A120872 a(n) is the value of k for row n of the fixed-k dispersion for Q = 8.

2, 1, 7, 4, 14, 9, 16, 7, 25, 14, 23, 8, 34, 17, 47, 28, 41, 18, 56, 31, 46, 17, 63, 32, 82, 49, 68, 31, 89, 50, 71, 28, 94, 49, 72, 23, 97, 46, 124, 71, 98, 41, 127, 68, 97, 34, 128, 63, 161, 94, 127, 56, 162, 89, 124, 47, 161, 82, 119, 36, 158, 73, 199, 112

Offset

1,1

Comments

This sequence results from A087059 by deleting duplicates.

Links

Table of n, a(n) for n=1..64.

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-k dispersion for Q=8, given by A120861, having northwest corner:
  1,  7,  41, 239, ...
  2, 12,  70, 408, ...
  3, 19, 111, 647, ...
  4, 24, 140, 816, ...
  ...
The pair (j,k) for each n, shown in the position occupied by n in the above array, is shown here:
  (1,2), (17,2),  (43,2),  (673,2), ...
  (4,1), (32,1), (196,1), (1152,1), ...
  (2,7), (46,7), (306,7), (1822,7), ...
  (7,4), (63,4), (391,4), (2303,4), ...
  ...
The fixed-k for row 1 is a(1) = 2;
the fixed-k for row 2 is a(2) = 1; etc.
(For example, (46 + 7 + 1)^2 - 4*7 = 8*19^2.)

Prog

(PARI) f(n) = 3*n + 2*sqrtint(2*n^2) + 2;
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) = (1 + sqrtint(2*n^2))^2 - 2*n^2; \\ A087059
lista(nn) = my(m=D(nn)); vector(nn, n, q(m[n, 1])); \\ Michel Marcus, Jul 10 2020

Crossrefs

Cf. A087059, A120861.

Sequence in context: A217458 A124048 A087059 * A204771 A141512 A143877

Adjacent sequences: A120869 A120870 A120871 * A120873 A120874 A120875

Keyword

nonn

Author

Clark Kimberling, Jul 10 2006

Extensions

More terms from Michel Marcus, Jul 10 2020

Status

approved