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 1-5 of 5 results.

A165137 a(n) is the number of patterns for n-character papaya words in an infinite alphabet.

Original entry on oeis.org

1, 1, 2, 4, 10, 21, 50, 99, 250, 454, 1242, 2223, 6394, 11389, 35002, 62034, 202010, 359483, 1233518, 2203507, 7944110, 14249736, 53811584, 96912709, 382289362, 691110821, 2841057442, 5154280744, 22033974854, 40105797777, 177946445580
Offset: 0

Views

Author

Sergei Bernstein and Tanya Khovanova, Sep 04 2009

Keywords

Comments

Papaya words are concatenations of two palindromes or palindromes themselves. A165136 is the number of papaya patterns for n-digit numbers. Thus a(n) coincides with A165136 for small n, and is greater than A165136 for larger n. The actual number of n-digit papaya numbers is A165135.
The first 19 terms of this sequence are the same as in A165136. A165137(20) = A165136(20)+10. - Tanya Khovanova, Oct 01 2009, corrected by Franklin T. Adams-Watters, Apr 10 2011

Examples

			There are two types of two-digit papaya numbers: aa, or ab. Hence a(2) = 2. There are four types of three-digit papaya numbers: aaa, aab, aba, abb. Hence a(3) = 4.

Links

Crossrefs

Cf. A165136, A165135, A165610, A165611, A188792, A007055.

Programs

  • Mathematica
    R[k_?EvenQ] := (1/2)*k*(BellB[1 + k/2] + BellB[k/2]);
R[k_?OddQ] := k*BellB[1 + (k - 1)/2];
a[0] = 1; a[n_] := a[n] = R[n] - Sum[EulerPhi[n/d]*a[d], {d, Most[Divisors[ n]]}];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Oct 08 2017, after Andrew Howroyd *)
  • Python
    from functools import lru_cache
from sympy import bell, totient, proper_divisors
@lru_cache(maxsize=None)
def A165137(n): return (n*bell((n>>1)+1) if n&1 else (a:=n>>1)*(bell(a)+bell(a+1)))-sum(totient(n//d)*A165137(d) for d in proper_divisors(n,generator=True)) if n else 1 # Chai Wah Wu, Feb 19 2024

Formula

a(n) = R(n) - Sum_{d|n,dA000110(k). - Andrew Howroyd, Mar 29 2016

Extensions

a(0) and a(7)-a(14) from Franklin T. Adams-Watters, Apr 10 2011
a(15)-a(30) from Andrew Howroyd, Mar 29 2016

A284873 Array read by antidiagonals: T(n,k) = number of double palindromes of length n using a maximum of k different symbols.

Original entry on oeis.org

1, 2, 1, 3, 4, 1, 4, 9, 8, 1, 5, 16, 21, 16, 1, 6, 25, 40, 57, 32, 1, 7, 36, 65, 136, 123, 52, 1, 8, 49, 96, 265, 304, 279, 100, 1, 9, 64, 133, 456, 605, 880, 549, 160, 1, 10, 81, 176, 721, 1056, 2125, 1768, 1209, 260, 1, 11, 100, 225, 1072, 1687, 4356, 4345, 4936, 2127, 424, 1
Offset: 1

Views

Author

Andrew Howroyd, Apr 04 2017

Keywords

Comments

A double palindrome is a concatenation of two palindromes.
Also, number of words of length n using a maximum of k different symbols that are rotations of their reversals.
The sequence A165135 (number of n-digit positive papaya numbers) is 9/10 of the value of column 10.
All rows are polynomials of degree 1 + floor(n/2). - Andrew Howroyd, Oct 10 2017
From Petros Hadjicostas, Oct 27 2017: (Start)
Following Kemp (1982), we note that the formula by A. Howroyd below is equivalent to r(n,k) = Sum_{d|n} phi(d)*T(n/d,k), where r(2d, k) = d*(k+1)*k^d and r(2d+1, k) = (2d+1)*k^(d+1). Inverting (according to the theory of Dirichlet convolutions) we get T(n,k) = Sum_{d|n} phi^{(-1)}(d)*r(n/d,k), where phi^{(-1)}(n) = A023900(n) is the Dirichlet inverse of Euler's totient function.
We can easily prove that Sum_{n>=1} r(n,k)*x^n = R(k,x) = k*x*(x+1)*(k*x+1)/(1-k*x^2)^2 (for each k>=1). We also have Sum_{n>=1} T(n,k)*x^n = Sum_{n>=1} Sum_{d|n} phi^{(-1)}(d)*r(n/d,k)*x^n. Letting m = n/d and noting that x^n = (x^d)^m, we can easily get the g.f. given in the formula section.
Note that r(n,1) = n, r(n,2) = A187272(n), r(n,3) = A187273(n), r(n,4) = A187274(n), and r(n,5) = A187275(n).
(End)

