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A186342
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 pentagonal numbers and the octagonal numbers. Complement of A186343.
4
1, 3, 5, 7, 8, 10, 12, 13, 15, 17, 18, 20, 22, 24, 25, 27, 29, 30, 32, 34, 36, 37, 39, 41, 42, 44, 46, 48, 49, 51, 53, 54, 56, 58, 59, 61, 63, 65, 66, 68, 70, 71, 73, 75, 77, 78, 80, 82, 83, 85, 87, 88, 90, 92, 94, 95, 97, 99, 100, 102, 104, 106, 107, 109, 111, 112, 114, 116, 118, 119, 121, 123, 124, 126, 128, 129, 131, 133, 135, 136, 138, 140, 141, 143, 145, 147, 148, 150, 152, 153, 155, 157, 158, 160, 162, 164, 165, 167, 169, 170
OFFSET
1,2
COMMENTS
See A186219 for a discussion of adjusted joint rank sequences.
EXAMPLE
First, write
1..5...12....22..35..... (pentagonal)
1....8....21........40.. (octagonal)
Then replace each number by its rank, where ties are settled by ranking the pentagonal number before the octagonal:
a=(1,3,5,7,8,10,12,13,15,...)=A186342
b=(2,4,6,9,11,14,16,19,21,...)=A186343.
MATHEMATICA
(* adjusted joint ranking; general formula *)
d=1/2; u=3/2; 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}] (* A186342 *)
Table[b[n], {n, 1, 100}] (* A186343 *)
CROSSREFS
A000326 (pentagonal), A000567 (octagonal).
Sequence in context: A221056 A288467 A276224 * A186315 A285074 A186219
KEYWORD
nonn
AUTHOR
Clark Kimberling, Feb 18 2011
STATUS
approved