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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A137747 Number of different strings of length n obtained from "abcdefg" by iteratively duplicating any substring.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 7, 34, 143, 555, 2050, 7343, 25809, 89642, 308986, 1059786, 3623524, 12365973, 42160774, 143701920, 489891138, 1670965268, 5703849531, 19488123707, 66652727622, 228212500386, 782258463295, 2684464903407, 9222805414564, 31722184749945, 109232421818064
Offset: 0

Views

Author

M. F. Hasler, Feb 10 2008

Keywords

Comments

See A137743 for more comments.

Examples

			a(k) = 0 for k<7, since no shorter string can be obtained by duplication of substrings.
a(7) = 1 = #{abcdefg},
a(8) = 7 = #{aabcdefg, abbcdefg, abccdefg, abcddefg, abcdeefg, abcdeffg, abcdefgg},
a(9) = 8*(8+1)/2-2 = 34:
for each letter we have one string of the form aaabcdefg;
for each 2-element subset {a,b}, {a,c}, ... we have the string with each of these two letters duplicated (i.e., aabbcdefg, aabccdefg, ...);
and for each of ab,bc,cd,...,fg we have the string with this substring duplicated (ababcdefg,...,abcdefgfg).
(See A137746 for the pattern.)

Links

Crossrefs

Cf. A137740-A137743, A135473, A137744-A137748.

Programs

Extensions

a(15)-a(20) from Lars Blomberg, Jan 12 2013
a(21) from Michael S. Branicky, Jan 07 2021
a(22)-a(23) from Bert Dobbelaere, Jun 11 2024
a(24)-a(32) from Martin Fuller, Jun 08 2025

A135475 Sorted list of strings that can be obtained by starting with 123 and repeatedly doubling any substring in place.

Original entry on oeis.org

123, 1123, 1223, 1233, 11123, 11223, 11233, 12123, 12223, 12233, 12323, 12333, 111123, 111223, 111233, 112123, 112223, 112233, 112323, 112333, 121123, 121223, 121233, 122123, 122223, 122233, 122323, 122333, 123123, 123223, 123233, 123323, 123333, 1111123
Offset: 1

Views

Author

N. J. A. Sloane, Feb 22 2008, Mar 16 2008

Keywords

Comments

The list contains A135473(n) strings of length n.

Links

Crossrefs

Cf. A135473, A130838, A135479, A135017.

A135156 a(n) = number of strings of length n that can be obtained by starting with abc and repeatedly doubling any substring of length >= 2 in place.

Original entry on oeis.org

1, 0, 2, 0, 3, 1, 4, 0, 5, 2, 6, 1, 7, 3, 8, 9, 9, 7, 10, 34, 11, 36, 12, 136, 13, 190, 14, 567, 15, 1018, 16, 2445, 17, 5474, 18, 11371, 19, 28233, 20, 57961, 21, 143391, 22, 308793, 23, 740519, 24, 1668316
Offset: 0

Views

Author

David Applegate and N. J. A. Sloane, Feb 15 2008

Keywords

Comments

A "weakly primitive" version of A135473. Cf. A135017.
Differs from A135017 in that the strings may contain repeated letters.

Links

A135157 Number of primitive (no repeated characters) and irreducible (not the concatenation of generatable strings) strings obtained from abc by iterated repetition of substrings in place.

Original entry on oeis.org

1, 0, 2, 0, 3, 1, 4, 0, 5, 2, 6, 0, 7, 3, 8, 0, 9, 6, 10, 0, 11, 12, 12, 4, 13, 24, 14, 23, 15, 54, 16, 91, 17, 149, 18, 326, 19, 484, 20, 1144, 21, 1741, 22, 4015, 23, 6589, 24, 14296, 25, 25299, 26, 52077, 27, 97264, 28, 193781, 29, 373996, 30, 733076
Offset: 0

Views

Author

David Applegate and N. J. A. Sloane, Feb 15 2008

Keywords

Links

Crossrefs

Cf. A135473, A135156.

A137738 Coefficients of the polynomial giving the n-th diagonal of A137743 * n!, read as an upper right triangle.

Original entry on oeis.org

1, 0, 1, -2, 3, 1, -24, 14, 9, 1, -288, 54, 95, 18, 1, -4320, -136, 1110, 315, 30, 1, -80640, -12300, 15064, 5775, 775, 45, 1
Offset: 0

Views

Author

M. F. Hasler, Mar 18 2008

Keywords

Comments

Let T(m,n) = number of different strings of length n obtained from "123...m" by iteratively duplicating any substring (A137743).
It can be shown that for given k>=0 and all n>k-2, T(n,n+k) = (1/k!) P[k](n), where P[k] is a monic polynomial of degree k with integer coefficients, whose coefficients are the k-th column of the upper right triangular array defined by present sequence A137738.
Here rows and columns start at 0 (which also motivated the chosen offset 0), i.e. a(0) = 1 means P[0] = 1, a(1..2) = 0,1 means P[1] = 0 + 1*X, a(3..6) = -2,3,1 means P[2] = -2 + 3*X + 1*X^2, etc.

Examples

			We have the following formulas for T(m,n) as given in A137743:
T(n,n) = 1, T(n,n+1) = n, T(n,n+2) = (n+1)(n+2)/2 - 2,
T(n,n+3) = A137742 = (1/6)*(n-1)*(n+6)*(n+4) for n>1, for n=1 this formula gives 0 instead of 1.
T(n,n+4) = A137741 = (1/24)*(n+3)*(n^3+15*n^2+50*n-96) for n>2, for n=2 this gives 15 instead of 16.
T(n,n+5) = A137740 = (1/5!)*(n+4)*(n^2+3*n-8)*(n^2+23*n+150)+4 for n>3, for n=3 this gives 137 instead of 138.
T(n,n+6) = A137739 = (1/6!)*(n+9)*(n^5+36*n^4+451*n^3+1716*n^2-380*n-8880)-1 for n>4, for n=4 this gives 1013 instead of 1014.
They satisfy the following relations:
T(n,n+2) = sum( T(k,k+1), k=0..n+1) - 2
T(n,n+3) = sum( T(k,k+2), k=1..n+1) - 5
T(n,n+4) = sum( T(k,k+3), k=2..n+1) - 12 - n
T(n,n+5) = sum( T(k,k+4), k=3..n+1) - 21 - 7n/2 - n^2/2
T(n,n+6) = sum( T(k,k+5), k=4..n+1) + 49 - 25n/3 - 5n^2/2 - n^3/6

Links

Crossrefs

Cf. A137739-A137743, A135473, A137744-A137748.

Formula

P[k](n) = k! T(n,n+k) for k>=0 and positive n>k-2, where T(m,n) is given in A137743.
P[k] = X^k + A045943(k) X^(k-1) + O(X^(k-2)) for k>=1.
For m>0, T(n,n+m+3) = sum( T(k,k+m+2), k=m+1..n+1) - (1/m!) Q[m](n), where Q[m] is a monic polynomial of degree <= m with integer coefficients (conjectured - see examples).
Previous Showing 11-15 of 15 results.

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

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