Examples

			Table starts:
  1   2    3     4     5      6      7       8       9      10 ...
  1   4    9    16    25     36     49      64      81     100 ...
  1   8   21    40    65     96    133     176     225     280 ...
  1  16   57   136   265    456    721    1072    1521    2080 ...
  1  32  123   304   605   1056   1687    2528    3609    4960 ...
  1  52  279   880  2125   4356   7987   13504   21465   32500 ...
  1 100  549  1768  4345   9036  16765   28624   45873   69940 ...
  1 160 1209  4936 14665  35736  75985  146224  260721  437680 ...
  1 260 2127  9112 27965  69756 150955  294512  530937  899380 ...
  1 424 4689 25216 93025 270936 670369 1471744 2948481 5494600 ...
From _Petros Hadjicostas_, Oct 27 2017: (Start)
We explain how to use the above formulae to find general expressions for some rows.
If p is an odd prime, then phi^{(-1)}(p) = 1-p. Since, also, phi^{(-1)}(1) = 1, we get T(p,k) = (1-p)*k+p*k^{(p+1)/2} for the p-th row above.
If m is a positive integer, then phi^{(-1)}(2^m) = -1, and so T(2^m,k) = 1+(k+1)*(2^{m-1}*k^{2^{m-1}}-1-Sum_{0<=s<=m-2} 2^s*k^{2^s}).
For example, if m=1, then T(2,k) = 1+(k+1)*(1*k-1-0) = k^2.
If m=2, then T(4,k) = 1+(k+1)*(2*k^2-1-k) = k*(2*k^2+k-2), which is the formula conjectured by C. Barker for sequence A187277 and verified by A. Howroyd.
(End)

Links

Crossrefs

Columns 2-5 are A007055, A007056, A007057, A007058.
Rows 3-4 are A000567, A187277.
Cf. A023900, A165137, A165135.

Programs

  • Mathematica
    r[d_, k_]:=If[OddQ[d], d*k^((d + 1)/2), (d/2)*(k + 1)*k^(d/2)]; a[n_, k_]:= r[n, k] - Sum[If[dIndranil Ghosh, Apr 07 2017 *)
  • PARI
    r(d,k)=if (d % 2 == 0, (d/2)*(k+1)*k^(d/2), d*k^((d+1)/2));
a(n,k) = r(n,k) - sumdiv(n,d, if (d
  • Python
    from sympy import totient, divisors
def r(d, k): return (d//2)*(k + 1)*k**(d//2) if d%2 == 0 else d*k**((d + 1)//2)
def a(n, k): return r(n, k) - sum([totient(n//d)*a(d, k) for d in divisors(n) if dIndranil Ghosh, Apr 07 2017

Formula

T(n, k) = r(n, k) - Sum_{d|n, d
From Petros Hadjicostas, Oct 27 2017: (Start)
T(n,k) = Sum_{d|n} phi^{(-1)}(d)*r(n/d,k), where r(n,k) is given above and phi^{(-1)}(n) = A023900(n) is the Dirichlet inverse of Euler's totient function.
G.f.: For each k>=1, Sum_{n>=1} T(n,k)*x^n = Sum_{d>=1} phi^{(-1)}(d)*R(k,x^d), where R(k,x) = k*x*(x+1)*(k*x+1)/(1-k*x^2)^2.
(End)
From Richard L. Ollerton, May 07 2021: (Start)
T(n,k) = Sum_{i=1..n} phi^{(-1)}(n/gcd(n,i))*r(gcd(n,i),k)/phi(n/gcd(n,i)).
T(n,k) = Sum_{i=1..n} phi^{(-1)}(gcd(n,i))*r(n/gcd(n,i),k)/phi(n/gcd(n,i)).
r(n,k) = Sum_{i=1..n} T(gcd(n,i),k). (End)

A165610 The number of patterns of non-papaya words.

Original entry on oeis.org

0, 0, 1, 5, 31, 153, 778, 3890, 20693, 114733, 676347, 4207203, 27633048, 190864320, 1382896511, 10479940137, 82864510321, 682075572641, 5832740001550, 51724150291262, 474869801907015, 4506715684635739, 44152005758171637, 445958868912515927, 4638590331538888532
Offset: 1

Views

Author

Tanya Khovanova, Sep 22 2009

Keywords

Comments

Papaya words are defined as palindromes or concatenations of two palindromes.

