A186220 -id:A186220 - cp's OEIS frontend

A186219 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f and g are the triangular numbers and squares. Complement of A186220.

1, 3, 5, 7, 8, 10, 12, 13, 15, 17, 19, 20, 22, 24, 25, 27, 29, 31, 32, 34, 36, 37, 39, 41, 43, 44, 46, 48, 49, 51, 53, 54, 56, 58, 60, 61, 63, 65, 66, 68, 70, 72, 73, 75, 77, 78, 80, 82, 83, 85, 87, 89, 90, 92, 94, 95, 97, 99, 101, 102, 104, 106, 107, 109, 111, 113, 114, 116, 118, 119, 121, 123, 124, 126, 128, 130, 131, 133, 135, 136, 138, 140, 142, 143, 145, 147, 148, 150, 152, 153, 155, 157, 159, 160, 162, 164, 165, 167, 169, 171

Offset: 1

Clark Kimberling, Feb 15 2011

nonn

Suppose that f and g are strictly increasing functions for which (f(i)) and (g(j)) are integer sequences.  If 0<|d|<1, the sets F={f(i): i>=1} and G={g(j)+d: j>=1} are clearly disjoint.  Let f^=(inverse of f) and g^=(inverse of g).  When the numbers in F and G are jointly ranked, the rank of f(n) is a(n):=n+floor(g^(f(n))-d), and the rank of g(n)+d is b(n):=n+floor(f^(g(n))+d).  Therefore, the sequences a and b are a complementary pair.
Although the sequences (f(i)) and (g(j)) may not be disjoint, the sequences (f(i)) and (g(j)+d) are disjoint, and this observation enables two types of adjusted joint rankings:
(1) if 0Using f(i)=ui^2+vi+w and g(j)=xj^2+yj+z, we can carry out adjusted joint rankings of any pair of polygonal sequences (triangular, square, pentagonal, etc.)  In this case,
  a(n)=n+floor((-y+sqrt(4x(un^2+vn+w-z-d)+y^2))/(2x)),
  b(n)=n+floor((-v+sqrt(4u(xn^2+yn+z-w+d)+v^2)/(2u)),
  where a(n) is the rank of un^2+vn+w and b(n) is the rank
  of xn^2+yn+z+d, where d must be chosen small enough, in
  absolute value, that the sets F and G are disjoint.
Example:  f=A000217 (triangular numbers) and g=A000290 (squares) yield adjusted rank sequences a=A186219 and b=A186220 for d=1/4 and a=A186221 and b=A186222 for d=-1/4.

			First, write
1..3...6..10..15...21..28..36..45...  (triangular)
1....4.. 9......16...25....36....49.. (square)
Replace each number by its rank, where ties are settled by ranking the triangular number before the square:
a=(1,3,5,7,8,10,12,13,...)
b=(2,4,6,9,11,14,16,18,...).

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A186145 (joint ranks of squares and cubes),
A000217 (triangular numbers),
A000290 (squares),
A186220 (complement of A186119)
A186221 ("after" instead of "before")
A186222 (complement of A186221).

[n + Floor(Sqrt((n^2 + n)/2 - 1/4)): n in [1..100]]; // G. C. Greubel, Aug 26 2018

(* adjusted joint ranking of triangular numbers and squares, using general formula *)
d=1/4; u=1/2; v=1/2; w=0; x=1; y=0; z=0;
h[n_]:=-y+(4x(u*n^2+v*n+w-z-d)+y^2)^(1/2);
a[n_]:=n+Floor[h[n]/(2x)]; (* rank of triangular n(n+1)/2 *)
k[n_]:=-v+(4u(x*n^2+y*n+z-w+d)+v^2)^(1/2);
b[n_]:=n+Floor[k[n]/(2u)]; (* rank of square n^2 *)
Table[a[n],{n,1,100}] (* A186219 *)
Table[b[n],{n,1,100}] (* A186220 *)

vector(100, n, n + floor(sqrt((n^2 + n)/2 - 1/4))) \\ G. C. Greubel, Aug 26 2018

a(n) = n + floor(sqrt((n^2+n)/2 - 1/4)), (A186219).

b(n) = n + floor((-1 + sqrt(8*n^2+3))/2), (A186220).

A186222 Adjusted joint rank sequence of (g(i)) and (f(j)) with f(i) after g(j) when f(i)=g(j), where f and g are the triangular numbers and squares. Complement of A186221.

