A186328 -id:A186328 - cp's OEIS frontend

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.

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

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.

Original entry on oeis.org

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 *)

A186331 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 A186330.

Original entry on oeis.org

1, 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, 150, 152, 155, 157, 159, 161, 163, 165, 167, 170, 172

Offset: 1

Clark Kimberling, Feb 17 2011

nonn

Does this differ from A186329 only at a(1)? - R. J. Mathar, Jan 09 2024

			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 after the hexagonal:
a=(2,3,5,7,9,11,13,15,16,....)=A186330
b=(1,4,6,8,10,12,14,17,19,...)=A186331.

Cf. A186328, A186329, A186330.

Mathematica
```
(See A186330.)
```

cp's OEIS Frontend

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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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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A186331 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 A186330.

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Author

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Comments

Examples

Crossrefs

Programs

Mathematica