A186222 -id:A186222 - cp's OEIS frontend

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.

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

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.

Original entry on oeis.org

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).

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

Original entry on oeis.org

2, 4, 6, 9, 11, 14, 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, 84, 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 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 before the square:
a=(1,3,5,7,8,10,12,13,...) = A186219;
b=(2,4,6,9,11,14,16,18,...) = A186220.

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

Cf. A186219, A186221, A186222.

[n + Floor((-1 + Sqrt(8*n^2 + 3))/2): n in [1..100]]; // G. C. Greubel, Aug 26 2018

(See A186219.)
Table[n + Floor[(-1 + Sqrt[8*n^2 + 3])/2], {n, 1, 100}] (* G. C. Greubel, Aug 26 2018 *)

vector(100, n, n + floor((-1 + sqrt(8*n^2 + 3))/2)) \\ G. C. Greubel, Aug 26 2018

See A186219.

A189457 a(n) = n+[ns/r]+[nt/r]; r=2, s=sqrt(2), t=1+sqrt(2).

Original entry on oeis.org

2, 5, 8, 10, 14, 17, 19, 22, 25, 29, 31, 34, 37, 39, 43, 46, 49, 51, 54, 58, 60, 63, 66, 68, 72, 75, 78, 80, 84, 87, 89, 92, 95, 99, 101, 104, 107, 109, 113, 116, 118, 121, 124, 128, 130, 133, 136, 138, 142, 145, 148, 150, 153, 157, 159, 162, 165, 169, 171, 174, 177, 179, 183, 186, 188, 191, 194, 198, 200, 203, 206, 208, 212, 215, 218, 220, 223, 227, 229, 232, 235, 237, 241, 244

Offset: 1

Clark Kimberling, Apr 22 2011

nonn

This is one of three sequences that partition the positive integers.  In general, suppose that r, s, t are positive real numbers for which the sets {i/r: i>=1}, {j/s: j>=1}, {k/t: k>=1} are pairwise disjoint.  Let a(n) be the rank of n/r when all the numbers in the three sets are jointly ranked.  Define b(n) and c(n) as the ranks of n/s and n/t.  It is easy to prove that
a(n) = n + [ns/r] + [nt/r],
b(n) = n + [nr/s] + [nt/s],
c(n) = n + [nr/t] + [ns/t], where []=floor.
With r=2, s=sqrt(2), t=1+sqrt(2), a=A189457, b=A189458, c=A186222 (conjectured).

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

Cf. A189458, A186222.

[Floor(n + Floor(n*Sqrt(2)/2) + Floor(n*(1+Sqrt(2))/2)): n in [1..120]]; // G. C. Greubel, Aug 19 2018

r=2; s=2^(1/2); t=1+2^(1/2);
a[n_] := n + Floor[n*s/r] + Floor[n*t/r];
b[n_] := n + Floor[n*r/s] + Floor[n*t/s];
c[n_] := n + Floor[n*r/t] + Floor[n*s/t];
Table[a[n], {n, 1, 120}]  (*A189457*)
Table[b[n], {n, 1, 120}]  (*A189458*)
Table[c[n], {n, 1, 120}]  (*A186222, conjectured*)

vector(120, n, floor(n+floor(n*sqrt(2)/2)+floor(n*(1+sqrt(2))/2))) \\ G. C. Greubel, Aug 19 2018

A189458 a(n) = n+[nr/s]+[nt/s]; r=2, s=sqrt(2), t=1+sqrt(2).

Original entry on oeis.org

3, 7, 12, 15, 20, 24, 27, 32, 36, 41, 44, 48, 53, 56, 61, 65, 70, 73, 77, 82, 85, 90, 94, 97, 102, 106, 111, 114, 119, 123, 126, 131, 135, 140, 143, 147, 152, 155, 160, 164, 167, 172, 176, 181, 184, 189, 193, 196, 201, 205, 210, 213, 217, 222, 225, 230, 234, 239, 242, 246, 251, 254, 259, 263, 266, 271, 275, 280, 283, 287, 292, 295, 300, 304, 309, 312, 316, 321, 324, 329, 333, 336, 341

Offset: 1

Clark Kimberling, Apr 22 2011

nonn

See A189457.

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

Cf. A189457, A186222.

[n+Floor((2*n)/Sqrt(2))+Floor((n*(1+Sqrt(2)))/Sqrt(2)): n in [1..120]]; // G. C. Greubel, Aug 19 2018

r=2; s=2^(1/2); t=1+2^(1/2);
a[n_] := n + Floor[n*s/r] + Floor[n*t/r];
b[n_] := n + Floor[n*r/s] + Floor[n*t/s];
c[n_] := n + Floor[n*r/t] + Floor[n*s/t];
Table[a[n], {n, 1, 120}]  (*A189457*)
Table[b[n], {n, 1, 120}]  (*A189458*)
Table[c[n], {n, 1, 120}]  (*A186222, conjectured*)
Table[n+Floor[(2n)/Sqrt[2]]+Floor[(n(1+Sqrt[2]))/Sqrt[2]],{n,90}] (* Harvey P. Dale, Feb 04 2015 *)

vector(120, n, n+floor((2*n)/sqrt(2))+floor((n*(1+sqrt(2)))/sqrt(2))) \\ G. C. Greubel, Aug 19 2018

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cp's OEIS Frontend

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

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A189457 a(n) = n+[ns/r]+[nt/r]; r=2, s=sqrt(2), t=1+sqrt(2).

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A189458 a(n) = n+[nr/s]+[nt/s]; r=2, s=sqrt(2), t=1+sqrt(2).

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