A186219 -id:A186219 - cp's OEIS frontend

A187224 Rank transform of the sequence floor(3*n/2).

1, 3, 5, 7, 8, 11, 12, 14, 16, 18, 19, 21, 23, 25, 27, 29, 30, 32, 34, 36, 38, 40, 41, 43, 45, 47, 48, 51, 52, 54, 56, 58, 60, 61, 63, 65, 67, 69, 70, 72, 74, 76, 78, 80, 81, 83, 85, 87, 89, 91, 92, 94, 96, 98, 100, 102, 103, 105, 107, 109, 110, 113, 114, 116, 118, 120, 121, 123, 125, 127, 129, 131, 132, 135, 136, 138, 140, 142, 143, 145, 147, 149, 151, 153, 154, 156, 158, 160, 162, 163, 165, 167, 169, 171, 172, 175, 176, 178, 180, 182, 183, 185, 187, 189, 191, 193, 194, 196, 198, 200

Offset: 1

Clark Kimberling, Mar 07 2011

nonn

Complement of A187225.
The notion of the rank transform of a sequence is introduced as follows.  Suppose that a=(a(n)), for n>=1, is a nondecreasing sequence of nonnegative integers, where a(1)<=1, and suppose that b=(b(n)), for n>=1, is an increasing sequence of positive integers.
Define h(1)=a(1), and for n>1, define h(n)=the number of numbers b(i) satisfying a(n-1)<=b(i)Define r(1)=1, and for n>1, define r(n)=b(n-1)+h(n)+1.
The sequence r is the adjusted rank sequence when a and b are jointly ranked, with a(i) before b(j) when a(i)=b(j).  (For a discussion of adjusted joint rank sequences, see A186219 and A186350.)
If r(n)=b(n) for all n>=1, we call r the rank transform of a and denote it by R(a).  To summarize,
  (1)  initial values: r(1)=1, h(1)=a(1);
  (2)  counting function: h(n)= # r(i) in the half-open
       interval [a(n-1),a(n));
  (3)  recurrence:  r(n)=r(n-1)+h(n)+1.
Assuming a unbounded, let c be the number of a(i)<=1, let c(1)=c+1, and for n>1, let c(n) be the rank of r(n) when all the numbers a(i)<=r(n) and r(1),...,r(n-1), r(n) are jointly ranked.  Then, clearly, a(n)<=r(n)<=c(n) for n>=1, and the sequences r and c are a complementary pair.
What conditions on the sequence a will ensure that R(a) exists?  That is, what conditions will ensure that the counting function in (2) can be determined inductively, so that the recurrence (3) can be used to self-generate the sequence r?  The answer is this:  a(n)<=c(n-1)+1; viz., if a(n)>c(n-1)+1, then c(n-1)+1=r(n), but then a(n)>r(n), a contradiction, but if a(n)<=c(n-1)+1, there is no such obstacle.
Examples:
R(A000012)=A000027
R(A000027)=A000201, the lower Wythoff sequence
R(A004526)=A026367
R(A005408)=A005408
Returning now to a and b as above, let (r(1,k)) be the adjusted joint rank sequence (AJRS) of a and b, with a(i) before b(j) when a(i)=b(j).  Let (r(2,k)) be the AJRS of a and (r(1,k)); and inductively, let (r(n,k)) be the AJRS of a and (r(n-1,k)).  If R(a) exists, then the limit of (r(n,k)) is R(a).
Thus, any choice of initial sequence b can be used to determine the first thousand terms of R(a).  In the Mathematica program below, b=(1,2,3,4,...)=A000027.

			a... 1..3..4..6..7...9...10..12..13..15..16..18..19..
r... 1..3..5..7..8...11..12..14..16..18..19..21..23..
c... 2..4..6..9..10..13..15..17..20..22..24..26..28..
h... 1..1..1..1..0...2...0...1...1...1...0...1...1...
The sequences which converge to R(a), starting with
a=A187224 and b=A000027:
a(k)....1..3..4..6..7...9...10..12..13..15...
b(k)....1..2..3..4..5...6...7...8...9...10...
r(1,k)..1..4..6..9..11..14..16..19..21..24...
r(2,k)..1..3..4..6..8...9...11..13..14..16...
r(3,k)..1..3..5..7..9...11..13..15..16..19...
r(4,k)..1..3..5..7..8...10..12..14..15..17...
r(5,k)..1..3..5..7..8...11..12..14..16..18...

