

A186219


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 squares. Complement of A186220.


35



1, 3, 5, 7, 8, 10, 12, 13, 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, 83, 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
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OFFSET

1,2


COMMENTS

Suppose that f and g are strictly increasing functions for which (f(i)) and (g(j)) are integer sequences. If 0<d<1, the sets F={f(i): i>=1} and G={g(j)+d: j>=1} are clearly disjoint. Let f^=(inverse of f) and g^=(inverse of g). When the numbers in F and G are jointly ranked, the rank of f(n) is a(n):=n+floor(g^(f(n))d), and the rank of g(n)+d is b(n):=n+floor(f^(g(n))+d). Therefore, the sequences a and b are a complementary pair.
Although the sequences (f(i)) and (g(j)) may not be disjoint, the sequences (f(i)) and (g(j)+d) are disjoint, and this observation enables two types of adjusted joint rankings:
(1) if 0<d<1, we call a and b the "adjusted joint rank sequences of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j)"; (2) if 1<d<0, we call a and b the "adjusted joint rank sequences of (f(i)) and (g(j)) with f(i) after g(j) when f(i)=g(j)".
Using f(i)=ui^2+vi+w and g(j)=xj^2+yj+z, we can carry out adjusted joint rankings of any pair of polygonal sequences (triangular, square, pentagonal, etc.) In this case,
a(n)=n+floor((y+sqrt(4x(un^2+vn+wzd)+y^2))/(2x)),
b(n)=n+floor((v+sqrt(4u(xn^2+yn+zw+d)+v^2)/(2u)),
where a(n) is the rank of un^2+vn+w and b(n) is the rank
of xn^2+yn+z+d, where d must be chosen small enough, in
absolute value, that the sets F and G are disjoint.
Example: f=A000217 (triangular numbers) and g=A000290 (squares) yield adjusted rank sequences a=A186219 and b=A186220 for d=1/4 and a=A186221 and b=A186222 for d=1/4.


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)), (A186219).
b(n) = n + floor((1 + sqrt(8*n^2+3))/2), (A186220).


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 before the square:
a=(1,3,5,7,8,10,12,13,...)
b=(2,4,6,9,11,14,16,18,...).


MATHEMATICA

(* adjusted joint ranking of triangular numbers and squares, using 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+wzd)+y^2)^(1/2);
a[n_]:=n+Floor[h[n]/(2x)]; (* rank of triangular n(n+1)/2 *)
k[n_]:=v+(4u(x*n^2+y*n+zw+d)+v^2)^(1/2);
b[n_]:=n+Floor[k[n]/(2u)]; (* rank of square n^2 *)
Table[a[n], {n, 1, 100}] (* A186219 *)
Table[b[n], {n, 1, 100}] (* A186220 *)


PROG

(PARI) vector(100, n, n + floor(sqrt((n^2 + n)/2  1/4))) \\ G. C. Greubel, Aug 26 2018
(Magma) [n + Floor(Sqrt((n^2 + n)/2  1/4)): n in [1..100]]; // G. C. Greubel, Aug 26 2018


CROSSREFS

Cf. A186145 (joint ranks of squares and cubes),
A000217 (triangular numbers),
A000290 (squares),
A186220 (complement of A186119)
A186221 ("after" instead of "before")
A186222 (complement of A186221).
Sequence in context: A186342 A186315 A285074 * A185050 A083034 A213908
Adjacent sequences: A186216 A186217 A186218 * A186220 A186221 A186222


KEYWORD

nonn


AUTHOR

Clark Kimberling, Feb 15 2011


STATUS

approved



