A120860 -id:A120860 - cp's OEIS frontend

A120861 Fixed-k dispersion for Q = 8: Square array D(g,h) (g, h >= 1), read by ascending antidiagonals.

1, 2, 7, 3, 12, 41, 4, 19, 70, 239, 5, 24, 111, 408, 1393, 6, 31, 140, 647, 2378, 8119, 8, 36, 181, 816, 3771, 13860, 47321, 9, 48, 210, 1055, 4756, 21979, 80782, 275807, 10, 53, 280, 1224, 6149, 27720, 128103, 470832, 1607521, 11, 60, 309, 1632, 7134

Offset: 1

Clark Kimberling, Jul 09 2006

For each positive integer n, there exists a unique pair (j,k) of positive integers such that (j + k + 1)^2 - 4*k = 8*n^2; in fact, j(n) = A087056(n) and k(n) = A087059(n).
Suppose g >= 1 and let k = k(g). The numbers in row g of array D are among those n for which (j + k + 1)^2 - 4*k = 8*n^2 for some j; that is, k stays fixed and j and n vary - hence the name "fixed-k dispersion". (The fixed-j dispersion for Q = 8 is A120860.)
Every positive integer occurs exactly once in array D and every pair of rows are mutually interspersed. That is, beginning at the first term of any row having greater initial term than that of another row, all the following terms individually separate the individual terms of the other row.

			Northwest corner:
  1,  7,  41,  239, 1393,  8119,  47321, ...
  2, 12,  70,  408, 2378, 13860,  80782, ...
  3, 19, 111,  647, 3771, 21979, 128103, ...
  4, 24, 140,  816, 4756, 27720, 161564, ...
  5, 31, 181, 1055, 6149, 35839, 208885, ...
  6, 36, 210, 1224, 7134, 41580, 242346, ...
... [Edited by _Petros Hadjicostas_, Jul 07 2020]

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.
N. J. A. Sloane, Classic Sequences.

Cf. A087056, A087059, A120858, A120859, A120860, A120862, A120863, A336109 (first column), A002315 (first row), A001542 (2nd row), A253811 (3rd row).

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;}; \\ Michel Marcus, Jul 08 2020

Define f(n) = 3*n + 2*floor(n*sqrt(2)) + 2. Let D(g,h) be the term in row g and column h of the array to be defined:
D(1,1) = 1; D(1,2) = f(1); and D(1,h) = 6*D(1,h-1) - D(1,h-2) for h >= 3.
For arbitrary g >= 1, once row g is defined, define D(g+1,1) = least positive integer not in rows 1,2,...,g; D(g+1,2) = f(D(g+1,1)); and D(g+1,h) = 6*D(g+1,h-1) - D(g+1,h-2) for h >= 3. All rows after row 1 are thus inductively defined. [Corrected by Petros Hadjicostas, Jul 07 2020]

Name edited by Petros Hadjicostas, Jul 07 2020

A120862 Fixed-j dispersion for Q = 13: array D(g,h) (g, h >= 1), read by ascending antidiagonals.

Original entry on oeis.org

1, 2, 10, 3, 20, 109, 4, 30, 218, 1189, 5, 43, 327, 2378, 12970, 6, 53, 469, 3567, 25940, 141481, 7, 63, 578, 5116, 38910, 282962, 1543321, 8, 76, 687, 6305, 55807, 424443, 3086642, 16835050, 9, 86, 829, 7494, 68777, 608761, 4629963, 33670100, 183642229

Offset: 1

Clark Kimberling, Jul 09 2006

For each positive integer n, there exists a unique pair (j,k) of positive integers such that (j+k+1)^2 - 4*k = 13*n^2; in fact, j(n) = A120869(n) and k(n) = A120870(n).
Suppose g >= 1 and let j = j(g). The numbers in row g of array D are among those n for which (j+k+1)^2 - 4*k = 13*n^2 for some k; that is, j stays fixed and k and n vary - hence the name "fixed-j dispersion". (The fixed-k dispersion for Q=13 is A120863.)
Every positive integer occurs exactly once in array D and every pair of rows are mutually interspersed. That is, beginning at the first term of any row having greater initial term than that of another row, all the following terms individually separate the individual terms of the other row.

