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 31-36 of 36 results.

A223087 Trajectory of 80 under the map n-> A006368(n).

Original entry on oeis.org

80, 120, 180, 270, 405, 304, 456, 684, 1026, 1539, 1154, 1731, 1298, 1947, 1460, 2190, 3285, 2464, 3696, 5544, 8316, 12474, 18711, 14033, 10525, 7894, 11841, 8881, 6661, 4996, 7494, 11241, 8431, 6323, 4742, 7113, 5335, 4001, 3001, 2251, 1688, 2532, 3798, 5697
Offset: 1

Views

Author

N. J. A. Sloane, Mar 22 2013

Keywords

Comments

It is conjectured that this trajectory does not close on itself.

Links

Crossrefs

Cf. A006369, A006368, A182205.
Trajectories under A006368 and A006369: A180853, A217218, A185590, A180864, A028393, A028394, A094328, A094329, A028396, A028395, A217729, A182205, A223083-A223088, A185589, A185590.

Programs

  • Maple
    f:=n-> if n mod 2 = 0 then 3*n/2 elif n mod 4 = 1 then (3*n+1)/4 else (3*n-1)/4; fi;
t1:=[80];
for n from 1 to 100 do t1:=[op(t1),f(t1[nops(t1)])]; od:
t1;
  • Mathematica
    t = {80}; While[n = t[[-1]]; s = If[EvenQ[n], 3*n/2, Round[3*n/4]]; Length[t] < 100 && ! MemberQ[t, s], AppendTo[t, s]]; t (* T. D. Noe, Mar 22 2013 *)
SubstitutionSystem[{n_ :> If[EvenQ[n], 3n/2, Round[3n/4]]}, {80}, 100] // Flatten (* Jean-François Alcover, Mar 01 2019 *)

A265672 a(n) = n + floor((n+1)/7)*(-1)^((n+1) mod 7).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 6, 9, 8, 11, 10, 13, 15, 12, 17, 14, 19, 16, 21, 23, 18, 25, 20, 27, 22, 29, 31, 24, 33, 26, 35, 28, 37, 39, 30, 41, 32, 43, 34, 45, 47, 36, 49, 38, 51, 40, 53, 55, 42, 57, 44, 59, 46, 61, 63, 48, 65, 50, 67, 52, 69, 71, 54, 73, 56, 75
Offset: 0

Views

Author

Paul Curtz, Dec 13 2015

Keywords

Comments

A permutation of A001477. This sequence, without the terms of the form 8*k+5, becomes A265228.
Similar sequences of the type n + floor((n+1)/k)*(-1)^((n+1) mod k):
k = 1: A005408;
k = 2: A014682;
k = 3: A006369 (permutation of A001477);
k = 4: 0, 1, 2, 4, 3, 6, 5, 9, 6, 11, 8, 14, ...;
k = 5: 0, 1, 2, 3, 5, 4, 7, 6, 9, 11, 8, 13, ... (permutation of A001477);
k = 6: 0, 1, 2, 3, 4, 6, 5, 8, 7, 10, 9, 13, ...;
k = 7: this sequence.

Examples

			-------------------------------------------------------------------------
0, 1, 2, 3, 4, 5, 6,  7, 8,  9, 10, 11, 12, 13, 14, 15, 16, 17, 18, ...
+  +  +  +  +  +  +   +  +   +   +   +   +   +   +   +   +   +   +
0, 0, 0, 0, 0, 0, 1, -1, 1, -1,  1, -1,  1,  2, -2,  2, -2,  2, -2, ...
-------------------------------------------------------------------------
0, 1, 2, 3, 4, 5, 7,  6, 9,  8, 11, 10, 13, 15, 12, 17, 14, 19, 16, ...
-------------------------------------------------------------------------

Links

Crossrefs

Cf. A001477, A006369, A265228, A265667, A265734.

