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.

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A356956 Numbers k such that the k-th composition in standard order is a gapless interval (in increasing order).

Original entry on oeis.org

0, 1, 2, 4, 6, 8, 16, 20, 32, 52, 64, 72, 128, 256, 272, 328, 512, 840, 1024, 1056, 2048, 2320, 4096, 4160, 8192, 10512, 16384, 16512, 17440, 26896, 32768, 65536, 65792, 131072, 135232, 148512, 262144, 262656, 524288, 672800, 1048576, 1049600, 1065088, 1721376
Offset: 1

Views

Author

Gus Wiseman, Sep 24 2022

Keywords

Comments

An interval such as {3,4,5} is a set of positive integers with all differences of adjacent elements equal to 1.
The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

Examples

			The terms and corresponding intervals begin:
        0: ()
        1: (1)
        2: (2)
        4: (3)
        6: (1,2)
        8: (4)
       16: (5)
       20: (2,3)
       32: (6)
       52: (1,2,3)
       64: (7)
       72: (3,4)
      128: (8)
      256: (9)
      272: (4,5)
      328: (2,3,4)
      512: (10)
      840: (1,2,3,4)

Links

Crossrefs

See link for sequences related to standard compositions.
These compositions are counted by A001227.
An unordered version is A073485, non-strict A073491 (complement A073492).
The initial version is A164894, non-strict A356843 (unordered A356845).
The non-strict version is A356841, initial A333217, counted by A107428.
A066311 lists gapless numbers.
A356230 ranks gapless factorization lengths, firsts A356603.
A356233 counts factorizations into gapless numbers.
A356844 ranks compositions with at least one 1.
Cf. A053251, A055932, A073493, A132747, A137921, A286470, A356224, A356842.

Programs

  • Mathematica
    stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
chQ[y_]:=Length[y]<=1||Union[Differences[y]]=={1};
Select[Range[0,1000],chQ[stc[#]]&]

A379836 Number of pairs of adjacent equal parts in all complete compositions of n.

Original entry on oeis.org

0, 0, 1, 2, 5, 12, 23, 54, 118, 258, 550, 1178, 2540, 5394, 11473, 24174, 51021, 107210, 225099, 471322, 985202, 2055542, 4281847, 8906676, 18500425, 38379246, 79516158, 164561560, 340179441, 702506576, 1449311429, 2987297778, 6151964642, 12658841766, 26027603925
Offset: 0

Views

Author

John Tyler Rascoe, Jan 14 2025

Keywords

Comments

An integer composition is complete if its set of parts covers an initial interval.

Examples

			The complete compositions of n = 4 are: (1,1,2), (1,2,1), (2,1,1), and (1,1,1,1); having a total of 5 pairs of equal adjacent parts giving a(4) = 5.

Crossrefs

Cf. A011782, A106356, A107428, A107429, A373306, A374147, A374726, A377823.

Programs

  • PARI
    C_xz(s,N) = {my(x='x+O('x^N), g=if(#s <1,1, sum(i=1,#s, C_xz(s[^i],N+1) * x^(s[i])/(1-(x^(s[i]))*(z-1)) )/(1-sum(i=1,#s, x^(s[i])/(1-(x^(s[i]))*(z-1)))))); return(g)}
B_xz(N) = {my(x='x+O('x^N), j=1, h=0); while((j*(j+1))/2 <= N, h += C_xz(vector(j, i, i), N+1); j+=1); h}
P_xz(N) = Pol(B_xz(N), {x})
B_x(N) = {my(cx = deriv(P_xz(N),z), z=1); Vecrev(eval(cx))}
B_x(20)

Formula

G.f.: B(x) = d/dz Sum_{k>0} C({1..k},x,z)|{z=1} where C({s},x,z) = Sum{i in {s}} ( C({s}-{i},x,z)*(x^i)/(1-(x^i)*(z-1)) )/(1 - Sum_{i in {s}} (x^i)/(1-(x^i)*(z-1))) with C({},x,z) = 1.

A380176 Number of pairs of adjacent equal parts in all gap-free compositions of n.

Original entry on oeis.org

0, 0, 1, 2, 6, 12, 26, 56, 124, 266, 563, 1204, 2573, 5468, 11559, 24370, 51281, 107720, 225867, 472660, 987378, 2059180, 4287932, 8916624, 18517398, 38406486, 79563118, 164636582, 340308519, 702713844, 1449664783, 2987870476, 6152930738, 12660419370, 26030245642
Offset: 0

Views

Author

John Tyler Rascoe, Jan 14 2025

Keywords

Comments

An integer composition is gap-free if its set of parts covers an interval.

Examples

			The gap-free compositions of n = 4 are: (4), (2,2), (1,1,2), (1,2,1), (2,1,1), and (1,1,1,1); having a total of 6 pairs of equal adjacent parts giving a(4) = 6.

Crossrefs

Cf. A011782, A106356, A107428, A107429, A373306, A374147, A374726, A377823.

Programs

  • PARI
    C_xz(s,N) = {my(x='x+O('x^N), g=if(#s <1,1, sum(i=1,#s, C_xz(s[^i],N+1) * x^(s[i])/(1-(x^(s[i]))*(z-1)) )/(1-sum(i=1,#s, x^(s[i])/(1-(x^(s[i]))*(z-1)))))); return(g)}
B_xz(N) = {my(x='x+O('x^N), j=1, h=0); while((j*(j+1))/2 <= N, for(k=0,N, h += C_xz([(1+k)..(j+k)], N+1)); j+=1); h}
P_xz(N) = Pol(B_xz(N), {x})
B_x(N) = {my(cx = deriv(P_xz(N),z), z=1); Vecrev(eval(cx))}
B_x(20)

Formula

G.f.: B(x) = d/dz Sum_{j>0} Sum_{k>=j} C({j..k},x,z)|{z=1} where C({s},x,z) = Sum{i in {s}} ( C({s}-{i},x,z)*(x^i)/(1-(x^i)*(z-1)) )/(1 - Sum_{i in {s}} (x^i)/(1-(x^i)*(z-1))) with C({},x,z) = 1.
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