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A186221
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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 squares. Complement of A186222.
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5
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2, 3, 5, 7, 8, 10, 12, 14, 15, 17, 19, 20, 22, 24, 25, 27, 29, 31, 32, 34, 36, 37, 39, 41, 43, 44, 46, 48, 49, 51, 53, 54, 56, 58, 60, 61, 63, 65, 66, 68, 70, 72, 73, 75, 77, 78, 80, 82, 84, 85, 87, 89, 90, 92, 94, 95, 97, 99, 101, 102, 104, 106, 107, 109, 111, 113, 114, 116, 118, 119, 121, 123, 124, 126, 128, 130, 131, 133, 135, 136, 138, 140, 142, 143, 145, 147, 148, 150, 152, 153, 155, 157, 159, 160, 162, 164, 165, 167, 169, 171
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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(sqrt((n^2+n)/2 + 1/4), A186221.
b(n) = n + floor(-1/2 + sqrt(2*n^2)), A186222.
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EXAMPLE
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First, write
1..3...6..10..15...21..28..36..45... (triangular)
1....4...9......16...25....36....49.. (square)
Replace each number by its rank, where ties are settled by ranking the triangular number after the square:
a=(2,3,5,7,8,10,12,14,...)
b=(1,4,6,9,11,13,16,18,...).
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MATHEMATICA
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(* adjusted joint ranking; general formula *)
d=-1/4; u=1/2; v=1/2; w=0; x=1; y=0; 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}] (* A186221 *)
Table[b[n], {n, 1, 100}] (* A186222 *)
a[ n_] := n + Floor[ Sqrt[ n (n + 1)/2]]; (* Michael Somos, Aug 19 2018 *)
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PROG
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(PARI) vector(120, n, n + floor(sqrt((n^2+n)/2 + 1/4))) \\ G. C. Greubel, Aug 18 2018
{a(n) = n + sqrtint( n * (n+1) \ 2)}; /* Michael Somos, Aug 19 2018 */
(Magma) [n + Floor(Sqrt((n^2+n)/2 + 1/4)): n in [1..120]]; // G. C. Greubel, Aug 18 2018
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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