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 A186159 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f and g are the triangular numbers and octagonal numbers.  Complement of A186274. 6
 1, 3, 4, 6, 7, 8, 10, 11, 13, 14, 16, 17, 18, 20, 21, 23, 24, 25, 27, 28, 30, 31, 32, 34, 35, 37, 38, 39, 41, 42, 44, 45, 47, 48, 49, 51, 52, 54, 55, 56, 58, 59, 61, 62, 63, 65, 66, 68, 69, 70, 72, 73, 75, 76, 77, 79, 80, 82, 83, 85, 86, 87, 89, 90, 92, 93, 94, 96, 97, 99, 100, 101, 103, 104, 106, 107, 108, 110, 111, 113, 114, 116, 117, 118, 120, 121, 123, 124, 125, 127, 128, 130, 131, 132, 134, 135, 137, 138, 139, 141 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS See A186219 for a discussion of adjusted joint rank sequences. LINKS EXAMPLE First, write the triangular and octagonal numbers: 1..3..6.....10..15..21..28 1........8..........21...... Then replace each by its rank, where ties are settled by ranking the triangular number before the octagonal: a=(1,3,4,6,7,8,10,11,13,...)=A186159. b=(2,5,9,12,15,19,22,26,...)=A186274. MATHEMATICA (* adjusted joint ranking; general formula *) d=1/2; u=1/2; v=1/2; w=0; x=3; y=-2; 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}] (* A186159 *) Table[b[n], {n, 1, 100}] (* A186274 *) CROSSREFS Cf. A186219, A186274, A186275, A186276, Cf. A000217 (triangular numbers). Cf. A000567 (octagonal numbers). Sequence in context: A062975 A276218 A047299 * A184578 A022846 A083922 Adjacent sequences:  A186156 A186157 A186158 * A186160 A186161 A186162 KEYWORD nonn AUTHOR Clark Kimberling, Feb 13 2011 STATUS approved

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Last modified April 12 21:47 EDT 2021. Contains 342933 sequences. (Running on oeis4.)