A186331 -id:A186331 - cp's OEIS frontend

A186330 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 pentagonal numbers and the hexagonal numbers. Complement of A186331.

2, 3, 5, 7, 9, 11, 13, 15, 16, 18, 20, 22, 24, 26, 28, 29, 31, 33, 35, 37, 39, 41, 43, 44, 46, 48, 50, 52, 54, 56, 57, 59, 61, 63, 65, 67, 69, 71, 72, 74, 76, 78, 80, 82, 84, 85, 87, 89, 91, 93, 95, 97, 99, 100, 102, 104, 106, 108, 110, 112, 113, 115, 117, 119, 121, 123, 125, 126, 128, 130, 132, 134, 136, 138, 140, 141, 143, 145, 147, 149, 151, 153, 154, 156, 158, 160, 162, 164, 166, 168, 169, 171, 173, 175, 177, 179, 181, 182, 184, 186

Offset: 1

Clark Kimberling, Feb 17 2011

nonn

Does this differ (apart from a(1)) from A186329 or A186328? - R. J. Mathar, Feb 25 2011

			First, write
1..5...12....22.....35......  (pentagonal)
1....6....15....28.......45.. (hexagonal)
Replace each number by its rank, where ties are settled by ranking the pentagonl number after the hexagonal:
a=(1,3,5,7,9,11,13,15,16,....)=A186330
b=(2,4,6,8,10,12,14,17,19,...)=A186331.

Cf. A186328, A186329, A186331.

(* adjusted joint ranking; general formula *)
d=-1/2; u=3/2; v=-1/2; 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}]  (* A186330 *)
Table[b[n], {n, 1, 100}]  (* A186331 *)

A186328 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 pentagonal numbers and the hexagonal numbers. Complement of A186329.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 16, 18, 20, 22, 24, 26, 28, 29, 31, 33, 35, 37, 39, 41, 43, 44, 46, 48, 50, 52, 54, 56, 57, 59, 61, 63, 65, 67, 69, 71, 72, 74, 76, 78, 80, 82, 84, 85, 87, 89, 91, 93, 95, 97, 99, 100, 102, 104, 106, 108, 110, 112, 113, 115, 117, 119, 121, 123, 125, 126, 128, 130, 132, 134, 136, 138, 140, 141, 143, 145, 147, 149, 151, 153, 154, 156, 158, 160, 162, 164, 166, 168, 169, 171, 173, 175, 177, 179, 181, 182, 184, 186

Offset: 1

Clark Kimberling, Feb 17 2011

nonn

See A186219 for a discussion of adjusted joint rank sequences.

			First, write
1..5...12....22.....35......  (pentagonal)
1....6....15....28.......45.. (hexagonal)
Then replace each number by its rank, where ties are settled by ranking the pentagonal number before the hexagonal:
a=(1,3,5,7,9,11,13,15,16,....)=A186328
b=(2,4,6,8,10,12,14,17,19,...)=A186329.

Cf. A186219, A186329, A186330, A186331,

A000384 (pentagonal), A000384 (hexagonal).

(* adjusted joint ranking; general formula *)
d=1/2; u=3/2; v=-1/2; 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}]  (* A186328 *)
Table[b[n], {n, 1, 100}]  (* A186329 *)

A186329 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 pentagonal numbers and the hexagonal numbers. Complement of A186328.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 17, 19, 21, 23, 25, 27, 30, 32, 34, 36, 38, 40, 42, 45, 47, 49, 51, 53, 55, 58, 60, 62, 64, 66, 68, 70, 73, 75, 77, 79, 81, 83, 86, 88, 90, 92, 94, 96, 98, 101, 103, 105, 107, 109, 111, 114, 116, 118, 120, 122, 124, 127, 129, 131, 133, 135, 137, 139, 142, 144, 146, 148

Offset: 1

Clark Kimberling, Feb 17 2011

nonn

			First, write
1..5...12....22.....35......  (pentagonal)
1....6....15....28.......45.. (hexagonal)
Then replace each number by its rank, where ties are settled by ranking the pentagonal number before the hexagonal:
a=(1,3,5,7,9,11,13,15,16,....)=A186328
b=(2,4,6,8,10,12,14,17,19,...)=A186329.

Cf. A186328, A186330, A186331.

Mathematica
```
(See A186328.)
```

Terms corrected by Clark Kimberling and R. J. Mathar, Jun 07 2011

cp's OEIS Frontend

A186330 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 pentagonal numbers and the hexagonal numbers. Complement of A186331.

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A186328 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 pentagonal numbers and the hexagonal numbers. Complement of A186329.

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A186329 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 pentagonal numbers and the hexagonal numbers. Complement of A186328.

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