			Northwest corner:
  1, 10, 109, 1189, ...
  2, 20, 218, 2378, ...
  3, 30, 327, 3567, ...
  4, 43, 469, 5116, ...
  5, 53, 578, 6305, ...
  6, 63, 687, 7494, ...
  ...

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.
N. J. A. Sloane, Classic Sequences.

Cf. A120858, A120859, A120860, A120861, A120863, A120869, A120870.

f(n) = floor((11 + 3*sqrt(13))/2*n) - floor(3*frac((1 + sqrt(13))*n/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 = 11*m[g, h-1] - m[g, h-2]; m[g, h] = t; listput(listus, t); ); ); m; };
lista(nb) = {my(m=D(nb)); for (n=1, nb, for (j=1, n, print1(m[n-j+1, j], ", ");););} \\ Michel Marcus, Jul 09 2020

Define f(n) = floor(r*n) - floor(3*F(n)), where r = (11 + 3*sqrt(13))/2 and F(n) is the fractional part of (1 + sqrt(13))*n/2. Let D(g,h) be the term in row g and column h of the array to be defined:
D(1,1) = 1; D(1,2) = f(1); and D(1,h) = 11*D(1,h-1) - D(1,h-2) for h >= 3.
For arbitrary g >= 1, once row g is defined, define D(g+1,1) = least positive integer not in rows 1,2,...,g; D(g+1,2) = f(D(g+1,1)); and D(g+1,h) = 11*D(g+1,h-1) - D(g+1,h-2) for h >= 3. All rows after row 1 are thus inductively defined. [Corrected using Conjecture 1 in Kimberling (2007) by Petros Hadjicostas, Jul 07 2020]

Name edited by Petros Hadjicostas, Jul 07 2020

More terms from Michel Marcus, Jul 09 2020

A120863 Fixed-k dispersion for Q = 13: array D(g,h) (g, h >= 1), read by ascending antidiagonals.

Original entry on oeis.org

1, 2, 13, 3, 23, 142, 4, 33, 251, 1549, 5, 46, 360, 2738, 16897, 6, 56, 502, 3927, 29867, 184318, 7, 66, 611, 5476, 42837, 325799, 2010601, 8, 79, 720, 6665, 59734, 467280, 3553922, 21932293, 9, 89, 862, 7854, 72704, 651598, 5097243, 38767343, 239244622

Offset: 1

Clark Kimberling, Jul 09 2006

For each positive integer n, there exists a unique pair (j,k) of positive integers such that (j+k+1)^2 - 4*k = 13*n^2; in fact, j(n) = A120869(n) and k(n) = A120870(n).
Suppose g >= 1 and let k = k(g). The numbers in row g of array D are among those n for which (j+k+1)^2 - 4*k = 13*n^2 for some j; that is, k stays fixed and j and n vary - hence the name "fixed-k dispersion". (The fixed-j dispersion for Q = 13 is A120862.)
Every positive integer occurs exactly once in the array D and every pair of rows are mutually interspersed. That is, beginning at the first term of any row of D having greater initial term than that of another row, all the following terms individually separate the individual terms of the other row.

			Northwest corner:
  1, 13, 142, 1549, ...
  2, 23, 251, 2738, ...
  3, 33, 360, 3927, ...
  4, 46, 502, 5476, ...
  5, 56, 611, 6665, ...
  6, 66, 720, 7854, ...
  ...

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.
N. J. A. Sloane, Classic Sequences.