Original entry on oeis.org

1, 4, 6, 9, 11, 13, 16, 18, 21, 23, 26, 28, 30, 33, 35, 38, 40, 42, 45, 47, 50, 52, 55, 57, 59, 62, 64, 67, 69, 71, 74, 76, 79, 81, 83, 86, 88, 91, 93, 96, 98, 100, 103, 105, 108, 110, 112, 115, 117, 120, 122, 125, 127, 129, 132, 134, 137, 139, 141, 144, 146, 149, 151, 154, 156, 158, 161, 163, 166, 168, 170, 173, 175, 178, 180, 182, 185, 187, 190, 192, 195, 197, 199, 202, 204, 207, 209, 211, 214, 216, 219, 221, 224, 226, 228, 231, 233, 236, 238, 240

Offset: 1

Clark Kimberling, Feb 15 2011

nonn

See A186221.

			See A186221.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A186219, A186220, A186221.

[n + Floor(-1/2 + Sqrt(2*n^2)): n in [1..120]]; // G. C. Greubel, Aug 18 2018

(* adjusted joint ranking; general formula *)
d=-1/4; u=1/2; v=1/2; w=0; x=1; y=0; z=0;
h[n_]:=-y+(4x(u*n^2+v*n+w-z-d)+y^2)^(1/2);
a[n_]:=n+Floor[h[n]/(2x)];
k[n_]:=-v+(4u(x*n^2+y*n+z-w+d)+v^2)^(1/2);
b[n_]:=n+Floor[k[n]/(2u)];
Table[a[n],{n,1,100}] (* A186221 *)
Table[b[n],{n,1,100}] (* A186222 *)
Table[n + Floor[Sqrt[2*n^2] - 1/2], {n, 1, 120}] (* G. C. Greubel, Aug 18 2018 *)

vector(120, n, n + floor(-1/2 + sqrt(2*n^2))) \\ G. C. Greubel, Aug 18 2018

a(n) = n + floor(-1/2 + sqrt(2*n^2)).

A186221 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) after g(j) when f(i)=g(j), where f and g are the triangular numbers and squares. Complement of A186222.

Original entry on oeis.org

2, 3, 5, 7, 8, 10, 12, 14, 15, 17, 19, 20, 22, 24, 25, 27, 29, 31, 32, 34, 36, 37, 39, 41, 43, 44, 46, 48, 49, 51, 53, 54, 56, 58, 60, 61, 63, 65, 66, 68, 70, 72, 73, 75, 77, 78, 80, 82, 84, 85, 87, 89, 90, 92, 94, 95, 97, 99, 101, 102, 104, 106, 107, 109, 111, 113, 114, 116, 118, 119, 121, 123, 124, 126, 128, 130, 131, 133, 135, 136, 138, 140, 142, 143, 145, 147, 148, 150, 152, 153, 155, 157, 159, 160, 162, 164, 165, 167, 169, 171

Offset: 1

Clark Kimberling, Feb 15 2011

See A186219.

			First, write
1..3...6..10..15...21..28..36..45...  (triangular)
1....4...9......16...25....36....49.. (square)
Replace each number by its rank, where ties are settled by ranking the triangular number after the square:
a=(2,3,5,7,8,10,12,14,...)
b=(1,4,6,9,11,13,16,18,...).

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A061288, A186219, A186220, A186222.

[n + Floor(Sqrt((n^2+n)/2 + 1/4)): n in [1..120]]; // G. C. Greubel, Aug 18 2018

(* adjusted joint ranking; general formula *)
d=-1/4; u=1/2; v=1/2; w=0; x=1; y=0; z=0;
h[n_]:=-y+(4x(u*n^2+v*n+w-z-d)+y^2)^(1/2);
a[n_]:=n+Floor[h[n]/(2x)];
k[n_]:=-v+(4u(x*n^2+y*n+z-w+d)+v^2)^(1/2);
b[n_]:=n+Floor[k[n]/(2u)];
Table[a[n],{n,1,100}] (* A186221 *)
Table[b[n],{n,1,100}] (* A186222 *)
a[ n_] := n + Floor[ Sqrt[ n (n + 1)/2]]; (* Michael Somos, Aug 19 2018 *)

vector(120, n, n + floor(sqrt((n^2+n)/2 + 1/4))) \\ G. C. Greubel, Aug 18 2018
{a(n) = n + sqrtint( n * (n+1) \ 2)}; /* Michael Somos, Aug 19 2018 */

a(n) = n + floor(sqrt((n^2+n)/2 + 1/4)).

a(n) = A061288(n) - n for all n in Z. - Michael Somos, Aug 19 2018

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cp's OEIS Frontend

A186219 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f and g are the triangular numbers and squares. Complement of A186220.

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A186222 Adjusted joint rank sequence of (g(i)) and (f(j)) with f(i) after g(j) when f(i)=g(j), where f and g are the triangular numbers and squares. Complement of A186221.

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A186221 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) after g(j) when f(i)=g(j), where f and g are the triangular numbers and squares. Complement of A186222.

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