Examples

			The only three-character non-papaya pattern is abc - words with all distinct letters. Four-character non-papaya patterns are: aabc, abbc, abcc, abca, abcd.

Links

Crossrefs

Cf. A165137, A165135, A165136, A165137, A165611.

Programs

  • Mathematica
    R[k_?EvenQ] := (1/2)*k*(BellB[1 + k/2] + BellB[k/2]);
R[k_?OddQ] := k*BellB[1 + (k - 1)/2];
b[0] = 1; b[n_] := b[n] = R[n] - Sum[EulerPhi[n/d]*b[d], {d, Most[ Divisors[n]]}];
a[n_] := BellB[n] - b[n];
Array[a, 25] (* Jean-François Alcover, Jul 02 2018, after Andrew Howroyd *)

Formula

a(n) = A000110(n) - A165137(n).

Extensions

a(7)-a(25) from Andrew Howroyd, Mar 29 2016

A165611 The number of n-digit non-papaya numbers.

Original entry on oeis.org

0, 0, 648, 7128, 85536, 870750, 8937054, 89606088, 899190558, 8995054860, 89990100090, 899940633714, 8999883000108, 89999307063540, 899998650010008, 8999992080398088, 89999984700000144, 899999910900869040, 8999999829000000162, 89999999010004961988
Offset: 1

Views

Author

Tanya Khovanova, Sep 22 2009

Keywords

Comments

These are numbers that are neither palindromes nor concatenations of two palindromes. Three digit numbers that are non-papaya are numbers with all distinct digits. There are 9*9*8 of them: 648.

Links

Crossrefs

Cf. A165135, A165136, A165137, A165610.

Formula

a(n) = A052268(n) - A165135(n).

Extensions

a(6)-a(20) from Andrew Howroyd, Mar 29 2016

A165136 a(n) is the number of patterns for n-digit papaya numbers.

Original entry on oeis.org

1, 2, 4, 10, 21, 50, 99, 250, 454, 1242, 2223, 6394, 11389, 35002, 62034, 202010, 359483, 1233518, 2203507, 7944100, 14249715, 53810836, 96911168, 382258438, 691048071, 2840120987, 5152403569, 22010733048, 40059670261, 177444599715
Offset: 1

Views

Author

Sergei Bernstein and Tanya Khovanova, Sep 04 2009

Keywords

Comments

Papaya numbers are concatenations of two palindromes or palindromes themselves (think of "papaya" as the concatenation of the palindromes "pap" and "aya").
The actual number of n-digit papaya numbers is A165135. If the pattern is "aa", for example, inserting digits 1 to 9 for "a" gives 9 positive 2-digit numbers, 11, 22, ..., 99. The pattern "ab" inserting a<>b gives 10, 12, ..., 98, that is 9*9 = 81 positive 2-digit numbers. (9 different choices for "a" because leading 0's are not allowed, and for each "a" 9 different choices of "b".) So the a(2) = 2 different patterns represent 9+81 = A165135(2) different 2-digit numbers.
The first 19 terms of this sequence are the same as in A165137. Then the sequences start to differ, because the number of patterns in an infinite alphabet can be larger than patterns in the 10-digits-alphabet of ordinary numbers: A165137(20) = a(20)+10. - Tanya Khovanova, Oct 01 2009 (Since at most 2 symbols in a papaya number can be present only once, to require 11 symbols takes a length of 2 + (11-2)*2 = 20. The 10 strings for A165137(20) not counted here are abcdefghijkjihgfedbc, abacdefghijkjihgfedc, abcbadefghijkjihgfed, ..., abcdefghijihgfedcbak. - Franklin T. Adams-Watters, Apr 10 2011)

Examples

			There are two types of two-digit papaya numbers: aa, or ab. Hence a(2) = 2.
There are four types of three-digit papaya numbers: aaa, aab, aba, abb. Hence a(3) = 4.
There is no pattern of the form "abcdefghijkl" contributing to a(12), because this requires 12 different letters in the alphabet, and the standard numbers alphabet provides only ten different digits 0-9.

Links

Crossrefs

Cf. A165135, A165137, A165610, A165611.

Formula

a(n) = R(n) - Sum_{d|n,dAndrew Howroyd, Mar 29 2016

Extensions

Three more terms from R. J. Mathar, Sep 25 2009
Keyword:base added, comment expanded - R. J. Mathar, Aug 29 2010
a(10)-a(14) from Franklin T. Adams-Watters, Apr 10 2011
a(15)-a(30) from Andrew Howroyd, Mar 29 2016
Showing 1-5 of 5 results.

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