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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A022471 Length of n-th term of A022470.

Original entry on oeis.org

1, 2, 4, 4, 6, 10, 12, 14, 22, 26, 30, 44, 56, 70, 98, 130, 162, 216, 292, 358, 470, 628, 792, 1050, 1384, 1788, 2334, 3072, 3974, 5162, 6784, 8786, 11420, 14992, 19484, 25388, 33160, 43262, 56252, 73392, 95798, 124496, 162556, 212048, 275976, 360154
Offset: 1

Views

Author

Clark Kimberling

Keywords

Comments

a(n) is the length of the n-th term of many sequences generated by methods A and B, including those shown here:
Method A, 1st term ... Method B, 1st term
A001155, 0
A006751, 2 ......... A022470, 2
A006715, 3 ......... A022499, 3
A001140, 4 ......... A022500, 4
A001141, 5 ......... A022501, 5
A001143, 6 ......... A022502, 6
A001145, 7 ......... A022503, 7
A001151, 8 ......... A022504, 8
A001154, 9 ......... A022505, 9
Clark Kimberling, Jun 14 2013

Links

Crossrefs

Cf. A022470.

Programs

  • Mathematica
    a[0] = 2; a[n_] := a[n] = FromDigits[Flatten[{First[#], Length[#]} & /@   Split[IntegerDigits[a[n - 1]]]]]; Map[Length[IntegerDigits[a[#]]] &, Range[0, 40]] (* Peter J. C. Moses, Jun 14 2013 *)
p = {9, -9, 6, -16, 5, 2, 0, -9, -1, -1, 20, 2, 6, -3, -15, -13, 15, 20, 15, -26, -28, 7, 6, 26, -27, -4, 9, -15, 3, 2, 8, 43, 9, -39, -24, -2, -24, 28, 9, 13, 13, -18, -12, -16, 14, 13, 16, 8, -36, 1, -6, -8, 15, 1, 14, 3, -6, -7, -3, 2, -2, 2, 2, 0, -1, -2, -1, 3, 3, -1, -1, -1}; q = {-6, 9, -9, 18, -16, 11, -14, 8, -1, 5, -7, -2, -8, 14, 5, 5, -19, -3, 6, 7, 6, -16, 7, -8, 22, -17, 12, -7, -5, -7, 8, -4, 7, 9, -13, 4, 6, -14, 14, -19, 7, 13, -2, 4, -18, 0, 1, 4, 12, -8, 5, 0, -8, -1, -7, 8, 5, 2, -3, -3, 0, 0, 0, 0, 2, 1, 0, -3, -1, 1, 1, 1, -1}; gf = Fold[x #1 + #2 &, 0, p]/Fold[x #1 + #2 &, 0, q]; CoefficientList[Series[gf, {x, 0, 100}], x] (* Peter J. C. Moses, Jun 16 2013 *)

A171773 This sequence is a relative of the audioactive sequences. We generate it by starting with a symbol * and describe sequentially: *, 1*, 111*, 311*, 13211*,...

Original entry on oeis.org

1, 111, 311, 13211, 111312211, 31131122211, 1321132132211, 111312211312111322211, 3113112221131112311332211, 13211321322113311213212322211, 1113122113121113222123211211131211121332211, 3113112221131112311332111213122112311311123112112322211
Offset: 1

Views

Author

Louis Hirsch Kauffman (kauffman(AT)uic.edu), Dec 18 2009

Keywords

Comments

The interest of this is that if A_{n} is the n-th term of this sequence then A_{n} is a truncate of A_{n+3}. Thus the sequence gives rise to a triple A,B,C of infinite sequences of 1,2,3 such that B describes A, C describes B and A describes C.
This sequence serves as the initial portion of A001155, A001140, A001141, A001143, A001145, A001151, and A001154, as it is those sequences with the 'seed value' removed. - James E Davis, Apr 28 2016

Examples

			The term after 311 is one-three, two-one, one: i.e. 13211. - _James E Davis_, Apr 28 2016

Links

Crossrefs

Cf. A005150, A001155, A001140, A001141, A001143, A001145, A001151, and A001154

Programs

  • Mathematica
    NestList[FromDigits@ Append[Flatten@ Map[{Length@ #, First@ #} &, Split@ IntegerDigits@ #], 1] &, 1, 10] (* Michael De Vlieger, Apr 28 2016 *)

Formula

Each term can be found by doing a look-and-say on the previous term and appending a 1. - James E Davis, Apr 28 2016

Extensions

More terms from James E Davis, Apr 28 2016

A265849 First differences of A006751.

Original entry on oeis.org

10, 1100, 2000, 129000, 1112990000, 310198100000, 12900010100000, 1113122099909791900000, 31130009089198002000100000, 132082082098921801009009900000, 11131221131211000108018890978199979090100000, 31131122211299991892189900998999891000999919009909900000
Offset: 1

Views

Author

Altug Alkan, Dec 16 2015

Keywords

Comments

Also first differences of A006715, A001140, A001141, A001143, A001145, A001151, A001154. - Michel Marcus, Dec 16 2015
Note that A005150 has really different first differences characteristic because of its initial term that is 1.

Examples

			a(1) = A006751(2) - A006751(1) = 12 - 2 = 10.
a(2) = A006751(3) - A006751(2) = 1112 - 12 = 1100.

Crossrefs

Cf. A006751, A006715, A001140, A001141, A001143, A001145, A001151, A001154.

Programs

  • Mathematica
    f[n_, d_: 1] := NestList[Flatten[Reverse /@ Map[Function[k, Through[{First, Length}@ k]], Split@ #]] &, {d}, n - 1]; Differences@ Array[FromDigits@ f[#, 2][[#]] &, {13}] (* Michael De Vlieger, Jan 03 2016, after Zerinvary Lajos at A006751 *)
  • PARI
    dpt(n) = {vd = []; d = digits(n); nbd = 0; old = -1; for (k=1, #d, if (d[k] == old, nbd ++, if (old != -1, vd = concat(vd, nbd); vd = concat(vd, old);); nbd = 1;); old = d[k];); vd = concat(vd, nbd); vd = concat(vd, old); subst(Pol(vd), x, 10);}
lista(nn, x=2) = {v = vector(nn); v[1] = x; for (n=2, nn, nx = dpt(x); v[n] = nx; x = nx;); vector(nn-1, n, v[n+1] - v[n]);} \\ 2nd param x can any value between 2 and 9 \\ Michel Marcus, Dec 16 2015

Formula

a(n) = A006751(n+1) - A006751(n).
a(n) mod 10^5 = 0, for n > 5.
a(2*n+2) - a(2*n) mod 10^6 = 0, for n > 3.
a(2*n+1) - a(2*n-1) mod 10^7 = 0, for n > 3.
Previous Showing 11-13 of 13 results.

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