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A186326 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 squares and octagonal numbers. Complement of A186327. 3
2, 3, 5, 6, 8, 9, 11, 12, 14, 16, 17, 19, 20, 22, 24, 25, 27, 28, 30, 31, 33, 35, 36, 38, 39, 41, 42, 44, 46, 47, 49, 50, 52, 53, 55, 57, 58, 60, 61, 63, 65, 66, 68, 69, 71, 72, 74, 76, 77, 79, 80, 82, 83, 85, 87, 88, 90, 91, 93, 94, 96, 98, 99, 101, 102, 104, 106, 107, 109, 110, 112, 113, 115, 117, 118, 120, 121, 123, 124, 126, 128, 129, 131, 132, 134, 135, 137, 139, 140, 142, 143, 145, 147, 148, 150, 151, 153, 154, 156, 158 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Matthias Christandl, Fulvio Gesmundo, Asger Kjærulff Jensen, Border rank is not multiplicative under the tensor product, arXiv:1801.04852 [math.AG], 2018.
EXAMPLE
First, write
1..4...9..16....25..36....49..64... (squares)
1....8.......21........40........65. (octagonal)
Replace each number by its rank, where ties are settled by ranking the square number after the octagonal:
a=(2,3,5,6,8,9,11,12,14,...)=A186326
b=(1,4,7,10,13,15,18,21,...)=A186327.
MATHEMATICA
(* adjusted joint ranking; general formula *)
d=-1/2; u=1; v=0; 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}] (* A186326 *)
Table[b[n], {n, 1, 100}] (* A186327 *)
CROSSREFS
Sequence in context: A248562 A117672 A194383 * A139364 A364000 A211265
KEYWORD
nonn
AUTHOR
Clark Kimberling, Feb 17 2011
STATUS
approved

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Last modified December 5 01:00 EST 2023. Contains 367565 sequences. (Running on oeis4.)