A186350 -id:A186350 - 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 *)

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).

A186384 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(j+1)/2 (triangular number). Complement of A186383.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 12, 15, 18, 21, 24, 27, 31, 35, 39, 43, 47, 52, 57, 62, 67, 72, 78, 84, 90, 96, 102, 109, 116, 123, 130, 137, 145, 153, 161, 169, 177, 186, 195, 204, 213, 222, 232, 242, 252, 262, 272, 283, 294, 305, 316, 327, 339, 351, 363, 375, 387, 400, 413, 426, 439, 452, 466, 480, 494, 508, 522, 537, 552, 567, 582, 597, 613, 629, 645, 661, 677, 694, 711, 728, 745, 762, 780, 798, 816, 834, 852, 871, 890

Offset: 1

Clark Kimberling, Feb 19 2011

nonn

			First, write
.....5...10..15..20..25..30.. (5i)
1..3..6..10..15....21..28.. (triangular)
Then replace each number by its rank, where ties are settled by ranking 5i before the triangular:
a=(3,5,7,9,11,13,14,16,17,..)=A186383
b=(1,2,4,6,8,10,12,15,18,...)=A186384.

Cf. A186350, A186383, A186385, A186386.

(* 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=1/2; y=1/2; (* 5i 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}]  (* A186383 *)
Table[b[n], {n, 1, 100}]  (* A186384 *)

A186346 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f(i)=8i and g(j)=j^2. Complement of A186347.

Original entry on oeis.org

3, 5, 7, 9, 11, 12, 14, 15, 17, 18, 20, 21, 23, 24, 25, 27, 28, 29, 31, 32, 33, 35, 36, 37, 39, 40, 41, 42, 44, 45, 46, 47, 49, 50, 51, 52, 54, 55, 56, 57, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 77, 78, 79, 80, 81, 83, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 97, 98, 99, 100, 101, 102, 104, 105, 106, 107, 108, 109, 111, 112, 113, 114, 115, 116, 117, 119, 120, 121, 122, 123, 124, 125, 127, 128, 129, 130, 131, 132, 133, 135, 136, 137, 138, 139, 140, 141

Offset: 1

Clark Kimberling, Feb 20 2011

nonn

See A186350 for a discussion of adjusted joint rank sequences.

			First, write
....8....16..24..32..40..48..56..64..72..80.. (8i)
1..4..9..16...25...36......49....64.......81 (squares)
Then replace each number by its rank, where ties are settled by ranking 8i before the square:
a=(3,5,7,9,11,12,14,15,17,..)=A186346
b=(1,2,4,6,8,10,13,16,19,...)=A186347.

Cf. A186350, A186347, A186348, A186349.

(* 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=8; v=0; x=1; y=0;
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}]  (* A186346 *)
Table[b[n],{n,1,100}]  (* A186347 *)

a(n)=n+floor(sqrt(8n-1/2))=A186346(n).

b(n)=n+floor((n^2+1/2)/8)=A186347(n).

A186352 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

2, 4, 5, 7, 8, 10, 11, 13, 14, 15, 17, 18, 19, 20, 22, 23, 24, 25, 27, 28, 29, 30, 32, 33, 34, 35, 36, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 52, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 88, 89, 90, 91, 92, 94, 95, 96, 97, 98, 99, 100, 101, 102, 104, 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, 137, 138, 139, 140, 141

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, A186353.

(* 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}]  (* A186352 *)
Table[b[n], {n, 1, 100}]  (* A186353 *)

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

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

A186379 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 A186380.

Original entry on oeis.org

3, 5, 7, 9, 10, 12, 13, 15, 16, 18, 19, 21, 22, 24, 25, 26, 28, 29, 30, 32, 33, 34, 36, 37, 38, 39, 41, 42, 43, 44, 46, 47, 48, 49, 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

See A186350.

			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. A186350, A186380, A186381, 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}]  (* A186379 *)
Table[b[n], {n, 1, 100}]  (* A186380 *)

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

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

A186383 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(j+1)/2 (triangular number). Complement of A186384.

Original entry on oeis.org

3, 5, 7, 9, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26, 28, 29, 30, 32, 33, 34, 36, 37, 38, 40, 41, 42, 44, 45, 46, 48, 49, 50, 51, 53, 54, 55, 56, 58, 59, 60, 61, 63, 64, 65, 66, 68, 69, 70, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 85, 86, 87, 88, 89, 91

Offset: 1

Clark Kimberling, Feb 19 2011

nonn

			First, write
.....5...10..15..20..25..30.. (5i)
1..3..6..10..15....21..28.. (triangular)
Then replace each number by its rank, where ties are settled by ranking 5i before the triangular:
a=(3,5,7,9,11,13,14,16,17,..)=A186383
b=(1,2,4,6,8,10,12,15,18,...)=A186384.

Cf. A186350, A186384, A186385, A186386.

(* 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=1/2; y=1/2; (* 5i 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}]  (* A186383 *)
Table[b[n], {n, 1, 100}]  (* A186384 *)

A186385 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(j+1)/2 (triangular number). Complement of A186386.

Original entry on oeis.org

3, 6, 8, 9, 11, 13, 14, 16, 18, 19, 21, 22, 23, 25, 26, 28, 29, 30, 32, 33, 35, 36, 37, 39, 40, 41, 42, 44, 45, 46, 48, 49, 50, 51, 53, 54, 55, 57, 58, 59, 60, 62, 63, 64, 65, 66, 68, 69, 70, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91

Offset: 1

Clark Kimberling, Feb 19 2011

nonn

			First, write
.....5...10..15..20..25..30.. (5*i)
1..3..6..10..15....21..28.. (triangular)
Then replace each number by its rank, where ties are settled by ranking 5*i after the triangular:
a=(3,6,8,9,11,13,14,16,18,..)=A186385
b=(1,2,4,5,7,10,12,15,17,...)=A186386.

Cf. A186350, A186383, A186384, A186386.

(* 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=1/2; y=1/2; (* 5i 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}]  (* A186385 *)
Table[b[n], {n, 1, 100}]  (* A186386 *)

A186387 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 A186388.

Original entry on oeis.org

3, 6, 8, 10, 12, 13, 15, 17, 18, 20, 21, 23, 24, 26, 27, 29, 30, 32, 33, 34, 36, 37, 39, 40, 41, 43, 44, 45, 47, 48, 49, 51, 52, 53, 54, 56, 57, 58, 60, 61, 62, 63, 65, 66, 67, 68, 70, 71, 72, 73, 75, 76, 77, 78, 80, 81, 82, 83, 85, 86, 87, 88, 89, 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. (6*i)
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, A186388, A186389, 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}]  (* A186387 *)
Table[b[n], {n, 1, 100}]  (* A186388 *)

Showing 1-10 of 19 results. Next

cp's OEIS Frontend

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

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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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A186384 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(j+1)/2 (triangular number). Complement of A186383.

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A186346 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f(i)=8i and g(j)=j^2. Complement of A186347.

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A186352 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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A186379 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 A186380.

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A186383 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(j+1)/2 (triangular number). Complement of A186384.

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A186385 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(j+1)/2 (triangular number). Complement of A186386.

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