Programs

  • Magma
    [n+Floor((n+1)/7)*(-1)^((n+1) mod 7): n in [0..80]]; // Bruno Berselli, Dec 26 2015
  • Maple
    A265672:=n->n + floor((n+1)/7)*(-1)^((n+1) mod 7): seq(A265672(n), n=0..100); # Wesley Ivan Hurt, Apr 09 2017
  • Mathematica
    Table[n + Floor[(n + 1)/7] (-1)^Mod[n + 1, 7], {n, 0, 80}] (* Bruno Berselli, Dec 22 2015 *)
  • PARI
    concat(0, Vec(x*(1 +x^2)*(1 +2*x +2*x^2 +2*x^3 +3*x^4 +5*x^5 +3*x^6 +2*x^7 +x^8 +3*x^9 +x^10) / ((1 -x)^2*(1 +x +x^2 +x^3 +x^4 +x^5 +x^6)^2) + O(x^100))) \\ Colin Barker, Dec 13 2015

Formula

a(n) = a(n-7) + (-1)^((n+1) mod 7) + 7 for n>6.
From Colin Barker, Dec 13 2015: (Start)
a(n) = 2*a(n-7) - a(n-14) for n>13.
G.f.: x*(1 +x^2)*(1 +2*x +2*x^2 +2*x^3 +3*x^4 +5*x^5 +3*x^6 +2*x^7 +x^8 +3*x^9 +x^10) / ((1 -x)^2*(1 +x +x^2 +x^3 +x^4 +x^5 +x^6)^2). (End)

Extensions

Edited by Bruno Berselli, Dec 22 2015

A069196 a(1)=1, a(2)=3; for n>1, a(n+2)=(a(n+1)+a(n))/3 if (a(n+1)+a(n)==0 (mod 3)); a(n+2)=(a(n+1)+a(n))/2 if (a(n+1)+a(n)==0 (mod 2)); and a(n+2)=a(n+1)+a(n) otherwise.

Original entry on oeis.org

1, 3, 2, 5, 7, 4, 11, 5, 8, 13, 7, 10, 17, 9, 13, 11, 8, 19, 9, 14, 23, 37, 20, 19, 13, 16, 29, 15, 22, 37, 59, 32, 91, 41, 44, 85, 43, 64, 107, 57, 82, 139, 221, 120, 341, 461, 401, 431, 416, 847, 421, 634, 1055, 563, 809, 686, 1495, 727, 1111, 919, 1015, 967, 991
Offset: 1

Views

Author

Benoit Cloitre, Apr 11 2002

Keywords

Comments

A Collatz-Fibonacci mixture. Does this sequence diverge to infinity?

Crossrefs

Cf. A006369.

A367286 Inverse of A342131.

Original entry on oeis.org

0, 2, 1, 4, 6, 3, 8, 10, 5, 12, 14, 7, 16, 18, 9, 20, 22, 11, 24, 26, 13, 28, 30, 15, 32, 34, 17, 36, 38, 19, 40, 42, 21, 44, 46, 23, 48, 50, 25, 52, 54, 27, 56, 58, 29, 60, 62, 31, 64, 66, 33, 68, 70, 35, 72, 74, 37, 76, 78, 39, 80, 82, 41, 84, 86, 43, 88, 90, 45, 92, 94, 47
Offset: 0

Views

Author

Philippe Deléham, Nov 12 2023

Keywords

Comments

Permutation of the nonnegative numbers.

Links

Crossrefs

Cf. A006369, A342131.

Programs

  • Mathematica
    LinearRecurrence[{0,0,2,0,0,-1},{0,2,1,4,6,3},100] (* Paolo Xausa, Nov 14 2023 *)

Formula

a(3*n) = 4*n, a(3*n+1) = 4*n+2, a(3*n+2) = 2*n+1.
a(n) = 2*a(n-3) - a(n-6) for n >= 6.
a(n) = A006369(n+1) - 1.
G.f.: x*(2+x+4*x^2+2*x^3+x^4)/(1-x^3)^2.
E.g.f.: exp(-x/2)*(exp(3*x/2)*(10*x + 1) + (2*x - 1)*cos(sqrt(3)*x/2) + sqrt(3)*(3 - 2*x)*sin(sqrt(3)*x/2))/9. - Stefano Spezia, Nov 12 2023

A272188 Triangle with 2*n+1 terms per row, read by rows: the first row is 1 (by decree), following rows contain 0 to 2n+1 but omitting 2n.

