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 1-4 of 4 results.

A186346 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f(i)=8i and g(j)=j^2. Complement of A186347.

Original entry on oeis.org

3, 5, 7, 9, 11, 12, 14, 15, 17, 18, 20, 21, 23, 24, 25, 27, 28, 29, 31, 32, 33, 35, 36, 37, 39, 40, 41, 42, 44, 45, 46, 47, 49, 50, 51, 52, 54, 55, 56, 57, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 77, 78, 79, 80, 81, 83, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 97, 98, 99, 100, 101, 102, 104, 105, 106, 107, 108, 109, 111, 112, 113, 114, 115, 116, 117, 119, 120, 121, 122, 123, 124, 125, 127, 128, 129, 130, 131, 132, 133, 135, 136, 137, 138, 139, 140, 141
Offset: 1

Views

Author

Clark Kimberling, Feb 20 2011

Keywords

Comments

See A186350 for a discussion of adjusted joint rank sequences.

Examples

			First, write
....8....16..24..32..40..48..56..64..72..80.. (8i)
1..4..9..16...25...36......49....64.......81 (squares)
Then replace each number by its rank, where ties are settled by ranking 8i before the square:
a=(3,5,7,9,11,12,14,15,17,..)=A186346
b=(1,2,4,6,8,10,13,16,19,...)=A186347.

Crossrefs

Cf. A186350, A186347, A186348, A186349.

Programs

  • Mathematica
    (* adjusted joint rank sequences a and b, using general formula for ranking 1st degree u*n+v and 2nd degree x*n^2+y*n+z *)
 d=1/2; u=8; v=0; x=1; y=0;
h[n_]:=(-y+(4x(u*n+v-d)+y^2)^(1/2))/(2x);
a[n_]:=n+Floor[h[n]];
k[n_]:=(x*n^2+y*n-v+d)/u;
b[n_]:=n+Floor[k[n]];
Table[a[n],{n,1,120}]  (* A186346 *)
Table[b[n],{n,1,100}]  (* A186347 *)

Formula

a(n)=n+floor(sqrt(8n-1/2))=A186346(n).
b(n)=n+floor((n^2+1/2)/8)=A186347(n).

A186349 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) after g(j) when f(i)=g(j), where f(i)=8i and g(j)=j^2. Complement of A186348.

Original entry on oeis.org

1, 2, 4, 5, 8, 10, 13, 15, 19, 22, 26, 29, 34, 38, 43, 47, 53, 58, 64, 69, 76, 82, 89, 95, 103, 110, 118, 125, 134, 142, 151, 159, 169, 178, 188, 197, 208, 218, 229, 239, 251, 262, 274, 285, 298, 310, 323, 335, 349, 362, 376, 389, 404, 418, 433, 447, 463, 478, 494, 509, 526, 542, 559, 575, 593, 610, 628, 645, 664, 682, 701, 719, 739, 758, 778, 797, 818, 838, 859, 879, 901, 922, 944, 965, 988, 1010
Offset: 1

Views

Author

Clark Kimberling, Feb 20 2011

Keywords

Examples

			First, write
.....8...16..24..32..40..48..56..64..72..80.. (8i)
1..4..9..16...25...36.....49.....64.......81. (squares)
Then replace each number by its rank, where ties are settled by ranking 8i after the square:
p = (3,6,7,9,11,12,14,16,17,...) = A186348 = n + floor(sqrt(8n+1/2)).
q = (1,2,4,5,8,10,13,15,19,...) = a(n).

Links

Crossrefs

Cf. A186346, A186347, A186348.

