A186350 -id:A186350 - cp's OEIS frontend

A186388 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f(i)=6i and g(j)=j(j+1)/2 (triangular number). Complement of A186387.

1, 2, 4, 5, 7, 9, 11, 14, 16, 19, 22, 25, 28, 31, 35, 38, 42, 46, 50, 55, 59, 64, 69, 74, 79, 84, 90, 95, 101, 107, 113, 120, 126, 133, 140, 147, 154, 161, 169, 176, 184, 192, 200, 209, 217, 226, 235, 244, 253, 262, 272, 281, 291, 301, 311, 322, 332, 343, 354, 365, 376, 387, 399, 410, 422, 434, 446, 459, 471, 484, 497, 510, 523, 536, 550, 563, 577, 591, 605, 620, 634, 649, 664, 679, 694, 709, 725, 740, 756, 772, 788, 805, 821

Offset: 1

Clark Kimberling, Feb 19 2011

nonn

			First, write
......6.....12..18....24..30. (6i)
1..3..6..10...15....21..28... (triangular)
Then replace each number by its rank, where ties are settled by ranking 6i before the triangular:
a=(3,6,8,10,12,13,15,17,...)=A186387
b=(1,2,4,5,7,9,11,14,16,...)=A186388.

Cf. A186350, A186387, A186389, A186390.

Mathematica
```
(See A186387.)
```

A186389 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) after g(j) when f(i)=g(j), where f(i)=6i and g(j)=j(j+1)/2 (triangular number). Complement of A186390.

Original entry on oeis.org

4, 6, 8, 10, 12, 14, 15, 17, 18, 20, 22, 23, 25, 26, 27, 29, 30, 32, 33, 35, 36, 37, 39, 40, 41, 43, 44, 45, 47, 48, 49, 51, 52, 53, 55, 56, 57, 58, 60, 61, 62, 63, 65, 66, 67, 69, 70, 71, 72, 74, 75, 76, 77, 78, 80, 81, 82, 83, 85, 86, 87, 88, 90, 91, 92

Offset: 1

Clark Kimberling, Feb 19 2011

nonn

See A186350 for a discussion of adjusted joint rank sequences.

			First, write
......6.....12..18....24..30. (6i)
1..3..6..10...15....21..28... (triangular)
Then replace each number by its rank, where ties are settled by ranking 6i after the triangular:
a=(4,6,8,10,12,14,15,17,...)=A186389
b=(1,2,3,5,7,9,11,13,16,...)=A186390.

Cf. A186350, A186387, A186388, A186390.

(* 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=6; v=0; x=1/2; y=1/2; (* 6i and triangular *)
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}]  (* A186389 *)
Table[b[n], {n, 1, 100}]  (* A186390 *)

A186493 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f(i)=5i and g(j)=j-th pentagonal number. Complement of A186494.

Original entry on oeis.org

2, 4, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 20, 22, 23, 24, 25, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83, 84, 85, 86, 87, 88, 89, 90, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 126, 127, 128, 129, 130, 131, 132, 133, 134

Offset: 1

Clark Kimberling, Feb 22 2011

nonn

See A186350 for a discussion of adjusted joint rank sequences.

			First, write
....5..10..15..20..25..30..35..40.. (5i),
1..5......12......22............35..(pentagonal numbers).
Then replace each number by its rank, where ties are settled by ranking 5i before the pentagonal number:
a=(2,4,6,7,9,10,11,13,14,15,17,...)=A186493,
b=(1,3,5,8,12,16,21,26,32,39,46,..)=A186494.

Cf. A186350, A186494, A186495, A186496.

(* 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=5; v=0; x=3/2; y=-1/2;
h[n_]:=(-y+(4x(u*n+v-d)+y^2)^(1/2))/(2x);
a[n_]:=n+Floor[h[n]];
k[n_]:=(x*n^2+y*n-v+d)/u;
b[n_]:=n+Floor[k[n]];
Table[a[n],{n,1,120}]  (* A186493 *)
Table[b[n],{n,1,100}]  (* A186494 *)

A186494 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f(i)=5i and g(j)=j-th pentagonal number. Complement of A186493.