Original entry on oeis.org

1, 0, 1, 3, 0, 1, 2, 3, 5, 0, 1, 2, 3, 4, 5, 7, 0, 1, 2, 3, 4, 5, 6, 7, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17
Offset: 0

Views

Author

Paul Curtz, Apr 22 2016

Keywords

Comments

Row n is row 2n+1 of A128138, a bisection.
The second bisection by rows
0, 2,
0, 1, 2, 4,
0, 1, 2, 3, 4, 6,
0, 1, 2, 3, 4, 5, 6, 8,
etc
is the basis of
0, 2, 4, 6, 8, 10, 12, ... the even numbers A005843(n)
0, 1, 2, 4, 3, 6, 8, 5, 10, ... a permutation of the nonnegative integers A265667(n).
0, 1, 2, 3, 4, 6, 5, 8, 7, 10, 12, ... a permutation of the nonnegative integers A265734(n)
etc.
A005843(n) - A005843(n-1) = 2, for n>0.
A265667(n) - A265667(n-3) = 4, 2, 4 (period 3), for n>2.
A265734(n) - A265734(n-5) = 6, 4, 6, 4, 6 (period 5), for n>4.
See A267654.
For
1, 3, 5, 7, 9, 11, 13 ... the odd numbers A005408(n),
0, 1, 3, 2, 5, 7, 4, 9, 11, ... a permutation of the nonnegative numbers A006369,
0, 1, 2, 3, 5, 4, 7, 6, 9, 11, 8, 13, 10, 15, ... another permutation,
a(n) must be extended with one term by row:
1, 3,
0, 1, 3, 2,
0, 1, 2, 3, 5, 4,

Examples

			Irregular triangle:
1,
0, 1, 3,
0, 1, 2, 3, 5,
0, 1, 2, 3, 4, 5, 7,
0, 1, 2, 3, 4, 5, 6, 7, 9,
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11,
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13,
etc.

Crossrefs

Cf. A001477, A005408, A005843, A006369, A028310 (main diagonal), A053186, A265667, A265734, A267654.

Programs

  • Mathematica
    Table[Delete[Range[0, 2 n + 1], 2 n + 1], {n, 0, 8}] // Flatten (* Michael De Vlieger, Apr 25 2016 *)

A367705 Coefficients of expansion of (1 + 5*x + 11*x^2 + 5*x^3 + 7*x^4 + x^5)/(1 - x^3)^2 in powers of x.

Original entry on oeis.org

1, 5, 11, 7, 17, 23, 13, 29, 35, 19, 41, 47, 25, 53, 59, 31, 65, 71, 37, 77, 83, 43, 89, 95, 49, 101, 107, 55, 113, 119, 61, 125, 131, 67, 137, 143, 73, 149, 155, 79, 161, 167, 85, 173, 179, 91, 185, 191, 97, 197, 203, 103, 209, 215, 109, 221, 227, 115, 233, 239
Offset: 0

Views

Author

Philippe Deléham, Nov 27 2023

Keywords

Comments

Based on an idea of Pierre CAMI.

Links

Crossrefs

Cf. A006369, A007310, A016921, A017581, A017653, A130196.

Programs

  • Mathematica
    CoefficientList[Series[(1 + 5*x + 11*x^2 + 5*x^3 + 7*x^4 + x^5)/(1 - x^3)^2, {x, 0, 60}], x] (* or *)
LinearRecurrence[{0, 0, 2, 0, 0, -1}, {1, 5, 11, 7, 17, 23}, 60] (* Amiram Eldar, Nov 28 2023 *)

Formula

a(n) = 3*A006369(n) + A130196(n).
a(n) = A007310(A006369(n) + 1).
a(n) = 2*a(n-3) - a(n-6) for n >= 6.
a(3*n) = 6*n+1, a(3*n+1) = 12*n+5, a(3*n+2) = 12*n+11.
Sum_{n>=0} (-1)^n/a(n) = ((2+sqrt(2))*Pi + sqrt(3)*log(7+4*sqrt(3)) + sqrt(6)*log(5-2*sqrt(6)))/12. - Amiram Eldar, Nov 28 2023
Previous Showing 31-36 of 36 results.

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