OFFSET
1,2
COMMENTS
This sequence results from A087056 by deleting duplicates.
LINKS
Clark Kimberling, The equation (j+k+1)^2 - 4*k = Q*n^2 and related dispersions, Journal of Integer Sequences, 10 (2007), Article #07.2.7.
EXAMPLE
For each positive integer n, there is a unique pair (j,k) of positive integers such that (j + k + 1)^2 - 4*k = 8*n^2. This representation is used to define the fixed-j dispersion for Q = 8, given by A120860, having northwest corner:
1, 5, 29, 169, ...
2, 10, 58, 338, ...
3, 17, 99, 577, ...
4, 22, 128, 746, ...
...
The pair (j,k) for each n, shown in the position occupied by n in the above array, is shown here:
(1,2), (1,14), (1,82), (1,478), ...
(4,1), (4,25), (4,161), (4,953), ...
(2,7), (2,47), (2,279), (2,1631), ...
(7,4), (7,56), (7,356), (7,2104), ...
...
The fixed-j for row 1 is a(1) = 1;
the fixed-j for row 2 is a(2) = 4; etc.
(For example, (4 + 25 + 1)^2 - 4*25 = 8*10^2.)
PROG
(PARI) f(n) = 3*n + 2*sqrtint(2*n^2); \\ A098021
unused(listus) = {my(v=vecsort(Vec(listus))); for (i=1, vecmax(v), if (!vecsearch(v, i), return (i)); ); };
D(nb) = {my(m = matrix(nb, nb), t); my(listus = List); for (g=1, nb, if (g==1, t = 1, t = unused(listus)); m[g, 1]=t; listput(listus, t); t = f(t); m[g, 2]=t; listput(listus, t); for (h=3, nb, t = 6*m[g, h-1] - m[g, h-2]; m[g, h] = t; listput(listus, t); ); ); m; }; \\ A120860
q(n) = 2*n^2 - sqrtint(2*n^2)^2; \\ A087056
lista(nn) = my(m=D(nn)); vector(nn, n, q(m[n, 1])); \\ Michel Marcus, Jul 09 2020
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Jul 10 2006
EXTENSIONS
More terms from Michel Marcus, Jul 09 2020
STATUS
approved