A005980 -id:A005980 - cp's OEIS frontend

A000293 a(n) = number of solid (i.e., three-dimensional) partitions of n.

1, 1, 4, 10, 26, 59, 140, 307, 684, 1464, 3122, 6500, 13426, 27248, 54804, 108802, 214071, 416849, 805124, 1541637, 2930329, 5528733, 10362312, 19295226, 35713454, 65715094, 120256653, 218893580, 396418699, 714399381, 1281403841, 2287986987, 4067428375, 7200210523, 12693890803, 22290727268, 38993410516, 67959010130, 118016656268, 204233654229, 352245710866, 605538866862, 1037668522922, 1772700955975, 3019333854177, 5127694484375, 8683676638832, 14665233966068, 24700752691832, 41495176877972, 69531305679518

Offset: 0

N. J. A. Sloane

An ordinary partition is a row of numbers in nondecreasing order whose sum is n. Here the numbers are in a three-dimensional pile, nondecreasing in the x-, y- and z-directions.
Finding a g.f. for this sequence is an unsolved problem. At first it was thought that it was given by A000294.
Equals A000041 convolved with A002836: [1, 0, 2, 5, 12, 24, 56, 113, ...] and row sums of the convolution triangle A161564. - Gary W. Adamson, Jun 13 2009

			Examples for n=2 and n=3.
a(2) = 4: 2; 11 where the first 1 is at the origin and the second 1 is in the x, y or z direction.
a(3) = 10: 3; 21 where the 2 is at the origin and the 1 is on the x, y or z axis; 111 (a row of 3 ones on the x, y or z axes); and three 1's with one 1 at the origin and the other two 1's on two of the three axes.
From _Gus Wiseman_, Jan 22 2019: (Start)
The a(1) = 1 through a(4) = 26 solid partitions, represented as chains of chains of integer partitions:
  ((1))  ((2))       ((3))            ((4))
         ((11))      ((21))           ((22))
         ((1)(1))    ((111))          ((31))
         ((1))((1))  ((2)(1))         ((211))
                     ((11)(1))        ((1111))
                     ((2))((1))       ((2)(2))
                     ((1)(1)(1))      ((3)(1))
                     ((11))((1))      ((21)(1))
                     ((1)(1))((1))    ((11)(11))
                     ((1))((1))((1))  ((111)(1))
                                      ((2))((2))
                                      ((3))((1))
                                      ((2)(1)(1))
                                      ((21))((1))
                                      ((11))((11))
                                      ((11)(1)(1))
                                      ((111))((1))
                                      ((2)(1))((1))
                                      ((1)(1)(1)(1))
                                      ((11)(1))((1))
                                      ((2))((1))((1))
                                      ((1)(1))((1)(1))
                                      ((1)(1)(1))((1))
                                      ((11))((1))((1))
                                      ((1)(1))((1))((1))
                                      ((1))((1))((1))((1))
(End)

P. A. MacMahon, Memoir on the theory of partitions of numbers - Part VI, Phil. Trans. Roal Soc., 211 (1912), 345-373.
P. A. MacMahon, Combinatory Analysis. Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 332.
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).

Suresh Govindarajan, Table of n, a(n) for n = 0..72
Alimzhan Amanov and Damir Yeliussizov, MacMahon's statistics on higher-dimensional partitions, arXiv:2009.00592 [math.CO], 2020. Mentions this sequence.
A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100. [Annotated scanned copy], DOI
Srivatsan Balakrishnan, Suresh Govindarajan and Naveen S. Prabhakar, On the asymptotics of higher-dimensional partitions, arXiv:1105.6231 [cond-mat.stat-mech], 2011.
P. Bratley and J. K. S. McKay, Algorithm 313: Multi-dimensional partition generator, Comm. ACM, 10 (Issue 10, 1967), p. 666.
Nicolas Destainville and Suresh Govindarajan, Estimating the asymptotics of solid partitions, arXiv:1406.5605 [cond-mat.stat-mech], 2014; J. Stat. Phys. 158 (2015) 950-967.
Suresh Govindarajan, Solid Partitions Project Dec 14, 2010.
D. E. Knuth, A Note on Solid Partitions, Math. Comp. 24, 955-961, 1970.
P. A. MacMahon, Combinatory analysis.
Ville Mustonen and R. Rajesh, Numerical Estimation of the Asymptotic Behaviour of Solid Partitions of an Integer, arXiv:cond-mat/0303607 [cond-mat.stat-mech], 2003; J. Phys. A 36 (2003), no. 24, 6651-6659.
S. P. Naveen, On The Asymptotics of Some Counting Problems in Physics, Thesis, Bachelor of Technology, Department of Physics, Indian Institute of Technology, Madras, May 2011.
Eric Weisstein's World of Mathematics, Solid Partition
Wikipedia, Solid partition
Damir Yeliussizov, Bounds on the number of higher-dimensional partitions, arXiv:2302.04799 [math.CO], 2023.