Cf. A120858, A120859, A120860, A120861, A120862, A120869, A120870.

f(n) = floor((11 + 3*sqrt(13))/2*n) - floor(3*frac((1 + sqrt(13))*n/2)) + 3;
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 = 11*m[g, h-1] - m[g, h-2]; m[g, h] = t; listput(listus, t); ); ); m; };
lista(nb) = {my(m=D(nb)); for (n=1, nb, for (j=1, n, print1(m[n-j+1, j], ", ");););} \\ Michel Marcus, Jul 09 2020

Define f(n) = floor(r*n) - floor(3*F(n)) + 3, where r = (11 + 3*sqrt(13))/2 and F(n) is the fractional part of (1 + sqrt(13))*n/2. Let D(g,h) be the term in row g and column h of the array to be defined:
D(1,1) = 1; D(1,2) = f(1); and D(1,h) = 11*D(1,h-1) - D(1,h-2) for h >= 3.
For arbitrary g >= 1, once row g is defined, define D(g+1,1) = least positive integer not in rows 1,2,...,g; D(g+1,2) = f(D(g+1,1)); and D(g+1,h) = 11*D(g+1,h-1) - D(g+1,h-2) for h >= 3. All rows after row 1 are thus inductively defined. [Corrected using Conjecture 2 in Kimberling (2007) by Petros Hadjicostas, Jul 07 2020]

Name edited by Petros Hadjicostas, Jul 07 2020

More terms from Michel Marcus, Jul 09 2020

A120858 Dispersion of the Beatty sequence ([r*n]: n >= 1), where r = 3 + 8^(1/2): square array D(n,m) (n, m >= 1), read by ascending antidiagonals.

Original entry on oeis.org

1, 2, 5, 3, 11, 29, 4, 17, 64, 169, 6, 23, 99, 373, 985, 7, 34, 134, 577, 2174, 5741, 8, 40, 198, 781, 3363, 12671, 33461, 9, 46, 233, 1154, 4552, 19601, 73852, 195025, 10, 52, 268, 1358, 6726, 26531, 114243, 430441, 1136689, 12, 58, 303, 1562

Offset: 1

Clark Kimberling, Jul 09 2006

Every positive integer occurs exactly once in array D and every pair of rows are mutually interspersed. That is, beginning at the first term of any row in D having greater initial term than that of another row, all the following terms individually separate the individual terms of the other row.

			Northwest corner:
  1,  5,  29,  169,  985, ...
  2, 11,  64,  373, 2174, ...
  3, 17,  99,  577, 3363, ...
  4, 23, 134,  781, 4552, ...
  6, 34, 198, 1154, 6726, ...
  ...
In row 1, we have 5 = [r], 29 = [5*r], 169 = [29*r], etc., where r = 3 +  8^(1/2); each new row starts with the least "new" number n, followed by [n*r], [[n*r]*r], [[[n*r]*r]*r], and so on.

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.
N. J. A. Sloane, Classic Sequences.
Eric Weisstein's World of Mathematics, Beatty sequence.
Wikipedia, Beatty sequence.

Cf. A120859, A120860, A120861, A120862, A120863.

tabls(nn)={default("realprecision", 1000); my(D=matrix(nn,nn));  r = 3 +  8^(1/2); s=r/(r-1); for(n=1, nn, D[n,1]=floor(s*n)); for(m=2, nn, for(n=1, nn, D[n,m]=floor(r*D[n,m-1]))); D}
/* To print the array flattened */
flat(nn)={D=tabls(nn); for(n=1, nn, for(m=1, n, print1(D[n+1-m,m],",")))}
/* To print the square array */
square(nn)={D=tabls(nn); for(n=1,nn, for(m=1,nn, print1(D[n,m], ",")); print())} \\ Petros Hadjicostas, Jul 07 2020

(1) Column 1 is the sequence ([s*n]: n >= 1) where 1/r + 1/s = 1. The numbers in all the other columns, arranged in increasing order, form the sequence ([r*n]: n >= 1).

(2) Every row satisfies these recurrences: x(n+1) = [r*x(n)] and x(n+2) = 6*x(n+1) - x(n). (Here [a] is the floor of number a.)

Name edited by Petros Hadjicostas, Jul 07 2020

A120859 Dispersion of the sequence ([r*n] + 1: n >= 1), where r = 3 + 8^(1/2): square array D(n,m) (n, m >= 1), read by ascending antidiagonals.

