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.

Previous Showing 31-33 of 33 results.

A325269 Number of integer partitions of n with 2 distinct parts or at least 3 parts.

Original entry on oeis.org

0, 0, 0, 2, 3, 6, 9, 14, 20, 29, 40, 55, 75, 100, 133, 175, 229, 296, 383, 489, 625, 791, 1000, 1254, 1573, 1957, 2434, 3009, 3716, 4564, 5602, 6841, 8347, 10142, 12308, 14882, 17975, 21636, 26013, 31184, 37336, 44582, 53172, 63260, 75173, 89133, 105556
Offset: 0

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Author

Gus Wiseman, Apr 18 2019

Keywords

Comments

The Heinz numbers of these partitions are given by A080257.
Partitions with 2 distinct parts are in A002133(n). Partitions with at least 3 parts are in A004250(n). Some partitions are in both subsets, so A002133(n)+A004250(n) >= a(n). - R. J. Mathar, Dec 13 2022

Examples

			The a(1) = 1 through a(8) = 20 partitions:
  (21)   (31)    (32)     (42)      (43)       (53)
  (111)  (211)   (41)     (51)      (52)       (62)
         (1111)  (221)    (222)     (61)       (71)
                 (311)    (321)     (322)      (332)
                 (2111)   (411)     (331)      (422)
                 (11111)  (2211)    (421)      (431)
                          (3111)    (511)      (521)
                          (21111)   (2221)     (611)
                          (111111)  (3211)     (2222)
                                    (4111)     (3221)
                                    (22111)    (3311)
                                    (31111)    (4211)
                                    (211111)   (5111)
                                    (1111111)  (22211)
                                               (32111)
                                               (41111)
                                               (221111)
                                               (311111)
                                               (2111111)
                                               (11111111)

Crossrefs

Cf. A001221, A001222, A001358, A001399, A007774, A008284, A060687, A080257, A090858, A116608, A325244.
Cf. A000041, A002133, A004250.

Programs

  • Maple
    A325269 := proc(n)
    local a,p,s ;
    a := 0 ;
    for p in combinat[partition](n) do
        s := convert(p,set) ;
        if nops(p) >= 3 or nops(s) = 2 then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc:
seq(A325269(n),n=0..40) ; # R. J. Mathar, Dec 13 2022
  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Length[Union[#]]==2||Length[#]>2&]],{n,0,30}]

Formula

conjecture: a(n) = A000041(n) - A000034(n-1), n>0. - R. J. Mathar, Dec 13 2022

A337507 Number of length-n sequences covering an initial interval of positive integers with exactly two maximal anti-runs, or with one pair of adjacent equal parts.

Original entry on oeis.org

0, 0, 1, 4, 24, 176, 1540, 15672, 181916, 2372512, 34348932, 546674120, 9486840748, 178285201008, 3607174453844, 78177409231768, 1806934004612220, 44367502983673664, 1153334584544496676, 31643148872573831016
Offset: 0

Views

Author

Gus Wiseman, Sep 06 2020

Keywords

Comments

An anti-run is a sequence with no adjacent equal parts. For example, the maximal anti-runs in (3,1,1,2,2,2,1) are ((3,1),(1,2),(2),(2,1)). In general, there is one more maximal anti-run than the number of pairs of adjacent equal parts.

Examples

			The a(4) = 24 sequences:
  (2,1,2,2)  (2,1,3,3)  (3,1,2,2)
  (2,2,1,2)  (2,3,3,1)  (3,2,2,1)
  (1,2,2,1)  (3,3,1,2)  (1,1,2,3)
  (2,1,1,2)  (3,3,2,1)  (1,1,3,2)
  (1,1,2,1)  (1,2,2,3)  (2,1,1,3)
  (1,2,1,1)  (1,3,2,2)  (2,3,1,1)
  (1,2,3,3)  (2,2,1,3)  (3,1,1,2)
  (1,3,3,2)  (2,2,3,1)  (3,2,1,1)

Crossrefs

A002133 is the version for runs in partitions.
A106357 is the version for compositions.
A337506 has this as column k = 2.
A000670 counts patterns.
A005649 counts anti-run patterns.
A003242 counts anti-run compositions.
A106356 counts compositions by number of maximal anti-runs.
A124762 counts adjacent equal terms in standard compositions.
A124767 counts maximal runs in standard compositions.
A238130/A238279/A333755 count maximal runs in compositions.
A333381 counts maximal anti-runs in standard compositions.
A333382 counts adjacent unequal terms in standard compositions.
A333489 ranks anti-run compositions.
A333769 gives maximal run lengths in standard compositions.
A337565 gives maximal anti-run lengths in standard compositions.
Cf. A019472, A052841, A060223, A106351, A269134, A335461, A337505, A337564.

Programs

  • Mathematica
    kv=2;
allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@Permutations/@allnorm[n],Length[Split[#,UnsameQ]]==kv&]],{n,0,6}]

Formula

a(n > 0) = (n - 1)*A005649(n - 2).

A377812 Number of quadruples of positive integers (x,y,a,b) such that a < b, gcd(a,b) = gcd(x,y) = 1 and a*x + b*y = n.

Original entry on oeis.org

0, 0, 1, 2, 5, 4, 11, 9, 15, 12, 27, 14, 37, 22, 32, 31, 59, 26, 71, 38, 58, 48, 97, 42, 99, 62, 93, 68, 141, 48, 157, 91, 120, 94, 150, 78, 207, 112, 154, 108, 241, 84, 259, 138, 170, 150, 295, 116, 289, 144, 232, 178, 353, 136, 304, 188, 274, 210, 413, 132
Offset: 1

Views

Author

Anshveer Bindra, Nov 08 2024

Keywords

Comments

Number of partitions of n into parts with exactly two different sizes, the sizes being relatively prime and also the multiplicities of the two part sizes being relatively prime. - Andrew Howroyd, Nov 10 2024

Crossrefs

Cf. A002133, A274108.

Programs

  • PARI
    a(n)={sum(b=2, n-1, sum(y=1, (n-1)\b, my(s=n-b*y); sumdiv(s, a, aAndrew Howroyd, Nov 10 2024
  • PARI
    seq(n)={my(v=Vec(sum(k=1, n-1, numdiv(k)*x^k, O(x^n))^2, -n), u=vector(n, n, moebius(n))); dirmul(dirmul(u,u), vector(#v, n, v[n]+numdiv(n)-sigma(n))/2)} \\ Andrew Howroyd, Nov 10 2024
  • Python
    def a(n):
    count = 0
    for a in range(1, n+1):
        for b in range(a + 1, n+1):
            if gcd(a, b) == 1:
                for x in range(1, n+1):
                    for y in range(1, n+1):
                        if gcd(x, y) == 1 and a * x + b * y == n:
                            count += 1
    return count
print([a(n) for n in range(1,21)])
  • Python
    from math import gcd
from sympy import divisors
def A377812(n): return sum(1 for ax in range(1,n-1) for a in divisors(ax,generator=True) for b in divisors(n-ax,generator=True) if aChai Wah Wu, Dec 11 2024

Formula

Moebius transform of A274108. - Andrew Howroyd, Nov 10 2024

Extensions

a(21) onwards from Andrew Howroyd, Nov 10 2024
Previous Showing 31-33 of 33 results.

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