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.

A186347 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 A186346.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 13, 16, 19, 22, 26, 30, 34, 38, 43, 48, 53, 58, 64, 70, 76, 82, 89, 96, 103, 110, 118, 126, 134, 142, 151, 160, 169, 178, 188, 198, 208, 218, 229, 240, 251, 262, 274, 286, 298, 310, 323, 336, 349, 362, 376, 390, 404, 418, 433, 448, 463, 478, 494, 510, 526, 542, 559, 576, 593, 610, 628, 646, 664, 682, 701, 720, 739, 758, 778, 798, 818, 838, 859, 880, 901, 922, 944, 966, 988, 1010
Offset: 1

Views

Author

Clark Kimberling, Feb 20 2011

Keywords

Comments

a(n) = a(-8-n) for all n in Z using the formula. - Michael Somos, Apr 05 2024

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.

Links

Crossrefs

Cf. A186346, A186348, A186349.

Programs

Formula

a(n)=n+floor(sqrt(8n-1/2))=A186346(n).
b(n)=n+floor((n^2+1/2)/8)=A186347(n).
G.f.: x*(1 + x^2 - x^4)/((1 - x)^2 * (1 - x^4)). - Michael Somos, Apr 05 2024

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)).
Showing 1-3 of 3 results.

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