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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.
4
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
1,1
COMMENTS
See A186350 for a discussion of adjusted joint rank sequences.
EXAMPLE
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.
MATHEMATICA
(* 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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Feb 22 2011
STATUS
approved