A186275 -id:A186275 - cp's OEIS frontend

A186276 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 octagonal numbers. Complement of A186275.

1, 5, 8, 12, 15, 19, 22, 26, 29, 33, 36, 40, 43, 46, 50, 53, 57, 60, 64, 67, 71, 74, 78, 81, 84, 88, 91, 95, 98, 102, 105, 109, 112, 115, 119, 122, 126, 129, 133, 136, 140, 143, 147, 150, 153, 157, 160, 164, 167, 171, 174, 178, 181, 184, 188, 191, 195, 198, 202, 205, 209, 212, 215, 219, 222, 226, 229, 233, 236, 240, 243, 247, 250, 253, 257, 260, 264, 267, 271, 274, 278, 281, 284, 288, 291, 295, 298, 302, 305, 309, 312, 316, 319, 322, 326, 329, 333, 336, 340, 343

Offset: 1

Clark Kimberling, Feb 16 2011

nonn

See A186275.

			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 after the octagonal:
a=(2,3,4,6,7,9,10,11,13,...)=A186275.
b=(1,5,8,12,15,19,22,26,...)=A186276.

Cf. A186159, A186274, A186275.

Mathematica
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(See A186275.)
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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.

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 *)

cp's OEIS Frontend

A186276 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 octagonal numbers. Complement of A186275.

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