A186352 -id:A186352 - cp's OEIS frontend

A186350 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 odd numbers and the triangular numbers. Complement of A186351.

1, 3, 5, 7, 8, 10, 11, 12, 14, 15, 16, 18, 19, 20, 22, 23, 24, 25, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 52, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 88, 89, 90, 91, 92, 93, 95, 96, 97, 98, 99, 100, 101, 102, 103, 105, 106, 107, 108, 109, 110, 111, 112, 113, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 138, 139, 140, 141

Offset: 1

Clark Kimberling, Feb 18 2011

nonn

Suppose that f and g are strictly increasing functions for which (f(i)) and (g(j)) are integer sequences.  If 0<|d|<1, the sets F={f(i): i>=1} and G={g(j)+d: j>=1} are clearly disjoint.  Let f^=(inverse of f) and g^=(inverse of g).  When the numbers in F and G are jointly ranked, the rank of f(n) is a(n):=n+floor(g^(f(n))-d), and the rank of g(n)+d is b(n):=n+floor(f^(g(n))+d).  Therefore, the sequences a and b are a complementary pair.
Although the sequences (f(i)) and (g(j)) may not be disjoint, the sequences (f(i)) and (g(j)+d) are disjoint, and this observation enables two types of adjusted joint rankings:
(1) if 0Using f(i)=ui+v, g(j)=xj^2+yj+z, we find a and b given by
  a(n)=n+floor((-y+sqrt(4x(un+v-d)+y^2))/(2x)),
  b(n)=n+floor((xn^2+yn-v+d)/(2u))),
  where a(n) is the rank of un+v and b(n) is the rank
  xn^2+yn+z+d, and d must be chosen small enough, in
  absolute value, that the sets F and G are disjoint.
Example:  f=A000217 (odd numbers) and g=A000290 (triangular numbers) yield adjusted joint rank sequences a=A186350 and b=A186351 for d=1/2 and a=A186352 and b=A186353 for d=-1/2.
For other classes of adjusted joint rank sequences, see A186145 and A186219.

			First, write
1..3..5..7..9..11..13..15..17..21..23.. (odds)
1..3....6.....10.......15......21.... (triangular)
Then replace each number by its rank, where ties are settled by ranking the odd number before the triangjular:
a=(1,3,5,7,8,10,11,12,14,....)=A186350
b=(2,4,6,9,13,17,21,26,32,...)=A186351.

Cf. A186145, A186219, A186351, A186352, A186353,

A005408 (odd numbers), A000217 (triangular numbers).

(* 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=2; v=-1; 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}]  (* A186350 *)
Table[b[n],{n,1,100}]  (* A186351 *)

a(n)=n+floor(-1/2+sqrt(4n-9/4))=A186350(n).

b(n)=n+floor((n^2+n+3)/4)=A186351(n).

A186351 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 odd numbers and the triangular numbers. Complement of A186350.

Original entry on oeis.org

2, 4, 6, 9, 13, 17, 21, 26, 32, 38, 44, 51, 59, 67, 75, 84, 94, 104, 114, 125, 137, 149, 161, 174, 188, 202, 216, 231, 247, 263, 279, 296, 314, 332, 350, 369, 389, 409, 429, 450, 472, 494, 516, 539, 563, 587, 611, 636, 662, 688, 714, 741, 769, 797, 825, 854, 884, 914, 944, 975, 1007, 1039, 1071, 1104, 1138, 1172, 1206, 1241, 1277, 1313, 1349, 1386, 1424, 1462, 1500

Offset: 1

Clark Kimberling, Feb 18 2011

nonn

See A186350.

			First, write
1..3..5..7..9..11..13..15..17..21..23.. (odds)
1..3....6.....10.......15......21.... (triangular)
Then replace each number by its rank, where ties are settled by ranking the odd number before the triangular:
a = (1,3,5,7,8,10,11,12,14,....) = A186350
b = (2,4,6,9,13,17,21,26,32,...) = A186351.

Cf. A186350, A186352, A186353, A227687.

Mathematica
```
(See A186350.)
```

See A186350.

A186353 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 odd numbers and the triangular numbers. Complement of A186353.

Original entry on oeis.org

1, 3, 6, 9, 12, 16, 21, 26, 31, 37, 44, 51, 58, 66, 75, 84, 93, 103, 114, 125, 136, 148, 161, 174, 187, 201, 216, 231, 246, 262, 279, 296, 313, 331, 350, 369, 388, 408, 429, 450, 471, 493, 516, 539, 562, 586, 611, 636, 661, 687, 714, 741, 768, 796, 825, 854, 883, 913, 944, 975, 1006, 1038, 1071, 1104, 1137, 1171, 1206, 1241, 1276, 1312, 1349, 1386, 1423, 1461, 1500, 1539, 1578, 1618, 1659, 1700, 1741, 1783, 1826, 1869, 1912, 1956, 2001, 2046, 2091

Offset: 1

Clark Kimberling, Feb 18 2011

nonn

			First, write
1..3..5..7..9..11..13..15..17..21..23.. (odds)
1..3....6.....10.......15......21.... (triangular)
Then replace each number by its rank, where ties are settled by ranking the odd number after the triangular:
a=(2,4,5,7,8,10,11,13,14,15,....)=A186352
b=(1,3,6,9,12,16,21,26,31,37,...)=A186353.

Cf. A186350, A186351, A186352.

Mathematica
```
(See A186352.)
```

a(n)=n+floor(-1/2+sqrt(4n-3/4))=A186352(n).

b(n)=n+floor((n^2+n+1)/4)=A186353(n).

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cp's OEIS Frontend

A186350 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 odd numbers and the triangular numbers. Complement of A186351.

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A186351 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 odd numbers and the triangular numbers. Complement of A186350.

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A186353 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 odd numbers and the triangular numbers. Complement of A186353.

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