Cf. A000041, A000219 (2-dim), A000294, A000334 (4-dim), A000390 (5-dim), A002835, A002836, A005980, A037452 (inverse Euler trans.), A080207, A007326, A000416 (6-dim), A000427 (7-dim), A179855 (8-dim).
Cf. A161564. - Gary W. Adamson, Jun 13 2009
Cf. A001970, A002974, A003293, A007713, A114736, A117433, A323657.

planePtns[n_]:=Join@@Table[Select[Tuples[IntegerPartitions/@ptn],And@@(GreaterEqual@@@Transpose[PadRight[#]])&],{ptn,IntegerPartitions[n]}];
solidPtns[n_]:=Join@@Table[Select[Tuples[planePtns/@y],And@@(GreaterEqual@@@Transpose[Join@@@(PadRight[#,{n,n}]&/@#)])&],{y,IntegerPartitions[n]}];
Table[Length[solidPtns[n]],{n,10}] (* Gus Wiseman, Jan 23 2019 *)

More terms from the Mustonen and Rajesh article, May 02 2003
a(51)-a(62) found by Suresh Govindarajan and students, Dec 14 2010
a(63)-a(68) found by Suresh Govindarajan and students, Jun 01 2011
a(69)-a(72) found by Suresh Govindarajan and Srivatsan Balakrishnan, Jan 03 2013

A096494 Largest value in the periodic part of the continued fraction of sqrt(prime(n)).

Original entry on oeis.org

2, 2, 4, 4, 6, 6, 8, 8, 8, 10, 10, 12, 12, 12, 12, 14, 14, 14, 16, 16, 16, 16, 18, 18, 18, 20, 20, 20, 20, 20, 22, 22, 22, 22, 24, 24, 24, 24, 24, 26, 26, 26, 26, 26, 28, 28, 28, 28, 30, 30, 30, 30, 30, 30, 32, 32, 32, 32, 32, 32, 32, 34, 34, 34, 34, 34, 36, 36, 36, 36, 36, 36

Offset: 1

Labos Elemer, Jun 29 2004

nonn

			n=31: prime(31) = 127, and the periodic part is {3,1,2,2,7,11,7,2,2,1,3,22}, so a(31)=22.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000006, A003285, A005980, A054269.
Cf. A096491, A096492, A096493, A096495, A096496.
Cf. A117767.

a096494 = (* 2) . a000006  -- Reinhard Zumkeller, Sep 20 2014

A096491 := proc(n)
if issqr(n) then
sqrt(n) ;
else
numtheory[cfrac](sqrt(n),'periodic','quotients') ;
%[2] ;
max(op(%)) ;
end if;
end proc:
A096494 := proc(n)
option remember ;
A096491(ithprime(n)) ;
end proc: # R. J. Mathar, Mar 18 2010

{te=Table[0, {m}], u=1}; Do[s=Max[Last[ContinuedFraction[Prime[n]^(1/2)]]]; te[[u]]=s;u=u+1, {n, 1, m}];te
a[n_]:=IntegerPart[Sqrt[Prime[n]]] 2 IntegerPart[Sqrt[#]]&/@Prime[Range[90]] (* Vincenzo Librandi, Aug 09 2015 *)

It seems that lim_{n->infinity} a(n)/n = 0. - Benoit Cloitre, Apr 19 2003

a(n) = 2*A000006(n). - Benoit Cloitre, Apr 19 2003

A002836 Let F(x) = 1 + x + 4x^2 + 10x^3 + ... = g.f. for A000293 (solid partitions) and expand (1-x)(1-x^2)(1-x^3)...*F(x) in powers of x.

Original entry on oeis.org

1, 0, 2, 5, 12, 24, 56, 113, 248, 503, 1043, 2080, 4169, 8145, 15897, 30545, 58402, 110461, 207802, 387561, 718875, 1324038, 2425473, 4416193, 7999516, 14411507, 25837198, 46092306, 81851250, 144691532, 254682865, 446399687, 779302305

Offset: 0

N. J. A. Sloane

nonn

Convolved with A000041 = A000293, solid partitions; and left border of the convolution triangle A161564. - Gary W. Adamson, Jun 13 2009

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).

