A186237 -id:A186237 - cp's OEIS frontend

A186238 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 heptagonal numbers. Complement of A186237.

1, 5, 8, 11, 14, 18, 21, 24, 27, 31, 34, 37, 40, 44, 47, 50, 53, 57, 60, 63, 66, 70, 73, 76, 79, 82, 86, 89, 92, 95, 99, 102, 105, 108, 112, 115, 118, 121, 125, 128, 131, 134, 137, 141, 144, 147, 150, 154, 157, 160, 163, 167, 170, 173, 176, 180, 183, 186, 189, 192, 196, 199, 202, 205, 209, 212, 215, 218, 222, 225, 228, 231, 235, 238, 241, 244, 248, 251, 254, 257, 260, 264, 267, 270, 273, 277, 280, 283, 286, 290, 293, 296, 299, 303, 306, 309

Offset: 1

Clark Kimberling, Feb 16 2011

nonn

			See A186227.

Cf. A186227, A186228, A186237.

Mathematica
```
(See A186237.)
```

A186227 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 heptagonal numbers. Complement of A186228.

Original entry on oeis.org

1, 3, 4, 6, 7, 9, 10, 12, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26, 28, 29, 30, 32, 33, 35, 36, 38, 39, 41, 42, 43, 45, 46, 48, 49, 51, 52, 54, 55, 56, 58, 59, 61, 62, 64, 65, 67, 68, 69, 71, 72, 74, 75, 77, 78, 80, 81, 83, 84, 85, 87, 88, 90, 91, 93, 94, 96, 97, 98, 100, 101, 103, 104, 106, 107, 109, 110, 111, 113, 114, 116, 117, 119, 120, 122, 123, 124, 126, 127, 129, 130, 132, 133, 135, 136, 138, 139, 140, 142, 143, 145

Offset: 1

Clark Kimberling, Feb 16 2011

nonn

See A186219 for a general discussion of adjusted joint rank sequences.

			First, write
1..3..6..10..15..21..28..36..45... (triangular)
1.......7......18......34.......55... (heptagonal)
Then replace each number by its rank, where ties are settled by ranking the triangular number before the heptagonal:
a=(1,3,4,6,7,9,10,12,...), A186227.
b=(2,5,8,11,15,18,21,...), A186228.

Cf. A186219, A186228, A186237, A186238,
Cf. A000217 (triangular numbers)
Cf. A000566 (heptagonal numbers)

(* adjusted joint ranking; general formula *)
d=1/2; u=1/2; v=1/2; w=0; x=5/2; y=-3/2; 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}]  (* A186227 *)
Table[b[n],{n,1,100}]  (* A186228 *)

A186228 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 heptagonal numbers. Complement of A186227.

Original entry on oeis.org

2, 5, 8, 11, 15, 18, 21, 24, 27, 31, 34, 37, 40, 44, 47, 50, 53, 57, 60, 63, 66, 70, 73, 76, 79, 82, 86, 89, 92, 95, 99, 102, 105, 108, 112, 115, 118, 121, 125, 128, 131, 134, 137, 141, 144, 147, 150, 154, 157, 160, 163, 167, 170, 173, 176, 180, 183, 186, 189, 192, 196, 199, 202, 205, 209, 212, 215, 218, 222, 225, 228, 231, 235, 238, 241, 244, 248, 251, 254, 257, 260, 264, 267, 270, 273, 277, 280, 283, 286, 290, 293, 296, 299, 303, 306, 309

Offset: 1

Clark Kimberling, Feb 16 2011

nonn

See A186227.

			See A186227.

Cf. A186219, A186227, A186237, A186238.

Mathematica
```
(See A186227.)
```

cp's OEIS Frontend

A186238 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 heptagonal numbers. Complement of A186237.

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A186227 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 heptagonal numbers. Complement of A186228.

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A186228 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 heptagonal numbers. Complement of A186227.

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Mathematica