A059811

Let g_n be the ball packing n-width for the manifold torus X interval; sequence gives numerator of (g_n/Pi)^2.

+10
3

1, 1, 1, 1, 4, 4, 9, 36, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

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

OFFSET

1,5

LINKS

Table of n, a(n) for n=1..93.

F. Miller Maley et al., Symplectic packings in cotangent bundles of tori, Experimental Mathematics, 9 (No. 3, 2000), 435-455.

Index entries for linear recurrences with constant coefficients, signature (1).

FORMULA

From Colin Barker, Nov 06 2019: (Start)

G.f.: x*(1 + 3*x^4 + 5*x^6 + 27*x^7 - 35*x^8) / (1 - x).

a(n) = a(n-1) for n>9.

a(n) = 1 for n>8.

(End)

EXAMPLE

1, 1/4, 1/4, 1/4, 4/25, 4/25, 9/64, 36/289, 1/9, 1/10, ...

CROSSREFS

Cf. A059812, A059813, A059814, A059815, A059816, A059817, A059818.

KEYWORD

nonn,frac,easy

AUTHOR

N. J. A. Sloane, Feb 24 2001

EXTENSIONS

Edited by N. J. A. Sloane, May 23 2014

STATUS

approved

A059815

Let g_n be the ball packing n-width for the manifold torus X square; sequence gives numerator of (g_n/Pi)^2.

+10
3

1, 1, 4, 4, 9, 16, 64, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1

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

OFFSET

1,3

LINKS

Table of n, a(n) for n=1..86.

F. Miller Maley et al., Symplectic packings in cotangent bundles of tori, Experimental Mathematics, 9 (No. 3, 2000), 435-455.

Index entries for linear recurrences with constant coefficients, signature (0, 1).

FORMULA

From Colin Barker, Nov 06 2019: (Start)

G.f.: x*(1 + x + 3*x^2 + 3*x^3 + 5*x^4 + 12*x^5 + 55*x^6 - 15*x^7 - 62*x^8) / ((1 - x^2)).

a(n) = a(n-2) for n>=10.

a(n) = (3 - (-1)^n) / 2 for n>=8.

(End)

a(n) / A059816(n) = 2 / n, for n >= 8 [from Maley et al.]. - Sean A. Irvine, Oct 11 2022

EXAMPLE

1, 1, 4/9, 4/9, 9/25, 16/49, 64/225, 1/4, ...

CROSSREFS

Cf. A059812, A059813, A059814, A059816, A059817, A059818.

KEYWORD

nonn,frac

AUTHOR

N. J. A. Sloane, Feb 24 2001

EXTENSIONS

Edited by N. J. A. Sloane, May 23 2014

Duplicated a(8) removed and entry revised by Sean A. Irvine, Oct 11 2022

STATUS

approved

A059816

Let g_n be the ball packing n-width for the manifold torus X square; sequence gives denominator of (g_n/Pi)^2.

+10
3

1, 1, 9, 9, 25, 49, 225, 4, 9, 5, 11, 6, 13, 7, 15, 8, 17, 9, 19, 10, 21, 11, 23, 12, 25, 13, 27, 14, 29, 15, 31, 16, 33, 17, 35, 18, 37, 19, 39, 20, 41, 21, 43, 22, 45, 23, 47, 24, 49, 25, 51, 26, 53, 27, 55, 28, 57, 29, 59, 30, 61, 31, 63, 32, 65, 33, 67, 34

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

OFFSET

1,3

LINKS

Table of n, a(n) for n=1..68.

F. Miller Maley et al., Symplectic packings in cotangent bundles of tori, Experimental Mathematics, 9 (No. 3, 2000), 435-455.

FORMULA

For n>=8, a(2n+1) = 2n+1, a(2n) = n. - Ralf Stephan, May 29 2004

From Colin Barker, Nov 06 2019: (Start)

G.f.: x*(1 + x + 7*x^2 + 7*x^3 + 8*x^4 + 32*x^5 + 184*x^6 - 85*x^7 - 416*x^8 + 46*x^9 + 218*x^10) / ((1 - x)^2*(1+x)^2).

a(n) = 2*a(n-2) - a(n-4) for n>=11.

a(n) = (1/4)*(3 - (-1)^n)*(n-1) for n>=8.

(End)

A059815(n) / a(n) = 2 / n, for n >= 8 [from Maley et al.]. - Sean A. Irvine, Oct 11 2022

EXAMPLE

1, 1, 4/9, 4/9, 9/25, 16/49, 64/225, 1/4, ...

CROSSREFS

Cf. A059812, A059813, A059814, A059815, A059817, A059818.

KEYWORD

nonn,frac

AUTHOR

N. J. A. Sloane, Feb 24 2001

EXTENSIONS

Edited by N. J. A. Sloane, May 23 2014

Duplicated a(8) removed and entry revised by Sean A. Irvine, Oct 11 2022

STATUS

approved

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