A010724 -id:A010724 - cp's OEIS frontend

A002348 Degree of rational Poncelet porism of n-gon.

1, 2, 3, 4, 6, 8, 9, 12, 15, 16, 21, 24, 24, 32, 36, 36, 45, 48, 48, 60, 66, 64, 75, 84, 81, 96, 105, 96, 120, 128, 120, 144, 144, 144, 171, 180, 168, 192, 210, 192, 231, 240, 216, 264, 276, 256, 294, 300, 288, 336, 351, 324, 360, 384, 360, 420, 435, 384, 465

Offset: 3

N. J. A. Sloane

			For a triangle the degree is 1, thus a(3) = 1. - _Michael Somos_, Dec 07 2018

Kerawala, S. M.; Poncelet Porism in Two Circles. Bull. Calcutta Math. Soc. 39, 85-105, 1947.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 3..10000
Eric Weisstein's World of Mathematics, Poncelet's Porism

Cf. A027748, A124010.

Cf. A007434, A209229, A007434, A328407, A007814, A000265, A008683, A010724.

a002348 n = product (zipWith d ps es) * 4 ^ e0 `div` 8 where
   d p e = (p ^ 2 - 1) * p ^ e
   e0 = if even n then head $ a124010_row n else 0
   es = map ((* 2) . subtract 1) $
            if even n then tail $ a124010_row n else a124010_row n
   ps = if even n then tail $ a027748_row n else a027748_row n
-- Reinhard Zumkeller, Mar 18 2012

Poncelet[ n_Integer /; n >= 3 ] := Module[ {p, a, i}, {p, a}=Transpose[ FactorInteger[ n ] ];
If[ p[[1]]==2, 4^a[[1]] Product[ p[[i]]^(2(a[[i]] - 1))(p[[i]]^2 - 1), {i, 2, Length[ p ]} ]/8, (* Else *) Product[ p[[i]]^(2(a[[i]] - 1))(p[[i]]^2 - 1), {i, Length[ p ]} ]/8 ] ]

{a(n) = my(p, e); if( n<3, 0, p=factor(n)~; e=p[2,]; p=p[1,]; if( p[1]==2, 4^e[1], 1) * prod(i=1 + (p[1]==2), length(p), p[i]^(2*(e[i] - 1)) * (p[i]^2 - 1)) / 8)}; /* Michael Somos, Dec 09 1999 */

From Ridouane Oudra, Jul 19 2025: (Start)
a(n) = (1/8) * Sum_{k=0..A007814(n)} A007434(n/2^k).
a(n) = (1/8) * Sum_{d|n} A209229(d)*A007434(n/d).
a(n) = (1/8) * Sum_{d|n} A328407(d)*mu(n/d).
a(n) = (n^2/8) * Prod_{p|n, p prime > 2} (1 - 1/p^2).
a(n) = (1/8) * 4^A007814(n) * A007434(A000265(n)).
a(n) = A007434(n)/A010724(n). (End)

Extended with Mathematica program by Eric W. Weisstein

A176401 Decimal expansion of (6+sqrt(39))/2.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 17 2010

Continued fraction expansion of (6+sqrt(39))/2 is A010724.

			(6+sqrt(39))/2 = 6.12249899919919910292...

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A010493 (decimal expansion of sqrt(39)), A010724 (repeat 6, 8).

A176456 Decimal expansion of (12+2*sqrt(39))/3.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 20 2010

Continued fraction expansion of (12+2*sqrt(39))/3 is A010724 preceded by 8.

			(12+2*sqrt(39))/3 = 8.16333199893226547056...

Cf. A010493 (decimal expansion of sqrt(39)), A010724 (repeat 6, 8).

RealDigits[(12+2*Sqrt[39])/3,10,120][[1]] (* Harvey P. Dale, Mar 08 2018 *)

A374921 Irregular triangle read by rows: T(n,k), n >= 0, k >= 1, in which if n is even then row n lists the first A008619(n) even indexed terms of A027336 otherwise if n is odd then row n lists the first A008619(n) odd indexed terms of A027336.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 2, 4, 1, 1, 3, 6, 1, 2, 4, 8, 1, 1, 3, 6, 11, 1, 2, 4, 8, 15, 1, 1, 3, 6, 11, 20, 1, 2, 4, 8, 15, 26, 1, 1, 3, 6, 11, 20, 35, 1, 2, 4, 8, 15, 26, 45, 1, 1, 3, 6, 11, 20, 35, 58, 1, 2, 4, 8, 15, 26, 45, 75, 1, 1, 3, 6, 11, 20, 35, 58, 96, 1, 2, 4, 8, 15, 26, 45, 75, 121

Offset: 0

Omar E. Pol, Aug 01 2024

The sum of row n equals the number of partitions of n.

			Triangle begins:
  1;
  1;
  1, 1;
  1, 2;
  1, 1, 3;
  1, 2, 4;
  1, 1, 3, 6;
  1, 2, 4, 8;
  1, 1, 3, 6, 11;
  1, 2, 4, 8, 15;
  1, 1, 3, 6, 11, 20;
  1, 2, 4, 8, 15, 26;
  1, 1, 3, 6, 11, 20, 35;
  1, 2, 4, 8, 15, 26, 45;
  1, 1, 3, 6, 11, 20, 35, 58;
  1, 2, 4, 8, 15, 26, 45, 75;
  1, 1, 3, 6, 11, 20, 35, 58, 96;
  1, 2, 4, 8, 15, 26, 45, 75, 121;
  ...
For n = 10 the sum of the 10th row is 1 + 1 + 3 + 6 + 11 + 20 = 42, the same as the number of partitions of 10.

Row sums give A000041.
Row lengths give A008619.
Right border gives A027336.
Columns 1..4: A000012, A000034, A010702, A010724.
Cf. A058695, A058696, A002865, A167430, A176206, A182844, A182845, A336811, A338156.

cp's OEIS Frontend

A002348 Degree of rational Poncelet porism of n-gon.

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A176401 Decimal expansion of (6+sqrt(39))/2.

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A176456 Decimal expansion of (12+2*sqrt(39))/3.

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A374921 Irregular triangle read by rows: T(n,k), n >= 0, k >= 1, in which if n is even then row n lists the first A008619(n) even indexed terms of A027336 otherwise if n is odd then row n lists the first A008619(n) odd indexed terms of A027336.

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