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 A186515 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) after g(j) when f(i)=g(j), where f(i)=i^2 and g(j)=4+5j^2. Complement of A186516. 4
 1, 2, 4, 5, 7, 8, 10, 11, 12, 14, 15, 17, 18, 20, 21, 23, 24, 26, 27, 28, 30, 31, 33, 34, 36, 37, 39, 40, 41, 43, 44, 46, 47, 49, 50, 52, 53, 54, 56, 57, 59, 60, 62, 63, 65, 66, 68, 69, 70, 72, 73, 75, 76, 78, 79, 81, 82, 83, 85, 86, 88, 89, 91, 92, 94, 95, 96, 98, 99, 101, 102, 104, 105, 107, 108, 109, 111, 112, 114, 115, 117, 118, 120, 121, 123, 124, 125, 127, 128, 130, 131, 133, 134, 136, 137, 138, 140, 141, 143, 144 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS See A186219 for a discussion of adjusted joint rank sequences. The pairs (i,j) for which i^2=4+5j^2 are (L(2h),F(2h)), where L=A000032 (Lucas numbers) and F=A000045 (Fibonacci numbers). LINKS Table of n, a(n) for n=1..100. FORMULA a(n)=n+floor(sqrt((n^2)/5-7/10))=A186515(n). b(n)=n+floor(sqrt(5n^2+7/2))=A186516(n). EXAMPLE First, write 1..4..9..16..25..36..49..... (i^2) ......9.....24.......49.. (4+5j^2) Then replace each number by its rank, where ties are settled by ranking i^2 after 4+5j^2: a=(1,2,4,5,7,8,10,11,12,14,15,17,..)=A186515 b=(3,6,9,13,16,19,22,25,29,32,35,..)=A186516. MATHEMATICA (* adjusted joint rank sequences a and b, using general formula for ranking ui^2+vi+w and xj^2+yj+z *) d = -1/2; u = 1; v = 0; w = 0; x = 5; y = 0; z = 4; h[n_] := -y + (4 x (u*n^2 + v*n + w - z - d) + y^2)^(1/2); a[n_] := n + Floor[h[n]/(2 x)]; k[n_] := -v + (4 u (x*n^2 + y*n + z - w + d) + v^2)^(1/2); b[n_] := n + Floor[k[n]/(2 u)]; Table[a[n], {n, 1, 100}] (* A186515 *) Table[b[n], {n, 1, 100}] (* A186516 *) CROSSREFS Cf. A186219, A186513, A186514, A186516. Sequence in context: A307913 A059539 A307513 * A078934 A026496 A056127 Adjacent sequences: A186512 A186513 A186514 * A186516 A186517 A186518 KEYWORD nonn AUTHOR Clark Kimberling, Feb 22 2011 STATUS approved

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Last modified February 21 10:30 EST 2024. Contains 370228 sequences. (Running on oeis4.)