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.

Previous Showing 11-16 of 16 results.

A101886 Smallest natural number sequence without any length 4 equidistant arithmetic subsequences.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 14, 15, 16, 18, 19, 20, 22, 24, 27, 28, 29, 31, 32, 35, 36, 37, 39, 41, 42, 43, 47, 48, 50, 51, 53, 55, 58, 60, 61, 63, 65, 66, 68, 70, 71, 72, 77, 78, 80, 82, 85, 86, 87, 89, 90, 91, 94, 95, 96, 98, 99, 100, 102, 103, 104, 107, 109, 110, 111, 114
Offset: 1

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Author

Douglas Stones (dssto1(AT)student.monash.edu.au), Dec 20 2004

Keywords

Examples

			4 is out because of 1,2,3,4. 13 is out because of 1,5,9,13.

Crossrefs

Cf. A101887, A101884, A101888.
A selection of sequences related to "no three-term arithmetic progression": A003002, A003003, A003278, A004793, A005047, A005487, A033157, A065825, A092482, A093678, A093679, A093680, A093681, A093682, A094870, A101884, A101886, A101888, A140577, A185256, A208746, A229037.

A101888 Smallest natural number sequence without any length 5 equidistant arithmetic subsequences.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 22, 23, 24, 25, 27, 28, 29, 30, 32, 33, 34, 35, 37, 38, 39, 40, 43, 44, 45, 46, 48, 49, 50, 51, 53, 54, 55, 56, 58, 59, 60, 61, 64, 65, 66, 67, 69, 70, 71, 72, 74, 75, 76, 77, 79, 80, 81, 82, 86, 87, 88, 90, 91, 92, 93, 95
Offset: 1

Views

Author

Douglas Stones (dssto1(AT)student.monash.edu.au), Dec 20 2004

Keywords

Examples

			5 is out because of 1,2,3,4,5. 21 is out because of 1,6,11,16,21.

Crossrefs

Cf. A101889, A101884, A101886.
A selection of sequences related to "no three-term arithmetic progression": A003002, A003003, A003278, A004793, A005047, A005487, A033157, A065825, A092482, A093678, A093679, A093680, A093681, A093682, A094870, A101884, A101886, A101888, A140577, A185256, A208746, A229037.

A236313 Recurrence: a(2n) = 3a(n)-1, a(2n+1) = 1.

Original entry on oeis.org

1, 2, 1, 5, 1, 2, 1, 14, 1, 2, 1, 5, 1, 2, 1, 41, 1, 2, 1, 5, 1, 2, 1, 14, 1, 2, 1, 5, 1, 2, 1, 122, 1, 2, 1, 5, 1, 2, 1, 14, 1, 2, 1, 5, 1, 2, 1, 41, 1, 2, 1, 5, 1, 2, 1, 14, 1, 2, 1, 5, 1, 2, 1, 365, 1, 2, 1, 5, 1, 2, 1, 14, 1, 2, 1, 5, 1, 2, 1, 41, 1, 2, 1, 5, 1, 2, 1, 14, 1, 2, 1, 5, 1, 2, 1, 122, 1, 2, 1, 5
Offset: 1

Views

Author

Ralf Stephan, Jan 22 2014

Keywords

Comments

First differences of Stanley sequence S(0,1) (A005836) and S(1,2) (A003278).
In the binary expansion of n, delete everything left of the rightmost 1 bit, then interpret as ternary, add one, and divide by 2.
A007051 is this sequence in strictly increasing order. - Max Barrentine, Sep 11 2015
Empirical: a(n) is the smallest natural number k such that no two adjacent subsequences t and u consisting of consecutive entries of (a(1), a(2), ..., a(n-1), k) are such that the sum of the entries of t is equal to the sum of the entries of u. For example, according to this definition, a(4) cannot be equal to 1, 2, 3, or 4. - John M. Campbell, Mar 20 2017

Links

Crossrefs

Cf. A003278, A005836, A006519, A007051, A007814, A061393, A093682.

Programs

  • Magma
    [(1+3^Valuation(n,2))/2: n in [1..100]]; // Bruno Berselli, Jan 22 2014
  • Mathematica
    t = {1}; Do[If[OddQ[n], AppendTo[t, 1], AppendTo[t, 3*t[[n/2]] - 1]], {n, 2, 100}]; t (* T. D. Noe, Apr 10 2014 *)
a[n_] := a[n] =If[OddQ@ n, 1, 3 a[n/2] - 1]; Array[a, 92] (* Robert G. Wilson v, Jul 31 2018 *)
  • PARI
    a(n)=(1+3^valuation(n,2))/2

Formula

a(n) = (1/2)*(1 + 3^A007814(n)) = A007051(A007814(n)).
a(n) = (1/2)*A061393(n), for n>=1.
Multiplicative with a(2^e) = (1 + 3^e)/2, a(p^e) = 1 for odd prime p. - Andrew Howroyd, Jul 31 2018
G.f.: Sum_{k>=0} 3^k * x^(2^k) / (1 + x^(2^k)). - Ilya Gutkovskiy, Dec 14 2020
Dirichlet g.f.: zeta(s)*(2^s-2)/(2^s-3). - Amiram Eldar, Dec 29 2022

A236269 First differences of Stanley sequence S(0,4) (A005487).

