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A120871
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a(n) is the value of j for row n of the fixed-j dispersion for Q = 8.
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1
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1, 4, 2, 7, 8, 17, 7, 18, 17, 32, 14, 31, 9, 28, 23, 46, 16, 41, 34, 63, 25, 56, 14, 47, 36, 73, 23, 62, 49, 92, 34, 79, 64, 113, 47, 98, 28, 81, 62, 119, 41, 100, 79, 142, 56, 121, 31, 98, 73, 144, 46, 119, 92, 169, 63, 142, 113, 196, 82, 167, 49, 136, 103, 194
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OFFSET
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1,2
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COMMENTS
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This sequence results from A087056 by deleting duplicates.
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LINKS
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EXAMPLE
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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.)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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