cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A008676 Expansion of 1/((1-x^3)*(1-x^5)).

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 4, 3, 4, 4, 3, 4, 4, 4, 4, 4, 4, 4, 5, 4, 4, 5, 4, 5, 5, 4, 5, 5, 5, 5, 5, 5, 5, 6, 5, 5
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

a(n) gives the number of partitions of n using only the parts 3 and 5. e.g. a(25)=2: 5+5+5+5+5 and 5+5+3+3+3+3+3+3. - Andrew Baxter, Jun 20 2011
a(n) gives the number of partitions of n+8 involving both a 3 and a 5. e.g. a(25)=2 and we may write 33 as 5+5+5+5+5+5+3 and 5+5+5+3+3+3+3+3+3. 11*3 doesn't count as no 5 is involved. - Jon Perry, Jul 03 2004
Conjecture: a(n) = Floor(2*(n + 3)/3) - Floor(3*(n + 3)/5). - John W. Layman, Sep 23 2009
Also, it appears that a(n) gives the number of distinct multisets of n-1 integers, each of which is -2, +3, or +4, such that the sum of the members of each multiset is 2. E.g., for n=5, the multiset {-2,-2,3,3}, and no others, of n-1=4 members, sums to 2, so a(5)=1. - John W. Layman, Sep 23 2009
Appears to be the number of ordered triples summing to n such that 2x = 3y + 4z, ranked by A357489. An unordered version appears to be A357849, ranked by A358102. - Gus Wiseman, Nov 04 2022

Links

Crossrefs

Cf. A103221.

Programs

  • GAP
    a:=[1,0,0,1,0,1,1,0];; for n in [9..100] do a[n]:=a[n-3]+a[n-5]-a[n-8]; od; a; # G. C. Greubel, Sep 08 2019
  • Magma
    R:=PowerSeriesRing(Integers(), 100); Coefficients(R!( 1/((1-x^3)*(1-x^5)) )); // G. C. Greubel, Sep 08 2019
  • Maple
    a := proc (n) option remember; if n < 0 then return 0 elif n = 0 then return 1 else return a(n-3)+a(n-5)-a(n-8) end if end proc
  • Mathematica
    CoefficientList[Series[1/((1-x^3)(1-x^5)), {x, 0, 100}], x] (* Vincenzo Librandi, Jun 23 2013 *)
  • PARI
    Vec(O(x^99)+1/(1-x^3)/(1-x^5)) \\ Charles R Greathouse IV, Jun 20 2011
  • Sage
    def A008676_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(1/((1-x^3)*(1-x^5))).list()
A008676_list(100) # G. C. Greubel, Sep 08 2019

Formula

G.f.: 1/( (1-x^3) * (1-x^5) ).
a(n) = a(n-3) + a(n-5) - a(n-8), a(0)=a(3)=a(5)=a(6)=1, a(1)=a(2)=a(4) =a(6)=a(7)=0.
a(n) = floor((2*n+5)/5) - floor((n+2)/3). - Tani Akinari, Aug 07 2013

Extensions

Edited by Andrew Baxter, Jun 20 2011
Typo in name fixed by Vincenzo Librandi, Jun 23 2013

A357849 Number of integer partitions (w,x,y) summing to n such that 2w = 3x + 4y.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 3, 2, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3
Offset: 0

Views

Author

Gus Wiseman, Nov 02 2022

Keywords

Examples

			The partitions for n = 34, 64, 89, 119, 144:
  (21,10,3)  (39,22,3)  (54,32,3)   (72,44,3)   (87,54,3)
             (40,16,8)  (55,26,8)   (73,38,8)   (88,48,8)
                        (56,20,13)  (74,32,13)  (89,42,13)
                                    (75,26,18)  (90,36,18)
                                                (91,30,23)

Crossrefs

Partitions are counted by A000041, strict A000009.
The ordered version appears to be A008676, ranked by A357489.
These partitions are ranked by A358102.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n,{3}],2*#[[1]]==3*#[[2]]+4*#[[3]]&]],{n,0,100}]
  • Python
    def A357849(n): return sum(1 for y in range(1,n-1) if (m:=2*n-6*y)>=5*y and 5*(n-y)>=2*m and m%5==0) # Chai Wah Wu, Nov 02 2022

A358102 Numbers of the form prime(w)*prime(x)*prime(y) with w >= x >= y such that 2w = 3x + 4y.

Original entry on oeis.org

66, 153, 266, 609, 806, 1295, 1599, 1634, 2107, 3021, 3055, 3422, 5254, 5369, 5795, 5829, 7138, 8769, 9443, 9581, 10585, 10706, 12337, 12513, 13298, 16465, 16511, 16849, 17013, 18602, 21983, 22145, 23241, 23542, 26159, 29014, 29607, 29945, 30943, 32623, 32809
Offset: 1

Views

Author

Gus Wiseman, Nov 02 2022

Keywords

Comments

Also Heinz numbers of integer partitions (w,x,y) summing to n such that 2w = 3x + 4y, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Examples

			The terms together with their prime indices begin:
     66: {1,2,5}
    153: {2,2,7}
    266: {1,4,8}
    609: {2,4,10}
    806: {1,6,11}
   1295: {3,4,12}
   1599: {2,6,13}
   1634: {1,8,14}
   2107: {4,4,14}
   3021: {2,8,16}
   3055: {3,6,15}
   3422: {1,10,17}
   5254: {1,12,20}
   5369: {4,6,17}
   5795: {3,8,18}
   5829: {2,10,19}
   7138: {1,14,23}
   8769: {2,12,22}

Crossrefs

The ordered version is A357489, apparently counted by A008676.
These partitions are counted by A357849.
A000040 lists the primes.
A000041 counts partitions, strict A000009.
A003963 multiplies prime indices.
A056239 adds up prime indices.
Cf. A000720, A001221, A001222, A215366, A296150, A318283.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],PrimeOmega[#]==3&&2*primeMS[#][[-1]]==3*primeMS[#][[-2]]+4*primeMS[#][[-3]]&]
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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