Original entry on oeis.org

1, 2, 6, 3, 12, 35, 4, 18, 70, 204, 5, 24, 105, 408, 1189, 7, 30, 140, 612, 2378, 6930, 8, 41, 175, 816, 3567, 13860, 40391, 9, 47, 239, 1020, 4756, 20790, 80782, 235416, 10, 53, 274, 1393, 5945, 27720, 121173, 470832, 1372105, 11, 59, 309, 1597, 8119

Offset: 1

Clark Kimberling, Jul 09 2006

Every positive integer occurs exactly once in array D and every pair of rows of D are mutually interspersed. That is, beginning at the first term of any row of array D having greater initial term than that of another row, all the following terms individually separate the individual terms of the other row.

			Northwest corner:
  1,  6,  35,  204, 1189, ...
  2, 12,  70,  408, 2378, ...
  3, 18, 105,  612, 3567, ...
  4, 24, 140,  816, 4756, ...
  5, 30, 175, 1020, 5945, ...
  ... [Corrected by _Petros Hadjicostas_, Jul 07 2020]
In row 1, we have 6 = [r] + 1, 35 = [6*r], 204 = [35*r] + 1, etc., where r = 3 + 8^(1/2); each new row starts with the least "new" number n, followed by [n*r] + 1, [[n*r + 1]*r + 1], [[[n*r + 1]*r + 1]*r] + 1, and so on.

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.
N. J. A. Sloane, Classic Sequences.
Eric Weisstein's World of Mathematics, Beatty sequence.
Wikipedia, Beatty sequence.

Cf. A120858, A120860, A120861, A120862, A120863.

tabls(nn)={default("realprecision", 1000); my(D=matrix(nn, nn));  r = 3 +  8^(1/2); s=r/(r-1); for(n=1, nn, D[n, 1]=floor(s*(n-1))+1); for(m=2, nn, for(n=1, nn, D[n, m]=floor(r*D[n, m-1])+1)); D}
/* To print the array flattened */
flat(nn)={D=tabls(nn); for(n=1, nn, for(m=1, n, print1(D[n+1-m, m], ", ")))}
/* To print the square array */
square(nn)={D=tabls(nn); for(n=1, nn, for(m=1, nn, print1(D[n, m], ", ")); print())} \\ Petros Hadjicostas, Jul 07 2020

(1) Column 1 is the sequence ([s*(n-1)] + 1: n >= 1), where 1/r + 1/s = 1. The numbers in all the other columns, arranged in increasing order, form the sequence ([r*n] + 1: n >= 1).

(2) Every row satisfies these recurrences: x(n+1) = [r*x(n)] + 1 and x(n+2) = 6*x(n+1) - x(n).

Name edited by Petros Hadjicostas, Jul 07 2020

A120871 a(n) is the value of j for row n of the fixed-j dispersion for Q = 8.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Jul 10 2006

nonn

This sequence results from A087056 by deleting duplicates.

			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.)

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.

Cf. A087056, A098021, A120860.

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

More terms from Michel Marcus, Jul 09 2020

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

Original entry on oeis.org

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

Clark Kimberling, Jul 10 2006

nonn

This sequence results from A087059 by deleting duplicates.

			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.)

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.

Cf. A087059, A120861.

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

More terms from Michel Marcus, Jul 10 2020

cp's OEIS Frontend

A120861 Fixed-k dispersion for Q = 8: Square array D(g,h) (g, h >= 1), read by ascending antidiagonals.

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A120862 Fixed-j dispersion for Q = 13: array D(g,h) (g, h >= 1), read by ascending antidiagonals.

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A120863 Fixed-k dispersion for Q = 13: array D(g,h) (g, h >= 1), read by ascending antidiagonals.

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A120858 Dispersion of the Beatty sequence ([r*n]: n >= 1), where r = 3 + 8^(1/2): square array D(n,m) (n, m >= 1), read by ascending antidiagonals.

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A120859 Dispersion of the sequence ([r*n] + 1: n >= 1), where r = 3 + 8^(1/2): square array D(n,m) (n, m >= 1), read by ascending antidiagonals.

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A120871 a(n) is the value of j for row n of the fixed-j dispersion for Q = 8.

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

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