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 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. 4
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS See A186219. LINKS G. C. Greubel, Table of n, a(n) for n = 1..10000 FORMULA a(n) = n + floor(sqrt((n^2+n)/2 + 1/4), A186221. b(n) = n + floor(-1/2 + sqrt(2*n^2)), A186222. a(n) = A061288(n) - n for all n in Z. - Michael Somos, Aug 19 2018 EXAMPLE 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,...). 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 *) PROG (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 CROSSREFS Cf. A061288, A186219, A186220, A186222. Sequence in context: A285077 A143826 A001912 * A083027 A060107 A159556 Adjacent sequences:  A186218 A186219 A186220 * A186222 A186223 A186224 KEYWORD nonn,changed AUTHOR Clark Kimberling, Feb 15 2011 STATUS approved

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Last modified August 20 22:36 EDT 2018. Contains 313928 sequences. (Running on oeis4.)