A186291 -id:A186291 - cp's OEIS frontend

A186290 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 pentagonal numbers. Complement of A186291.

2, 3, 5, 7, 9, 11, 12, 14, 16, 18, 20, 21, 23, 25, 27, 29, 31, 32, 34, 36, 38, 40, 41, 43, 45, 47, 49, 51, 52, 54, 56, 58, 60, 61, 63, 65, 67, 69, 71, 72, 74, 76, 78, 80, 81, 83, 85, 87, 89, 90, 92, 94, 96, 98, 100, 101, 103, 105, 107, 109, 110, 112, 114, 116, 118, 120, 121, 123, 125, 127, 129, 130, 132, 134, 136, 138, 140, 141, 143, 145, 147, 149, 150, 152, 154, 156, 158, 160, 161, 163, 165, 167, 169, 170, 172, 174, 176, 178, 180, 181

Offset: 1

Clark Kimberling, Feb 17 2011

nonn

See A186219 for a discussion of adjusted joint rank sequences.

			First, write
1..4...9....16....25..36..49..... (squares)
1....5...12....22....35......51.. (pentagonal)
Replace each number by its rank, where ties are settled by ranking the square number after the pentagonal:
a=(2,3,5,7,9,11,12,14,....)=A186290.
b=(1,4,6,8,10,13,15,17,...)=A186291.

(* adjusted joint ranking; general formula *)
d=-1/2; u=1; v=0; w=0; x=3/2; y=-1/2; 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}]  (* A186290 *)
Table[b[n], {n, 1, 100}]  (* A186291 *)

A186288 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 pentagonal numbers. Complement of A186289.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 12, 14, 16, 18, 20, 21, 23, 25, 27, 29, 31, 32, 34, 36, 38, 40, 41, 43, 45, 47, 49, 51, 52, 54, 56, 58, 60, 61, 63, 65, 67, 69, 71, 72, 74, 76, 78, 80, 81, 83, 85, 87, 89, 90, 92, 94, 96, 98, 100, 101, 103, 105, 107, 109, 110, 112, 114, 116, 118, 120, 121, 123, 125, 127, 129, 130, 132, 134, 136, 138, 140, 141, 143, 145, 147, 149, 150, 152, 154, 156, 158, 160, 161, 163, 165, 167, 169, 170, 172, 174, 176, 178, 179, 181

Offset: 1

Clark Kimberling, Feb 17 2011

nonn

See A186219 for a discussion of adjusted rank sequences.

			First, write
1..4...9....16....25..36..49..... (squares)
1....5...12....22....35......51.. (pentagonal)
Replace each number by its rank, where ties are settled by ranking the square number before the pentagonal:
a=(1,3,5,7,9,11,12,14,....)=A186288.
b=(2,4,6,8,10,13,15,17,...)=A186289.

Cf. A186219, A186289, A186290, A186291, A000290 (squares), A000326 (pentagonal).

(* adjusted joint ranking; general formula *)
d=1/2; u=1; v=0; w=0; x=3/2; y=-1/2; 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}]  (* A186288 *)
Table[b[n],{n,1,100}]  (* A186289 *)

A186289 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 pentagonal numbers. Complement of A186288.

Original entry on oeis.org

2, 4, 6, 8, 10, 13, 15, 17, 19, 22, 24, 26, 28, 30, 33, 35, 37, 39, 42, 44, 46, 48, 50, 53, 55, 57, 59, 62, 64, 66, 68, 70, 73, 75, 77, 79, 82, 84, 86, 88, 91, 93, 95, 97, 99, 102, 104, 106, 108, 111, 113, 115, 117, 119, 122, 124, 126, 128, 131, 133, 135, 137, 139, 142, 144, 146, 148, 151, 153, 155, 157, 159, 162, 164, 166, 168, 171, 173, 175, 177, 180, 182, 184, 186, 188, 191, 193, 195, 197, 200

Offset: 1

Clark Kimberling, Feb 17 2011

nonn

See A186288.

			First, write
1..4...9....16....25..36..49..... (squares)
1....5...12....22....35......51.. (pentagonal)
Replace each number by its rank, where ties are settled by ranking the square number before the pentagonal:
a=(1,3,5,7,9,11,12,14,....)=A186288.
b=(2,4,6,8,10,13,15,17,...)=A186289.

Cf. A186288, A186290, A186291.

Mathematica
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(See A186288.)
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cp's OEIS Frontend

A186290 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 pentagonal numbers. Complement of A186291.

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A186288 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 pentagonal numbers. Complement of A186289.

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Mathematica

A186289 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 pentagonal numbers. Complement of A186288.

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Author

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Comments

Examples

Crossrefs

Programs

Mathematica