A186221 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.
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
Offset: 1
Examples
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,...).
Links
- G. C. Greubel, Table of n, a(n) for n = 1..10000
Programs
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Magma
[n + Floor(Sqrt((n^2+n)/2 + 1/4)): n in [1..120]]; // G. C. Greubel, Aug 18 2018
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Mathematica
(* 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 *) -
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 */
Formula
a(n) = n + floor(sqrt((n^2+n)/2 + 1/4)).
a(n) = A061288(n) - n for all n in Z. - Michael Somos, Aug 19 2018
Comments