Cf. A186219, A186350, A187225.

seqA=Table[Floor[3*n/2], {n,1,220}]     (* A032766 *)
seqB=Table[n, {n,1,120}];               (* A000027 *)
jointRank[{seqA_,seqB_}]:={Flatten@Position[#1,{,1}],Flatten@Position[#1,{,2}]}&[Sort@Flatten[{{#1,1}&/@seqA,{#1,2}&/@seqB},1]];
limseqU=FixedPoint[jointRank[{seqA,#1[[1]]}]&,jointRank[{seqA,seqB}]][[1]]                     (* A187224 *)
Complement[Range[Length[seqA]],limseqU] (* A187225 *)
(* by Peter J. C. Moses, Mar 07 2011 *)

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.

Original entry on oeis.org

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).

A186159 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 triangular numbers and octagonal numbers. Complement of A186274.

Original entry on oeis.org

1, 3, 4, 6, 7, 8, 10, 11, 13, 14, 16, 17, 18, 20, 21, 23, 24, 25, 27, 28, 30, 31, 32, 34, 35, 37, 38, 39, 41, 42, 44, 45, 47, 48, 49, 51, 52, 54, 55, 56, 58, 59, 61, 62, 63, 65, 66, 68, 69, 70, 72, 73, 75, 76, 77, 79, 80, 82, 83, 85, 86, 87, 89, 90, 92, 93, 94, 96, 97, 99, 100, 101, 103, 104, 106, 107, 108, 110, 111, 113, 114, 116, 117, 118, 120, 121, 123, 124, 125, 127, 128, 130, 131, 132, 134, 135, 137, 138, 139, 141

Offset: 1

Clark Kimberling, Feb 13 2011

nonn

See A186219 for a discussion of adjusted joint rank sequences.

			First, write the triangular and octagonal numbers:
1..3..6.....10..15..21..28
1........8..........21......
Then replace each by its rank, where ties are settled by ranking the triangular number before the octagonal:
a=(1,3,4,6,7,8,10,11,13,...)=A186159.
b=(2,5,9,12,15,19,22,26,...)=A186274.

Cf. A186219, A186274, A186275, A186276,
Cf. A000217 (triangular numbers).
Cf. A000567 (octagonal numbers).

(* adjusted joint ranking; general formula *)
d=1/2; u=1/2; v=1/2; w=0; x=3; y=-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}] (* A186159 *)
Table[b[n],{n,1,100}] (* A186274 *)

A186222 Adjusted joint rank sequence of (g(i)) and (f(j)) with f(i) after g(j) when f(i)=g(j), where f and g are the triangular numbers and squares. Complement of A186221.

Original entry on oeis.org

1, 4, 6, 9, 11, 13, 16, 18, 21, 23, 26, 28, 30, 33, 35, 38, 40, 42, 45, 47, 50, 52, 55, 57, 59, 62, 64, 67, 69, 71, 74, 76, 79, 81, 83, 86, 88, 91, 93, 96, 98, 100, 103, 105, 108, 110, 112, 115, 117, 120, 122, 125, 127, 129, 132, 134, 137, 139, 141, 144, 146, 149, 151, 154, 156, 158, 161, 163, 166, 168, 170, 173, 175, 178, 180, 182, 185, 187, 190, 192, 195, 197, 199, 202, 204, 207, 209, 211, 214, 216, 219, 221, 224, 226, 228, 231, 233, 236, 238, 240

Offset: 1

Clark Kimberling, Feb 15 2011

nonn

See A186221.

			See A186221.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A186219, A186220, A186221.

[n + Floor(-1/2 + Sqrt(2*n^2)): n in [1..120]]; // G. C. Greubel, Aug 18 2018

(* adjusted joint ranking; general formula *)
d=-1/4; u=1/2; v=1/2; w=0; x=1; y=0; 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}] (* A186221 *)
Table[b[n],{n,1,100}] (* A186222 *)
Table[n + Floor[Sqrt[2*n^2] - 1/2], {n, 1, 120}] (* G. C. Greubel, Aug 18 2018 *)

vector(120, n, n + floor(-1/2 + sqrt(2*n^2))) \\ G. C. Greubel, Aug 18 2018

a(n) = n + floor(-1/2 + sqrt(2*n^2)).

A186499 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f(i)=i^2 and g(j)=-4+5j^2. Complement of A186500.

Original entry on oeis.org

1, 3, 4, 5, 7, 8, 10, 11, 13, 14, 15, 17, 18, 20, 21, 23, 24, 26, 27, 28, 30, 31, 33, 34, 36, 37, 39, 40, 41, 43, 44, 46, 47, 49, 50, 52, 53, 55, 56, 57, 59, 60, 62, 63, 65, 66, 68, 69, 70, 72, 73, 75, 76, 78, 79, 81, 82, 83, 85, 86, 88, 89, 91, 92, 94, 95, 96, 98, 99, 101, 102, 104, 105, 107, 108, 109, 111, 112, 114, 115, 117, 118, 120, 121, 123, 124, 125, 127, 128, 130, 131, 133, 134, 136, 137, 138, 140, 141, 143, 144

Offset: 1

Clark Kimberling, Feb 22 2011

nonn

See A186219 for a discussion of adjusted joint rank sequences.