Original entry on oeis.org

1, 3, 5, 8, 12, 16, 21, 26, 32, 39, 46, 54, 62, 71, 81, 91, 102, 113, 125, 138, 151, 165, 179, 194, 210, 226, 243, 260, 278, 297, 316, 336, 356, 377, 399, 421, 444, 467, 491, 516, 541, 567, 593, 620, 648, 676, 705, 734, 764, 795, 826, 858, 890, 923, 957, 991, 1026, 1061, 1097, 1134, 1171, 1209, 1247, 1286, 1326, 1366, 1407, 1448, 1490, 1533, 1576, 1620, 1664, 1709, 1755, 1801, 1848, 1895, 1943, 1992, 2041, 2091, 2141

Offset: 1

Clark Kimberling, Feb 22 2011

nonn

			First, write
....5..10..15..20..25..30..35..40.. (5i),
1..5......12......22............35..(pentagonal numbers).
Then replace each number by its rank, where ties are settled by ranking 5i before the pentagonal number:
a=(2,4,6,7,9,10,11,13,14,15,17,...)=A186493,
b=(1,3,5,8,12,16,21,26,32,39,46,..)=A186494.

Cf. A186350, A186493, A186495, A186496.

Mathematica
```
(See A186493.)
```

A186495 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) after g(j) when f(i)=g(j), where f(i)=5i (A008587) and g(j)=j-th pentagonal number (A000326). Complement of A186496.

Original entry on oeis.org

3, 4, 6, 7, 9, 10, 12, 13, 14, 15, 17, 18, 19, 21, 22, 23, 24, 25, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 39, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 92, 93, 94, 95, 96, 97, 98, 99, 100, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 138, 139, 140

Offset: 1

Clark Kimberling, Feb 22 2011

nonn

			First, write
...5..10..15..20..25..30..35..40... (5i),
1..5......12......22............35..(pentagonal numbers).
Then replace each number by its rank, where ties are settled by ranking 5i after the pentagonal number:
a=(3,4,6,7,9,10,12,13,14,15,17,...)=A186495,
b=(1,2,5,8,11,16,20,26,32,38,46,..)=A186496.

Cf. A000326, A008587, A186350, A186493, A186494, A186496.

(* 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=5; v=0; x=3/2; y=-1/2;
h[n_]:=(-y+(4x(u*n+v-d)+y^2)^(1/2))/(2x);
a[n_]:=n+Floor[h[n]];
k[n_]:=(x*n^2+y*n-v+d)/u;
b[n_]:=n+Floor[k[n]];
Table[a[n],{n,1,120}]  (* A186495 *)
Table[b[n],{n,1,100}]  (* A186496 *)

A186496 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) after g(j) when f(i)=g(j), where f(i)=5i and g(j)=j-th pentagonal number. Complement of A186495.

Original entry on oeis.org

1, 2, 5, 8, 11, 16, 20, 26, 32, 38, 46, 53, 62, 71, 80, 91, 101, 113, 125, 137, 151, 164, 179, 194, 209, 226, 242, 260, 278, 296, 316, 335, 356, 377, 398, 421, 443, 467, 491, 515, 541, 566, 593, 620, 647, 676, 704, 734, 764, 794, 826, 857, 890, 923, 956, 991, 1025, 1061, 1097, 1133, 1171, 1208, 1247, 1286, 1325, 1366, 1406, 1448, 1490, 1532, 1576, 1619, 1664, 1709, 1754, 1801, 1847, 1895, 1943, 1991, 2041

Offset: 1

Clark Kimberling, Feb 22 2011

nonn

			First, write
...5..10..15..20..25..30..35..40... (5i),
1..5......12......22............35..(pentagonal numbers).
Then replace each number by its rank, where ties are settled by ranking 5i after the pentagonal number:
a=(3,4,6,7,9,10,12,13,14,15,17,...)=A186495,
b=(1,2,5,8,11,16,20,26,32,38,46,..)=A186496.

Cf. A186350, A186493, A186494, A186495.

Mathematica
```
(See A186495.)
```

A186354 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 A186355.

Original entry on oeis.org

2, 4, 6, 8, 9, 11, 12, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 34, 35, 36, 37, 39, 40, 41, 42, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 67, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 83, 84, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 96, 97, 99, 100, 101, 102, 103, 104, 105, 107, 108, 109, 110, 111, 112, 113, 114, 116, 117, 118, 119, 120, 121, 122, 123, 125, 126, 127, 128, 129, 130, 131, 132, 134, 135

Offset: 1

Clark Kimberling, Feb 18 2011

nonn

See A186350.

			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. A186550, A186555, A186556, 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; (* odds and triangular *)
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}]  (* A186354 *)
Table[b[n],{n,1,100}]  (* A186355 *)

A186497 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-2 and g(j)=j-th triangular number. Complement of A186498.

Original entry on oeis.org

1, 4, 6, 7, 9, 11, 12, 14, 15, 16, 18, 19, 21, 22, 23, 25, 26, 27, 28, 30, 31, 32, 34, 35, 36, 37, 39, 40, 41, 42, 43, 45, 46, 47, 48, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 82, 84, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 96, 97, 99, 100, 101, 102, 103, 104, 105, 106, 108, 109, 110, 111, 112, 113, 114, 116, 117, 118, 119, 120, 121, 122, 123, 125, 126, 127, 128, 129, 130, 131, 132, 133, 135, 136, 137, 138, 139, 140, 141, 142, 144, 145, 146

Offset: 1

Clark Kimberling, Feb 22 2011

nonn

See A186350 for a discussion of adjusted joint rank sequences.

			First, write
1..4..7.10..13..16..19..22..25..28..31. (3i-2),
1.3..6..10....15.......21.......28.....(j(j+1)/2).
Then replace each number by its rank, where ties are settled by ranking 3i-2 before j(j+1)/2:
a=(1,4,6,7,9,11,12,14,15,16,18,...)=A186497,
b=(2,3,5,8,10,13,17,20,24,29,33,..)=A186498.

Cf. A186350, A186498, A186499, A186500.

(* 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=-2; 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]];
k[n_]:=(x*n^2+y*n-v+d)/u;
b[n_]:=n+Floor[k[n]];
Table[a[n],{n,1,120}]  (* A186497 *)
Table[b[n],{n,1,100}]  (* A186498 *)

A186498 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-2 and g(j)=j-th triangular number. Complement of A186497.

Original entry on oeis.org

2, 3, 5, 8, 10, 13, 17, 20, 24, 29, 33, 38, 44, 49, 55, 62, 68, 75, 83, 90, 98, 107, 115, 124, 134, 143, 153, 164, 174, 185, 197, 208, 220, 233, 245, 258, 272, 285, 299, 314, 328, 343, 359, 374, 390, 407, 423, 440, 458, 475, 493, 512, 530, 549, 569, 588, 608, 629, 649, 670, 692, 713, 735, 758, 780, 803, 827, 850, 874, 899, 923, 948, 974, 999, 1025, 1052, 1078, 1105, 1133, 1160, 1188, 1217, 1245, 1274, 1304

Offset: 1

Clark Kimberling, Feb 22 2011

nonn

			First, write
1..4..7.10..13..16..19..22..25..28..31. (3i-2),
1.3..6..10....15.......21.......28.....(j(j+1)/2).
Then replace each number by its rank, where ties are settled by ranking 3i-2 before j(j+1)/2:
a=(1,4,6,7,9,11,12,14,15,16,18,...)=A186497,
b=(2,3,5,8,10,13,17,20,24,29,33,..)=A186498.

Cf. A186350, A186497, A186499, A186500.

Mathematica
```
(See A186497.)
```

cp's OEIS Frontend

A186388 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f(i)=6i and g(j)=j(j+1)/2 (triangular number). Complement of A186387.

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A186389 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) after g(j) when f(i)=g(j), where f(i)=6i and g(j)=j(j+1)/2 (triangular number). Complement of A186390.

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A186493 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f(i)=5i and g(j)=j-th pentagonal number. Complement of A186494.

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Mathematica

A186494 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f(i)=5i and g(j)=j-th pentagonal number. Complement of A186493.

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A186495 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) after g(j) when f(i)=g(j), where f(i)=5i (A008587) and g(j)=j-th pentagonal number (A000326). Complement of A186496.

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A186496 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) after g(j) when f(i)=g(j), where f(i)=5i and g(j)=j-th pentagonal number. Complement of A186495.

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A186354 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 A186355.

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A186497 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-2 and g(j)=j-th triangular number. Complement of A186498.

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A186498 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-2 and g(j)=j-th triangular number. Complement of A186497.

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