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.

A232739 Iterates of Hofstadter's A030124: start with a(1) = A030124(1) = 2, thereafter a(n) = A030124(a(n-1)).

Original entry on oeis.org

2, 4, 6, 9, 13, 17, 22, 28, 34, 41, 49, 58, 67, 77, 88, 100, 112, 125, 139, 154, 169, 185, 202, 220, 239, 258, 278, 299, 321, 344, 367, 391, 416, 442, 469, 497, 525, 554, 584, 615, 647, 680, 713, 747, 782, 818, 855, 893, 931, 970, 1010, 1051, 1093, 1136, 1179
Offset: 1

Views

Author

Antti Karttunen, Dec 04 2013

Keywords

Comments

Is the ratio A005228(n)/a(n) converging towards 1 or some larger value? (Cf. the graph drawn by the OEIS Server's plot2-link).
Cf. also M. F. Hasler's comment in A030124.

Links

Crossrefs

Cf. A005228, A030124, A232751, A232752, A000225, A225850.
Concerning the ratio A005228/A232739 see also A232740, A232750, A232753.

Formula

a(1) = 2, and for n> 1, a(n) = A030124(a(n-1)).
For all n >= 1, A232751(a(n)) = A000225(n+1) and a(n) = A232752(A000225(n+1)) [This is just a consequence of how the permutation pair A232751/A232752 has been defined].
For all n >= 1, a(n) = A225850(a(n+1))/2.

A232746 n occurs A030124(n) times; a(n) = one less than the least k such that A005228(k) > n.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Nov 30 2013

Keywords

Comments

The characteristic function for Hofstadter's A005228 is given by X_A005228(1)=1, and for n>1, X_A005228(n) = a(n)-a(n-1).
The characteristic function for Hofstadter's A030124 is given by X_A030124(1)=0, and for n>1, X_A030124(n) = 1-(a(n)-a(n-1)).
Useful when computing A232747, A232748, A232750 & A225850.

Links

Crossrefs

Cf. A005228, A030124, A232747, A232748, A232750, A232753, A225850.

Programs

  • Mathematica
    nmax = 100; A5228 = {1}; Module[{d = 2, k = 1}, Do[While[MemberQ[A5228, d], d++]; k += d; d++; AppendTo[A5228, k], {n, 1, nmax}]];
a[n_] := For[k = 1, True, k++, If[A5228[[k]] > n, Return[k-1]]];
Array[a, nmax] (* Jean-François Alcover, Dec 09 2021 *)

Formula

a(n) = one less than the least k such that A005228(k) > n.

A232740 a(n) = Number of terms of A232739 which occur between each consecutive terms of A005228, in range A005228(n)..A005228(n+1).

Original entry on oeis.org

1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1
Offset: 1

Views

Author

Antti Karttunen, Dec 04 2013

Keywords

Comments

Do any other values appear than 1 and 2? The 2's seem to be getting rarer, as zeros correspondingly get rarer in A232750. This has some implications about how the ratio A005228(n)/A232739(n) will develop. Please see also the comments and graph-drawing link in A232739.

Examples

			The two sequences begin as:
A005228: 1, 3,    7, 12,      18,   26,     35,  45,   56,      69,   83, ...
A232739:  2,  4,6,  9,  13,17,   22,   28,34,  41,  49,   58,67,   77,   ...
Grouping together the terms of A232739 that occur between two successive terms of A232739, we get {2}, {4,6}, {9}, {13,17}, {22}, {28,34}, {41}, {49}, {58,67}, {77}, ... and counting how many terms are in each such group, we get 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, ..., the first terms of this sequence.

Links

Crossrefs

Cf. A005228, A232739, A232750, A232753.

Programs

Formula

a(n) = A232753(A005228(n+1)) - A232753(A005228(n)).
Showing 1-3 of 3 results.

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