The pairs (i,j) for which i^2=-4+5j^2 are (L(2h-2),F(2h-1)), where L=A000032 (Lucas numbers) and F=A000045 (Fibonacci numbers); compare this with the comment at A186511.

			First, write
1..4..9..16..25..36..49..... (i^2)
1........16........41........(-4+5j^2)
Then replace each number by its rank, where ties are settled by ranking i^2 before -4+5j^2:
a=(1,3,4,5,7,8,10,11,13,14,15,17,18...)=A186499
b=(2,6,9,12,16,19,22,25,29,32,35,38,..)=A186500.

Cf. A186219, A186500, A186511, A186512.

(* adjusted joint rank sequences a and b, using general formula for ranking ui^2+vi+w and xj^2+yj+z *)
d=1/2; u=1; v=0; w=0; x=5; y=0; z=4;
h[n_]:=-y+(4x(u*n^2+v*n+w-z-d)+y^2)^(1/2);
a[n_]:=n+Floor[h[n]/(2 x)];
k[n_]:=-v+(4u(x*n^2+y*n+z-w+d)+v^2)^(1/2);
b[n_]:=n + Floor[k[n]/(2 u)];
Table[a[n], {n, 1, 100}]  (* A186499 *)
Table[b[n], {n, 1, 100}]  (* A186500 *)

a(n)=n+floor((1/10)(sqrt(2n^2+7)))=A186499(n).

b(n)=n+floor(sqrt(5n^2-7/2))=A186500(n).

A186500 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f(i)=i^2 and g(j)=-4+5j^2. Complement of A186499.

Original entry on oeis.org

2, 6, 9, 12, 16, 19, 22, 25, 29, 32, 35, 38, 42, 45, 48, 51, 54, 58, 61, 64, 67, 71, 74, 77, 80, 84, 87, 90, 93, 97, 100, 103, 106, 110, 113, 116, 119, 122, 126, 129, 132, 135, 139, 142, 145, 148, 152, 155, 158, 161, 165, 168, 171, 174, 177, 181, 184, 187, 190, 194, 197, 200, 203, 207, 210, 213, 216, 220, 223, 226, 229, 232, 236, 239, 242, 245, 249, 252, 255, 258, 262, 265

Offset: 1

Clark Kimberling, Feb 22 2011

nonn

See A186219 for a discussion of adjusted joint rank sequences.

The pairs (i,j) for which i^2=-4+5j^2 are (L(2h-2),F(2h-1)), where L=A000032 (Lucas numbers) and F=A000045 (Fibonacci numbers).

			First, write
1..4..9..16..25..36..49..... (i^2)
1........16........41........(-4+5j^2)
Then replace each number by its rank, where ties are settled by ranking i^2 before -4+5j^2:
a=(1,3,4,5,7,8,10,11,13,14,15,17,18...)=A186499
b=(2,6,9,12,16,19,22,25,29,32,35,38,.)=A186500.

Cf. A186219, A186499, A186511, A186512.

Mathematica
```
(See A186499.)
```

a(n)=n+floor((1/10)(sqrt(2n^2+7)))=A186499(n).

b(n)=n+floor(sqrt(5n^2-7/2))=A186500(n).

A186221 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 triangular numbers and squares. Complement of A186222.

Original entry on oeis.org

2, 3, 5, 7, 8, 10, 12, 14, 15, 17, 19, 20, 22, 24, 25, 27, 29, 31, 32, 34, 36, 37, 39, 41, 43, 44, 46, 48, 49, 51, 53, 54, 56, 58, 60, 61, 63, 65, 66, 68, 70, 72, 73, 75, 77, 78, 80, 82, 84, 85, 87, 89, 90, 92, 94, 95, 97, 99, 101, 102, 104, 106, 107, 109, 111, 113, 114, 116, 118, 119, 121, 123, 124, 126, 128, 130, 131, 133, 135, 136, 138, 140, 142, 143, 145, 147, 148, 150, 152, 153, 155, 157, 159, 160, 162, 164, 165, 167, 169, 171

Offset: 1

Clark Kimberling, Feb 15 2011

See A186219.

			First, write
1..3...6..10..15...21..28..36..45...  (triangular)
1....4...9......16...25....36....49.. (square)
Replace each number by its rank, where ties are settled by ranking the triangular number after the square:
a=(2,3,5,7,8,10,12,14,...)
b=(1,4,6,9,11,13,16,18,...).

