A186342 -id:A186342 - cp's OEIS frontend

A186343 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 octagonal numbers. Complement of A186342.

2, 4, 6, 9, 11, 14, 16, 19, 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, 89, 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, 159, 161, 163, 166, 168, 171, 173, 175, 178, 180, 183, 185, 188, 190, 192, 195, 197, 200, 202, 204, 207

Offset: 1

Clark Kimberling, Feb 18 2011

nonn

			First, write
1..5...12....22..35..... (pentagonal)
1....8....21........40.. (octagonal)
Then replace each number by its rank, where ties are settled by ranking the pentagonal number before the octagonal:
a=(1,3,5,7,8,10,12,13,15,...)=A186342
b=(2,4,6,9,11,14,16,19,21,...)=A186343.

Cf. A186342, A186344, A186345.

Mathematica
```
(See A186342.)
```

A186344 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 octagonal numbers. Complement of A186345.

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, 59, 61, 63, 65, 66, 68, 70, 71, 73, 75, 77, 78, 80, 82, 83, 85, 87, 88, 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, 158, 160, 162, 164, 165, 167, 169, 170

Offset: 1

Clark Kimberling, Feb 18 2011

nonn

			First, write
1..5...12....22..35..... (pentagonal)
1....8....21........40.. (octagonal)
Then replace each number by its rank, where ties are settled by ranking the pentagonal number after the octagonal:
a=(2,3,5,7,8,10,12,13,15,....)=A186344
b=(1,4,6,9,11,14,16,19,21,...)=A186345.

Cf. A186342, A186343, A186345.

(* adjusted joint ranking; general formula *)
d=-1/2; u=3/2; v=-1/2; w=0; x=3; y=-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}]  (* A186344 *)
Table[b[n], {n, 1, 100}]  (* A186345 *)

A186345 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 octagonal numbers. Complement of A186344.

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, 60, 62, 64, 67, 69, 72, 74, 76, 79, 81, 84, 86, 89, 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, 159, 161, 163, 166, 168, 171, 173, 175, 178, 180, 183, 185, 188, 190, 192, 195, 197, 200, 202, 204

Offset: 1

Clark Kimberling, Feb 18 2011

nonn

			First, write
1..5...12....22..35..... (pentagonal)
1....8....21........40.. (octagonal)
Then replace each number by its rank, where ties are settled by ranking the pentagonal number after the octagonal:
a=(2,3,5,7,8,10,12,13,15,....)=A186344
b=(1,4,6,9,11,14,16,19,21,...)=A186345.

Cf. A186342, A186343, A186344.

Mathematica
```
(See A186344.)
```

cp's OEIS Frontend

A186343 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 octagonal numbers. Complement of A186342.

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Mathematica

A186344 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 octagonal numbers. Complement of A186345.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A186345 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 octagonal numbers. Complement of A186344.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica