A186379 -id:A186379 - cp's OEIS frontend

A186380 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f(i)=4i and g(j)=j(j+1)/2 (triangular number). Complement of A186379.

1, 2, 4, 6, 8, 11, 14, 17, 20, 23, 27, 31, 35, 40, 45, 50, 55, 60, 66, 72, 78, 85, 92, 99, 106, 113, 121, 129, 137, 146, 155, 164, 173, 182, 192, 202, 212, 223, 234, 245, 256, 267, 279, 291, 303, 316, 329, 342, 355, 368, 382, 396, 410, 425, 440, 455, 470, 485, 501, 517, 533, 550, 567, 584, 601, 618, 636, 654, 672, 691, 710, 729, 748, 767, 787, 807, 827, 848, 869, 890, 911, 932, 954, 976, 998, 1021, 1044

Offset: 1

Clark Kimberling, Feb 19 2011

nonn

See A186379.

			First, write
.....4..8..12..16..20..24..28.. (4*i)
1..3..6..10..15.....21.....28.. (triangular)
Then replace each number by its rank, where ties are settled by ranking 4i before the triangular:
a=(3,5,7,9,10,12,13,15,16,..)=A186379
b=(1,2,4,6,8,11,14,17,20,...)=A186380.

Cf. A186379, A186381, A186382.

Mathematica
```
(See A186379.)
```

a(n)=n+floor(-1/2+sqrt(8n-3/4))=A186379(n).

b(n)=n+floor((n^2+n+1)/8)=A186380(n).

A186381 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) after g(j) when f(i)=g(j), where f(i)=4i and g(j)=j(j+1)/2 (triangular number). Complement of A186382.

Original entry on oeis.org

3, 5, 7, 9, 10, 12, 14, 15, 17, 18, 19, 21, 22, 24, 25, 26, 28, 29, 30, 32, 33, 34, 36, 37, 38, 39, 41, 42, 43, 45, 46, 47, 48, 50, 51, 52, 53, 54, 56, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 84, 86, 87, 88

Offset: 1

Clark Kimberling, Feb 19 2011

nonn

			First, write
.....4..8..12..16..20..24..28.. (4*i)
1..3..6..10..15.....21.....28.. (triangular)
Then replace each number by its rank, where ties are settled by ranking 4i after the triangular:
a=(3,5,7,9,10,12,14,15,17,..)=A186381
b=(1,2,4,6,8,11,13,16,20,...)=A186382.

Cf. A186379, A186380, A186382.

(* 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=4; v=0; x=1/2; y=1/2; (* 4i 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}]  (* A186381 *)
Table[b[n], {n, 1, 100}]  (* A186382 *)

A186382 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) after g(j) when f(i)=g(j), where f(i)=4i and g(j)=j(j+1)/2 (triangular number). Complement of A186381.

Original entry on oeis.org

1, 2, 4, 6, 8, 11, 13, 16, 20, 23, 27, 31, 35, 40, 44, 49, 55, 60, 66, 72, 78, 85, 91, 98, 106, 113, 121, 129, 137, 146, 154, 163, 173, 182, 192, 202, 212, 223, 233, 244, 256, 267, 279, 291, 303, 316, 328, 341, 355, 368, 382, 396, 410, 425, 439, 454, 470, 485, 501, 517, 533, 550, 566, 583, 601, 618, 636, 654, 672, 691, 709, 728, 748, 767, 787, 807, 827, 848, 868, 889, 911, 932, 954, 976, 998, 1021, 1043, 1066, 1090

Offset: 1

Clark Kimberling, Feb 19 2011

nonn

See A186381.

			See A186381.

Cf. A186379, A186380, A186381.

Mathematica
```
(See A186381.)
```

cp's OEIS Frontend

A186380 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f(i)=4i and g(j)=j(j+1)/2 (triangular number). Complement of A186379.

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A186381 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) after g(j) when f(i)=g(j), where f(i)=4i and g(j)=j(j+1)/2 (triangular number). Complement of A186382.

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Mathematica

A186382 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) after g(j) when f(i)=g(j), where f(i)=4i and g(j)=j(j+1)/2 (triangular number). Complement of A186381.

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Mathematica