A186315 -id:A186315 - cp's OEIS frontend

A186316 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 squares and hexagonal numbers. Complement of A186315.

2, 4, 6, 9, 11, 14, 16, 18, 21, 23, 26, 28, 31, 33, 35, 38, 40, 43, 45, 47, 50, 52, 55, 57, 60, 62, 64, 67, 69, 72, 74, 76, 79, 81, 84, 86, 88, 91, 93, 96, 98, 101, 103, 105, 108, 110, 113, 115, 117, 120, 122, 125, 127, 130, 132, 134, 137, 139, 142, 144, 146, 149, 151, 154, 156, 158, 161, 163, 166, 168, 171, 173, 175, 178, 180, 183, 185, 187, 190, 192, 195, 197, 200, 202, 204, 207, 209, 212, 214, 216, 219, 221, 224, 226, 228, 231, 233, 236, 238, 241

Offset: 1

Clark Kimberling, Feb 17 2011

nonn

			First, write
1..4...9...16..25....36....49. (squares)
1....6...15.......28....45.... (hexagonal)
Replace each number by its rank, where ties are settled by ranking the square number before the hexagonal:
a=(1,3,5,7,8,10,12,13,...)=A186315.
b=(2,4,6,9,11,14,16,18,...)=A186316.

Cf. A186315, A186317, A186318.

Mathematica
```
(See A186315.)
```

A186317 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 squares and hexagonal numbers. Complement of A186318.

Original entry on oeis.org

2, 3, 5, 7, 8, 10, 12, 13, 15, 17, 19, 20, 22, 24, 25, 27, 29, 30, 32, 34, 36, 37, 39, 41, 42, 44, 46, 48, 49, 51, 53, 54, 56, 58, 60, 61, 63, 65, 66, 68, 70, 71, 73, 75, 77, 78, 80, 82, 83, 85, 87, 89, 90, 92, 94, 95, 97, 99, 100, 102, 104, 106, 107, 109, 111, 112, 114, 116, 118, 119, 121, 123, 124, 126, 128, 129, 131, 133, 135, 136, 138, 140, 141, 143, 145, 147, 148, 150, 152, 153, 155, 157, 159, 160, 162, 164, 165, 167, 169, 170

Offset: 1

Clark Kimberling, Feb 17 2011

nonn

			First, write
1..4...9...16..25....36....49. (squares)
1....6...15.......28....45.... (hexagonals)
Replace each number by its rank, where ties are settled by ranking the square number after the hexagonal:
a=(2,3,5,7,8,10,12,13,...)=A186317.
b=(1,4,6,9,11,14,16,18,...)=A186318.

Cf. A186315, A186316, A186318.

(* adjusted joint ranking; general formula *)
d=-1/2; u=1; v=0; w=0; x=2; y=-1; 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}]  (* A186317 *)
Table[b[n], {n, 1, 100}]  (* A186318 *)

A186318 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 squares and hexagonal numbers. Complement of A186317.

Original entry on oeis.org

1, 4, 6, 9, 11, 14, 16, 18, 21, 23, 26, 28, 31, 33, 35, 38, 40, 43, 45, 47, 50, 52, 55, 57, 59, 62, 64, 67, 69, 72, 74, 76, 79, 81, 84, 86, 88, 91, 93, 96, 98, 101, 103, 105, 108, 110, 113, 115, 117, 120, 122, 125, 127, 130, 132, 134, 137, 139, 142, 144, 146, 149, 151, 154, 156, 158, 161, 163, 166, 168, 171, 173, 175, 178, 180, 183, 185, 187, 190, 192, 195, 197, 200

Offset: 1

Clark Kimberling, Feb 17 2011

nonn

			First, write
1..4...9...16..25....36....49. (squares)
1....6...15.......28....45.... (hexagonal)
Replace each number by its rank, where ties are settled by ranking the square number after the hexagonal:
a=(2,3,5,7,8,10,12,13,...)=A186317.
b=(1,4,6,9,11,14,16,18,...)=A186318.

Cf. A186315, A186316, A186318.

Mathematica
```
(See A186317.)
```

cp's OEIS Frontend

A186316 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 squares and hexagonal numbers. Complement of A186315.

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Mathematica

A186317 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 squares and hexagonal numbers. Complement of A186318.

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Mathematica

A186318 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 squares and hexagonal numbers. Complement of A186317.

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Mathematica