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A186379
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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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4
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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
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OFFSET
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1,1
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COMMENTS
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LINKS
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FORMULA
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a(n)=n+floor(-1/2+sqrt(8n-3/4))=A186379(n).
b(n)=n+floor((n^2+n+1)/8)=A186380(n).
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EXAMPLE
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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.
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MATHEMATICA
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(* 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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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