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A061288, A186219, A186220, A186222.

[n + Floor(Sqrt((n^2+n)/2 + 1/4)): n in [1..120]]; // G. C. Greubel, Aug 18 2018

(* adjusted joint ranking; general formula *)
d=-1/4; u=1/2; v=1/2; w=0; x=1; y=0; 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}] (* A186221 *)
Table[b[n],{n,1,100}] (* A186222 *)
a[ n_] := n + Floor[ Sqrt[ n (n + 1)/2]]; (* Michael Somos, Aug 19 2018 *)

vector(120, n, n + floor(sqrt((n^2+n)/2 + 1/4))) \\ G. C. Greubel, Aug 18 2018
{a(n) = n + sqrtint( n * (n+1) \ 2)}; /* Michael Somos, Aug 19 2018 */

a(n) = n + floor(sqrt((n^2+n)/2 + 1/4)).

a(n) = A061288(n) - n for all n in Z. - Michael Somos, Aug 19 2018

A186223 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 triangular numbers and pentagonal numbers. Complement of A186224.

Original entry on oeis.org

1, 3, 5, 6, 8, 9, 11, 13, 14, 16, 17, 19, 20, 22, 24, 25, 27, 28, 30, 31, 33, 35, 36, 38, 39, 41, 43, 44, 46, 47, 49, 50, 52, 54, 55, 57, 58, 60, 61, 63, 65, 66, 68, 69, 71, 73, 74, 76, 77, 79, 80, 82, 84, 85, 87, 88, 90, 91, 93, 95, 96, 98, 99, 101, 102, 104, 106, 107, 109, 110, 112, 114, 115, 117, 118, 120, 121, 123, 125, 126, 128, 129, 131, 132, 134, 136, 137, 139, 140, 142, 143, 145, 147, 148, 150, 151, 153, 155, 156, 158

Offset: 1

Clark Kimberling, Feb 15 2011

nonn

See A186219 for a general description.

			First, write
1..3...6..10....15...21.....28......36...45...  (triangular)
1....5.........12...........22......35........... (pentagonal)
Replace each number by its rank, where ties are settled by ranking the triangular number before the pentagonal:
a=(1,3,5,6,8,9,11,13,...)
b=(2,4,7,10,12,15,18,...).

Cf. A186219, A186224, A186225, A186226,

A000217 (triangular), A000326 (pentagonal).

d=1/2; u=1/2; v=1/2; w=0; x=3/2; y=-1/2; z=0;
(* triangular & pentagonal *)
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}]  (* A186223 *)
Table[b[n],{n,1,100}]  (* A186224 *)

A186225 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 triangular numbers and pentagonal numbers. Complement of A186226.

Original entry on oeis.org

2, 3, 5, 6, 8, 9, 11, 13, 14, 16, 17, 19, 20, 22, 24, 25, 27, 28, 30, 32, 33, 35, 36, 38, 39, 41, 43, 44, 46, 47, 49, 50, 52, 54, 55, 57, 58, 60, 61, 63, 65, 66, 68, 69, 71, 73, 74, 76, 77, 79, 80, 82, 84, 85, 87, 88, 90, 91, 93, 95, 96, 98, 99, 101, 102, 104, 106, 107, 109, 110, 112, 114, 115, 117, 118, 120, 121, 123, 125, 126, 128, 129, 131, 132, 134, 136, 137, 139, 140, 142, 143, 145, 147, 148, 150, 151, 153, 155, 156, 158

Offset: 1

Clark Kimberling, Feb 15 2011

nonn

			See A186223.

A186219, A186223, A186224, A186226.

(* Program for adjusted rank sequences as described at A186219 *)
d=-1/2; u=1/2; v=1/2; w=0; x=3/2; y=-1/2; z=0; (* triangular & pentagonal *)
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}]  (* A186225 *)
Table[b[n],{n,1,100}]  (* A186226 *)

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A186220 Adjusted joint rank sequence of (g(i)) and (f(j)) with f(i) before g(j) when f(i)=g(j), where f and g are the triangular numbers and squares. Complement of A186219.

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A187224 Rank transform of the sequence floor(3*n/2).

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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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A186159 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 triangular numbers and octagonal numbers. Complement of A186274.

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A186222 Adjusted joint rank sequence of (g(i)) and (f(j)) with f(i) after g(j) when f(i)=g(j), where f and g are the triangular numbers and squares. Complement of A186221.

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A186499 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f(i)=i^2 and g(j)=-4+5j^2. Complement of A186500.

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A186500 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f(i)=i^2 and g(j)=-4+5j^2. Complement of A186499.

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A186221 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 triangular numbers and squares. Complement of A186222.

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