Original entry on oeis.org

4, 1, 2, 4, 1, 4, 7, 3, 5, 2, 4, 1, 6, 5, 7, 17, 5, 2, 5, 10, 4, 7, 18, 4, 3, 5, 7, 26, 19, 9, 6, 17, 1, 5, 1, 11, 9, 12, 10, 7, 8, 3, 15, 6, 2, 3, 6, 18, 48, 7, 5, 25, 12, 21, 11, 4, 21, 2, 6, 5, 50, 5, 21, 18, 30, 1, 6, 5, 4, 6, 4, 1, 2, 20, 10, 4, 24, 3, 13, 5
Offset: 1

Views

Author

Ralf Stephan, Jan 21 2014

Keywords

Comments

Also first differences of Stanley sequence S(1,5) (A033158).
While there are conjectures about formulas for S(0,m), m=1,2,3,6,9... (see A093682), m=4 is the first case where the first differences look almost random.
Records are 4, 7, 17, 18, 26, 48, 50, 55, 76, 87, 92, 93, 165, 175,...
Positions of records are 1, 7, 16, 23, 28, 49, 61, 81, 83, 101, 147, 165, 185, 250, 400,...
Positions where a(n)=1: 2, 5, 12, 33, 35, 66, 72, 94, 125, 160, 189, 288, 307, 327,...

Links

Programs

  • PARI
    NAP(sv,N)=local(v,vv,m,k,l,sl,vvl);sl=length(sv);vvl=min(N*N,10^5);v=vector(N);vv=vector(vvl);for(k=1,sl,v[k]=sv[k];for(l=1,k-1,vv[2*v[k]-v[l]]=1));m=v[sl]+1;for(k=sl+1,N,while(m<=vvl&&vv[m],m=m+1);if(m>vvl,return(v));for(l=1,k-1,sl=2*m-v[l];if(sl<=vvl,vv[sl]=1));vv[m]=1;v[k]=m);v
S04(n)=N=1000;NAP([0,4],N)[n]
a(n)=S04(n+1)-S04(n)

A231990 First differences of Stanley sequence S(0,5) (A187843).

Original entry on oeis.org

5, 1, 2, 1, 5, 1, 2, 10, 4, 1, 4, 2, 4, 1, 8, 14, 8, 1, 8, 7, 11, 1, 6, 2, 14, 29, 2, 11, 9, 3, 2, 11, 8, 13, 5, 4, 7, 6, 3, 24, 59, 9, 7, 1, 20, 5, 1, 8, 3, 33, 27, 11, 9, 12, 3, 2, 11, 9, 27, 4, 2, 5, 12, 1, 5, 16, 54, 80, 37, 15, 1, 6, 3, 33, 3, 17, 5, 9, 17, 1, 35, 32, 2, 72, 10, 3, 82, 6, 4, 6, 5, 2, 1, 5, 31, 14, 13
Offset: 1

Views

Author

Ralf Stephan, Jan 30 2014

Keywords

Comments

For known formulas of differences of other Stanley sequences see A093682.

Links

Crossrefs

Cf. A236269.

Programs

  • PARI
    NAP(sv,N)=local(v,vv,m,k,l,sl,vvl);sl=length(sv);vvl=min(N*N,10^5);v=vector(N);vv=vector(vvl);for(k=1,sl,v[k]=sv[k];for(l=1,k-1,vv[2*v[k]-v[l]]=1));m=v[sl]+1;for(k=sl+1,N,while(m<=vvl&&vv[m],m=m+1);if(m>vvl,return(v));for(l=1,k-1,sl=2*m-v[l];if(sl<=vvl,vv[sl]=1));vv[m]=1;v[k]=m);v
S05(n)=N=1000;NAP([0,5],N)[n]
a(n)=S05(n+1)-S05(n)

A236697 First differences of A131741.

Original entry on oeis.org

1, 2, 6, 2, 16, 2, 6, 4, 26, 6, 10, 6, 12, 6, 20, 12, 18, 22, 14, 34, 6, 30, 8, 10, 26, 24, 6, 42, 10, 8, 4, 8, 22, 2, 34, 24, 8, 10, 54, 8, 42, 28, 6, 96, 26, 40, 14, 60, 4, 20, 30, 46, 26, 12, 42, 28, 2, 70, 8, 126, 4, 26, 34, 6, 42, 18, 96, 26, 48, 4
Offset: 1

Views

Author

Zak Seidov, Jan 30 2014

Keywords

Comments

Among first 10000 terms, the largest is a(7790) = 17412.

Links

Crossrefs

Cf. A003002, A003003, A003278, A004793, A005047, A005487, A033157, A065825, A092482, A093678, A093679, A093680, A093681, A093682, A094870, A101884, A101886, A101888, A131741, A140577, A185256, A208746, A229037.

Formula

a(n) = A131741(n+1) - A131741(n).
Previous Showing 11-16 of 16 results.

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