Suresh Govindarajan, Table of n, a(n) for n = 0..72
D. E. Knuth, A Note on Solid Partitions, Math. Comp. 24, 955-961, 1970.
Physics enthusiasts at IIT Madras, The Solid Partitions Project

Cf. A000293, A005980, A000293, A000041, A161564.

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 08 2004

More terms from Srivatsan Balakrishnan and Suresh Govindarajan, Jan 03 2013

A096495 Number of distinct terms in the periodic part of the continued fraction for sqrt(prime(n)).

Original entry on oeis.org

1, 2, 1, 2, 2, 2, 1, 4, 3, 3, 4, 1, 2, 4, 3, 3, 4, 5, 5, 4, 3, 3, 2, 3, 3, 1, 5, 4, 6, 3, 6, 4, 3, 6, 5, 7, 5, 6, 3, 3, 6, 6, 6, 5, 1, 7, 8, 3, 2, 3, 3, 6, 5, 5, 1, 4, 2, 7, 7, 5, 6, 3, 6, 6, 6, 5, 8, 6, 5, 4, 4, 3, 7, 3, 9, 4, 3, 7, 1, 6, 6, 8, 7, 6, 3, 2, 5, 7, 5, 9, 4, 6, 9, 8, 4, 4, 6, 6, 8, 9, 8, 2, 4, 6, 10

Offset: 1

Labos Elemer, Jun 29 2004

nonn

			n = 31: prime(31) = 127, and the periodic part is {3,1,2,2,7,11,7,2,2,1,3,22}, so a(31) = 6.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A003285, A028832, A054269, A005980, A096491, A096492, A096493, A096494, A096496.

{te=Table[0, {m}], u=1}; Do[s=Length[Union[Last[ContinuedFraction[Prime[n]^(1/2)]]]]; te[[u]]=s;u=u+1, {n, 1, m}];te
Table[Length[Union[ContinuedFraction[Sqrt[Prime[n]]][[2]]]],{n,110}] (* Harvey P. Dale, Jun 22 2017 *)

a(n) = A028832(A000040(n)). - Amiram Eldar, Nov 10 2021

A096496 Number of distinct primes in the periodic part of the continued fraction for sqrt(prime(n)).

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 0, 2, 1, 1, 2, 0, 1, 2, 1, 1, 2, 2, 3, 2, 1, 1, 0, 2, 1, 0, 1, 1, 2, 1, 4, 2, 1, 4, 2, 4, 3, 4, 1, 0, 4, 1, 3, 2, 0, 3, 4, 1, 0, 1, 1, 2, 2, 2, 0, 0, 1, 1, 3, 1, 1, 0, 4, 3, 3, 1, 5, 3, 2, 2, 2, 1, 3, 2, 4, 2, 1, 2, 0, 3, 4, 5, 5, 3, 1, 0, 3, 4, 1, 4, 1, 3, 3, 2, 1, 1, 2, 2, 2, 4, 4, 0, 2, 3, 4

Offset: 1

Labos Elemer, Jun 29 2004

nonn

			n=31: prime(31) = 127, and the periodic part of the continued fraction of sqrt(127) is {3,1,2,2,7,11,7,2,2,1,3,22}, so a(31) = 4.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A003285, A054269, A005980, A096491, A096492, A096493, A096494, A096495.

{te=Table[0, {m}], u=1}; Do[s=Count[PrimeQ[Union[Last[ContinuedFraction[f[n]^(1/2)]]]], True]; te[[u]]=s;u=u+1, {n, 1, m}];te
Count[Union[ContinuedFraction[Sqrt[#]][[2]]],?PrimeQ]&/@Prime[ Range[ 110]] (* _Harvey P. Dale, Apr 27 2016 *)

cp's OEIS Frontend

A000293 a(n) = number of solid (i.e., three-dimensional) partitions of n.

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A096494 Largest value in the periodic part of the continued fraction of sqrt(prime(n)).

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A002836 Let F(x) = 1 + x + 4x^2 + 10x^3 + ... = g.f. for A000293 (solid partitions) and expand (1-x)(1-x^2)(1-x^3)...*F(x) in powers of x.

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Crossrefs

Extensions

A096495 Number of distinct terms in the periodic part of the continued fraction for sqrt(prime(n)).

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Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A096496 Number of distinct primes in the periodic part of the continued fraction for sqrt(prime(n)).

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Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica