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.

A319701 Filter sequence for sequences that are constant for all odd terms >= 3.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 3, 6, 3, 7, 3, 8, 3, 9, 3, 10, 3, 11, 3, 12, 3, 13, 3, 14, 3, 15, 3, 16, 3, 17, 3, 18, 3, 19, 3, 20, 3, 21, 3, 22, 3, 23, 3, 24, 3, 25, 3, 26, 3, 27, 3, 28, 3, 29, 3, 30, 3, 31, 3, 32, 3, 33, 3, 34, 3, 35, 3, 36, 3, 37, 3, 38, 3, 39, 3, 40, 3, 41, 3, 42, 3, 43, 3, 44, 3, 45, 3, 46, 3, 47, 3, 48, 3, 49, 3, 50, 3, 51, 3, 52, 3, 53, 3, 54, 3, 55, 3, 56, 3, 57, 3, 58, 3, 59, 3, 60, 3, 61, 3, 62
Offset: 1

Views

Author

Antti Karttunen, Oct 02 2018

Keywords

Comments

For all i, j:
A305801(i) = A305801(j) => A305890(i) = A305890(j) => a(i) = a(j).
a(i) = a(j) => A007814(i) = A007814(j) => A000035(i) = A000035(j).

Links

Crossrefs

Cf. A305801, A305890, A319702.

Programs

  • PARI
    A319701(n) = if(n<=2, n, if(n%2, 3, 2+(n/2)));

Formula

a(1) = 1, and for n > 1, if n is odd, a(n) = 3, otherwise [when n is even], a(n) = 2+(n/2).

A141310 The odd numbers interlaced with the constant-2 sequence.

Original entry on oeis.org

1, 2, 3, 2, 5, 2, 7, 2, 9, 2, 11, 2, 13, 2, 15, 2, 17, 2, 19, 2, 21, 2, 23, 2, 25, 2, 27, 2, 29, 2, 31, 2, 33, 2, 35, 2, 37, 2, 39, 2, 41, 2, 43, 2, 45, 2, 47, 2, 49, 2, 51, 2, 53, 2, 55, 2, 57, 2, 59, 2, 61, 2, 63, 2, 65, 2, 67, 2, 69, 2, 71, 2, 73, 2, 75, 2, 77, 2, 79, 2, 81, 2, 83, 2, 85, 2, 87, 2, 89, 2, 91, 2, 93, 2, 95, 2, 97
Offset: 0

Views

Author

Paul Curtz, Aug 02 2008

Keywords

Comments

Similarly, the principle of interlacing a sequence and its first differences leads from A000012 and its differences A000004 to A059841, or from A140811 and its first differences A017593 to a sequence -1, 6, 5, 18, ...
If n is even then a(n) = n + 1 ; otherwise a(n) = 2. - Wesley Ivan Hurt, Jun 05 2013
Denominators of floor((n+1)/2) / (n+1), n > 0. - Wesley Ivan Hurt, Jun 14 2013
a(n) is also the number of minimum total dominating sets in the (n+1)-gear graph for n>1. - Eric W. Weisstein, Apr 11 2018
a(n) is also the number of minimum total dominating sets in the (n+1)-sun graph for n>1. - Eric W. Weisstein, Sep 09 2021
Denominators of Cesàro means sequence of A114112, corresponding numerators are in A354008. - Bernard Schott, May 14 2022
Also, denominators of Cesàro means sequence of A237420, corresponding numerators are in A354280. - Bernard Schott, May 22 2022

Links

Crossrefs

Cf. A000004, A000012, A000142, A005408, A008586, A017593, A059841, A060747, A109613, A140811, A211374, A319702 (rgs-transform).
Cf. A114112, A354008.
Cf. A237420, A354280.

Programs

  • Maple
    a:= n-> n+1-(n-1)*(n mod 2): seq(a(n), n=0..96); # Wesley Ivan Hurt, Jun 05 2013
  • Mathematica
    Flatten[Table[{2 n - 1, 2}, {n, 40}]] (* Alonso del Arte, Jun 15 2013 *)
Riffle[Range[1, 79, 2], 2] (* Alonso del Arte, Jun 14 2013 *)
Table[((-1)^n (n - 1) + n + 3)/2, {n, 0, 20}] (* Eric W. Weisstein, Apr 11 2018 *)
Table[Floor[(n + 1)/2]/(n + 1), {n, 0, 20}] // Denominator (* Eric W. Weisstein, Apr 11 2018 *)
LinearRecurrence[{0, 2, 0, -1}, {2, 3, 2, 5}, {0, 20}] (* Eric W. Weisstein, Apr 11 2018 *)
CoefficientList[Series[(1 + 2 x + x^2 - 2 x^3)/(-1 + x^2)^2, {x, 0, 20}], x] (* Eric W. Weisstein, Apr 11 2018 *)
  • PARI
    A141310(n) = if(n%2,2,1+n); \\ (for offset=0 version) - Antti Karttunen, Oct 02 2018
  • PARI
    A141310off1(n) = if(n%2,n,2); \\ (for offset=1 version) - Antti Karttunen, Oct 02 2018
  • Python
    def A141310(n): return 2 if n % 2 else n + 1 # Chai Wah Wu, May 24 2022

Formula

a(2n) = A005408(n). a(2n+1) = 2.
First differences: a(n+1) - a(n) = (-1)^(n+1)*A109613(n-1), n > 0.
b(2n) = -A008586(n), and b(2n+1) = A060747(n), where b(n) = a(n+1) - 2*a(n).
a(n) = 2*a(n-2) - a(n-4). - R. J. Mathar, Feb 23 2009
G.f.: (1+2*x+x^2-2*x^3)/((x-1)^2*(1+x)^2). - R. J. Mathar, Feb 23 2009
From Wesley Ivan Hurt, Jun 05 2013: (Start)
a(n) = n + 1 - (n - 1)*(n mod 2).
a(n) = (n + 1) * (n - floor((n+1)/2))! / floor((n+1)/2)!.
a(n) = A000142(n+1) / A211374(n+1). (End)

Extensions

Edited by R. J. Mathar, Feb 23 2009
Term a(45) corrected, and more terms added by Antti Karttunen, Oct 02 2018

A323236 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(1) = 0, f(n) = -1 if n is an even number > 2, and f(n) = A323234(n) for odd numbers >= 3.

Original entry on oeis.org

1, 2, 3, 2, 4, 2, 5, 2, 4, 2, 6, 2, 7, 2, 8, 2, 4, 2, 6, 2, 9, 2, 10, 2, 11, 2, 12, 2, 13, 2, 14, 2, 4, 2, 6, 2, 9, 2, 10, 2, 15, 2, 16, 2, 17, 2, 18, 2, 19, 2, 20, 2, 21, 2, 22, 2, 23, 2, 24, 2, 25, 2, 26, 2, 4, 2, 6, 2, 9, 2, 10, 2, 15, 2, 16, 2, 17, 2, 18, 2, 27, 2, 28, 2, 29, 2, 30, 2, 31, 2, 32, 2, 33, 2, 34, 2, 35, 2, 36, 2, 37, 2, 38, 2, 39
Offset: 1

Views

Author

Antti Karttunen, Jan 08 2019

Keywords

Comments

For all i, j:
A319702(i) = A319702(j) => a(i) = a(j),
A323234(i) = A323234(j) => a(i) = a(j).

Links

Crossrefs

Cf. A053645, A079944, A319702, A323234.
Cf. also A323242 (somewhat analogous filter sequence for prime factorization).

Programs

  • PARI
    up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A053644(n) = { my(k=1); while(k<=n, k<<=1); (k>>1); }; \\ From A053644
A053645(n) = (n-A053644(n));
A079944off0(n) = (1==binary(2+n)[2]);
A323236aux(n) = if(1==n,0,if(!(n%2),-1,[A053645(n), A079944off0(n-2)]));
v323236 = rgs_transform(vector(up_to,n,A323236aux(n)));
A323236(n) = v323236[n];
Showing 1-3 of 3 results.

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