A186495 -id:A186495 - cp's OEIS frontend

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.

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.)
```

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.)
```

cp's OEIS Frontend

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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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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Examples

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Mathematica