A186554 -id:A186554 - cp's OEIS frontend

A186355 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f(i)=3i and g(j)=j(j+1)/2 (triangular number). Complement of A186354.

1, 3, 5, 7, 10, 13, 16, 20, 24, 28, 33, 38, 43, 49, 55, 61, 68, 75, 82, 90, 98, 106, 115, 124, 133, 143, 153, 163, 174, 185, 196, 208, 220, 232, 245, 258, 271, 285, 299, 313, 328, 343, 358, 374, 390, 406, 423, 440, 457, 475, 493, 511, 530, 549, 568, 588, 608, 628, 649, 670, 691, 713, 735, 757, 780, 803, 826, 850, 874, 898, 923, 948, 973, 999, 1025, 1051, 1078, 1105, 1132, 1160, 1188, 1216, 1245, 1274, 1303

Offset: 1

Clark Kimberling, Feb 18 2011

nonn

See A186554.

Does this differ (apart from the additional 1) from A089108? - R. J. Mathar, Feb 25 2011

			First, write
...3..6..9....12..15..18..21..24.. (3*i)
1..3..6....10.....15......21.... (triangular)
Then replace each number by its rank, where ties are settled by ranking 3i before the triangular:
a=(2,4,6,8,9,11,12,14,15,17,....)=A186354
b=(1,3,5,7,10,13,16,20,24,28,...)=A186355.

Cf. A186554, A186556, A186557.

Mathematica
```
(See A186554.)
```

A186356 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) after g(j) when f(i)=g(j), where f(i)=3i and g(j)=j(j+1)/2 (triangular number). Complement of A186357.

Original entry on oeis.org

3, 5, 6, 8, 10, 11, 13, 14, 15, 17, 18, 20, 21, 22, 24, 25, 26, 27, 29, 30, 31, 33, 34, 35, 36, 38, 39, 40, 41, 42, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 80, 81, 83, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 101, 102, 103, 104, 105, 107, 108, 109, 110, 111, 112, 113, 115, 116, 117, 118, 119, 120, 121, 122, 124, 125, 126, 127, 128, 129, 130, 131, 132, 134, 135, 136, 137, 138, 139, 140, 141, 143, 144, 145, 146

Offset: 1

Clark Kimberling, Feb 18 2011

nonn

			First, write
...3..6..9....12..15..18..21..24.. (3*i)
1..3..6....10.....15......21.... (triangular)
Then replace each number by its rank, where ties are settled by ranking 3i after the triangular:
a=(3,5,6,8,10,11,13,14,15,..)=A186356
b=(1,2,4,7,9,12,16,19,23,...)=A186357.

Cf. A186554, A186555, A186557.

(* adjusted joint rank sequences a and b, using general formula for ranking 1st degree u*n+v and 2nd degree x*n^2+y*n+z *)
d=1/2; u=3; v=0; x=1/2; y=1/2;
h[n_]:=(-y+(4x(u*n+v-d)+y^2)^(1/2))/(2x);
a[n_]:=n+Floor[h[n]]; (* rank of u*n+v *)
k[n_]:=(x*n^2+y*n-v+d)/u;
b[n_]:=n+Floor[k[n]]; (* rank of x*n^2+y*n+d *)
Table[a[n],{n,1,120}]  (* A186356 *)
Table[b[n],{n,1,100}]  (* A186357 *)

A186357 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) after g(j) when f(i)=g(j), where f(i)=3i and g(j)=j(j+1)/2 (triangular number). Complement of A186357.

Original entry on oeis.org

1, 2, 4, 7, 9, 12, 16, 19, 23, 28, 32, 37, 43, 48, 54, 61, 67, 74, 82, 89, 97, 106, 114, 123, 133, 142, 152, 163, 173, 184, 196, 207, 219, 232, 244, 257, 271, 284, 298, 313, 327, 342, 358, 373, 389, 406, 422, 439, 457, 474, 492, 511, 529, 548, 568, 587, 607, 628, 648, 669, 691, 712, 734, 757, 779, 802, 826, 849, 873, 898, 922, 947, 973, 998, 1024, 1051, 1077, 1104, 1132, 1159, 1187, 1216, 1244, 1273, 1303, 1332, 1362, 1393, 1423, 1454

Offset: 1

Clark Kimberling, Feb 18 2011

nonn

			First, write
...3..6..9....12..15..18..21..24.. (3*i)
1..3..6....10.....15......21.... (triangular)
Then replace each number by its rank, where ties are settled by ranking 3i after the triangular:
a=(3,5,6,8,10,11,13,14,15,..)=A186356
b=(1,2,4,7,9,12,16,19,23,...)=A186357.

Cf. A186554, A186555, A186556.

Mathematica
```
(See A186556.)
```

cp's OEIS Frontend

A186355 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f(i)=3i and g(j)=j(j+1)/2 (triangular number). Complement of A186354.

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A186356 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) after g(j) when f(i)=g(j), where f(i)=3i and g(j)=j(j+1)/2 (triangular number). Complement of A186357.

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Author

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Examples

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Mathematica

A186357 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) after g(j) when f(i)=g(j), where f(i)=3i and g(j)=j(j+1)/2 (triangular number). Complement of A186357.

Views

Author

Keywords

Examples

Crossrefs

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Mathematica