Programs

  • Magma
    m:=90; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+x^2-x^3+x^4-x^5)/((1+x)*(1+x^2)*(1-x)^3))); // Bruno Berselli, Jul 05 2013
  • Maple
    seq(k+ceil(k^2/8)-1,k=1..100); # Wesley Ivan Hurt, Jun 28 2013
  • Mathematica
    (* adjusted joint rank sequences p and q (=a(n)), using general formula for ranking 1st degree u*n+v and 2nd degree x*n^2 + y*n + z *)
d=-1/2; u=8; v=0; x=1; y=0;
k[n_]:=(x*n^2+y*n-v+d)/u;
a[n_]:=n+Floor[k[n]];
Table[a[n], {n, 1, 100}]
  • Maxima
    makelist((2*n*(n+8)-(1+(-1)^n)*(5+2*%i^(n*(n+1)))-2)/16, n, 1, 90); /* Bruno Berselli, Jul 05 2013 */
  • PARI
    a(n)=(n^2-1)\8+n \\ Charles R Greathouse IV, Jul 05 2013

Formula

a(n) = n + floor((n^2 - 1)/8).
a(n) = n + ceiling(n^2/8) - 1. - Wesley Ivan Hurt, Jun 28 2013
From Bruno Berselli, Jul 05 2013: (Start)
G.f.: x*(1 + x^2 - x^3 + x^4 - x^5)/((1+x)*(1+x^2)*(1-x)^3).
a(n) = (2*n*(n+8) - (1+(-1)^n)*(5+2*i^(n*(n+1))) - 2)/16 where i=sqrt(-1). (End)
E.g.f.: (8 - 2*cos(x) + (x^2 + 9*x - 6)*cosh(x) + (x^2 + 9*x - 1)*sinh(x))/8. - Stefano Spezia, Apr 06 2024

A186348 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) after g(j) when f(i)=g(j), where f(i)=8i and g(j)=j^2. Complement of A186349.

Original entry on oeis.org

3, 6, 7, 9, 11, 12, 14, 16, 17, 18, 20, 21, 23, 24, 25, 27, 28, 30, 31, 32, 33, 35, 36, 37, 39, 40, 41, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 59, 60, 61, 62, 63, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 77, 78, 79, 80, 81, 83, 84, 85, 86, 87, 88
Offset: 1

Views

Author

Clark Kimberling, Feb 20 2011

Keywords

Examples

			First, write
....8....16..24..32..40..48..56..64..72..80.. (8i)
1..4..9..16...25...36......49....64.......81 (squares)
Then replace each number by its rank, where ties are settled by ranking 8i after the square:
p=(3,6,7,9,11,12,14,16,17,..)=A186348=a(n).
q=(1,2,4,5,8,10,13,15,19,...)=A186349=n+floor((n^2-1)/8).

Crossrefs

Cf. A186346, A186347, A186349.

Programs

  • Mathematica
    (* adjusted joint rank sequences p and q, using general formula for ranking 1st degree u*n+v and 2nd degree x*n^2+y*n+z *)
d=-1/2; u=8; v=0; x=1; y=0;
h[n_]:=(-y+(4x(u*n+v-d)+y^2)^(1/2))/(2x);
a[n_]:=n+Floor[h[n]];
Table[a[n],{n,1,120}]  (* A186348 *)
  • PARI
    a(n)=n+sqrtint(8*n) \\ Charles R Greathouse IV, Jul 05 2013

Formula

a(n) = n+floor(sqrt(8n)).

A128929 Diameter of a special type of regular graph of degree 4 whose order maintain an increase in form of an arithmetic progression.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 19, 20, 20, 20, 20, 21, 21, 21
Offset: 4

Views

Author

Aminu Alhaji Ibrahim, Apr 25 2007

Keywords

Examples

			f(D4,5)=1 when order=4, f(D4,5)=1 when order=5, f(D)=f(D4,5)+1=1+1=2 when order is 5+1=6

References

  • Claude C.S. and Dinneen M.J (1998), Group-theoretic methods for designing networks, Discrete mathematics and theoretical computer science, Research report
  • Comellas, F. and Gomez, J. (1995), New large graphs with given degree and diameter, in Proceedings of the seventh quadrennial international conference on the theory and applications of graphs, Volume 1: pp. 222-233
  • Ibrahim, A., A. (2007), A stable variety of Cayley graphs (in preparation)

Links

Crossrefs

Cf. A123642.
First differences of A186347.

Formula

f(D4,5)=1: Order =4,5; f(D)= f(D4,5)+n: order=5+n, n=1,2,...
I am assuming this sequence is just Floor[(n+5)/4]... [From Eric W. Weisstein, Sep 09 2008]
Showing 1-4 of 4 results.

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

Content is available under The OEIS End-User License Agreement.