cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A283961 Rank array, R, of (golden ratio)^2, by antidiagonals.

Original entry on oeis.org

1, 2, 4, 3, 6, 10, 5, 8, 13, 18, 7, 11, 16, 22, 29, 9, 14, 20, 26, 34, 43, 12, 17, 24, 31, 39, 49, 59, 15, 21, 28, 36, 45, 55, 66, 78, 19, 25, 33, 41, 51, 62, 73, 86, 99, 23, 30, 38, 47, 57, 69, 81, 94, 108, 123, 27, 35, 44, 53, 64, 76, 89, 103, 117, 133
Offset: 1

Views

Author

Clark Kimberling, Mar 18 2017

Keywords

Comments

Antidiagonals n = 1..60, flattened.
Every row intersperses all other rows, and every column intersperses all other columns. The array is the dispersion of the complement of column 1, where column 1 is given by c(n) = c(n-1) + 1 + U(n), where U = upper Wythoff sequence (A001950).

Examples

			Northwest corner of R:
1  2  3  5  7  9  12 15
4  6  8  11 14 17 21 25
10 13 16 20 24 28 33 38
18 22 26 31 36 41 47 53
29 34 39 45 51 57 64 71
43 49 55 62 69 76 84 92
Let t = (golden ratio)^2 = (3 + sqrt(5))/2; then R(i,j) = rank of (j,i) when all nonnegative integer pairs (a,b) are ranked by the relation << defined as follows: (a,b) << (c,d) if a + b*t < c + d*t, and also (a,b) << (c,d) if a + b*t = c + d*t and b < d. Thus R(2,0) = 10 is the rank of (0,2) in the list (0,0) << (1,0) << (2,0) << (0,1) << (3,0) << (1,1) << (4,0) << (2,1) << (5,0) << (0,2).
From _Indranil Ghosh_, Mar 19 2017: (Start)
Triangle formed when the array is read by antidiagonals:
   1;
   2,  4;
   3,  6, 10;
   5,  8, 13, 18;
   7, 11, 16, 22, 29;
   9, 14, 20, 26, 34, 43;
  12, 17, 24, 31, 39, 49, 59;
  15, 21, 28, 36, 45, 55, 66, 78;
  19, 25, 33, 41, 51, 62, 73, 86, 99;
  23, 30, 38, 47, 57, 69, 81, 94, 108, 123;
  ...
(End)

Links

Crossrefs

Cf. A001622, A118276, A283961, A283968, A283969.

Programs

  • Mathematica
    r = GoldenRatio^2; z = 100;
s[0] = 1; s[n_] := s[n] = s[n - 1] + 1 + Floor[n*r];
u = Table[n + 1 + Sum[Floor[(n - k)/r], {k, 0, n}], {n, 0, z}]; (* A283968, row 1 of A283961 *)
v = Table[s[n], {n, 0, z}]; (* A283969, col 1 of A283961 *)
w[i_, j_] := v[[i]] + u[[j]] + (i - 1)*(j - 1) - 1;
Grid[Table[w[i, j], {i, 1, 10}, {j, 1, 10}]] (* A283961 *)
v1 = Flatten[Table[w[k, n - k + 1], {n, 1, 60}, {k, 1, n}]] (* A283961,sequence *)
  • PARI
    \\ This code produces the triangle mentioned in the example section
r = (3 +sqrt(5))/2;
z = 100;
s(n) = if(n<1, 1, s(n - 1) + 1 + floor(n*r));
p(n) = n + 1 + sum(k=0, n, floor((n - k)/r));
u = v = vector(z + 1);
for(n=1, 101, (v[n] = s(n - 1)));
for(n=1, 101, (u[n] = p(n - 1)));
w(i,j) = v[i] + u[j] + (i - 1) * (j - 1) - 1;
tabl(nn) = {for(n=1, nn, for(k=1, n, print1(w(k, n - k + 1),", ");); print(););};
tabl(10) \\ Indranil Ghosh, Mar 19 2017

Formula

R(i,j) = R(i,0) + R(0,j) + i*j - 1, for i>=1, j>=1.

A283969 a(n) = n + 1 + Sum_{k=0..n} floor((n-k)/r), where r = (3+sqrt(5))/2.

Original entry on oeis.org

1, 4, 10, 18, 29, 43, 59, 78, 99, 123, 150, 179, 211, 246, 283, 323, 365, 410, 458, 508, 561, 616, 674, 735, 798, 864, 933, 1004, 1078, 1154, 1233, 1315, 1399, 1486, 1576, 1668, 1763, 1860, 1960, 2063, 2168, 2276, 2386, 2499, 2615, 2733, 2854, 2978, 3104
Offset: 0

Views

Author

Clark Kimberling, Mar 18 2017

Keywords

Comments

This is column 1 of the transposable interspersion A283938.

Links

Crossrefs

Cf. A104457, A283968, A283938, A283961.
Partial sums of A026352.

Programs

  • Mathematica
    r = GoldenRatio^2; z = 120;
s[0] = 1; s[n_] := s[n] = s[n - 1] + 1 + Floor[n*r];
Table[n + 1 + Sum[Floor[(n - k)/r], {k, 0, n}], {n, 0, z}] (* A283968 *)
Table[s[n], {n, 0, z}] (* A283969 *)
  • PARI
    a(n) = if(n<1, 1, a(n - 1) + 1 + floor(n*(3 + sqrt(5))/2));
for(n = 0, 50, print1(a(n),", ")) \\ Indranil Ghosh, Mar 19 2017
  • Python
    import math
from sympy import sqrt
def a(n):
    return 1 if n<1 else a(n - 1) + 1 + int(math.floor(n*(3 + sqrt(5))/2))
print([a(n) for n in range(51)]) # Indranil Ghosh, Mar 19 2017

Formula

a(n) = n + 1 + Sum_{k=0..n} floor((n-k)/r), where r = (3+sqrt(5))/2.

A283938 Interspersion of the signature sequence of tau^2, where tau = (1 + sqrt(5))/2 = golden ratio.

Original entry on oeis.org

1, 4, 2, 10, 6, 3, 18, 13, 8, 5, 29, 22, 16, 11, 7, 43, 34, 26, 20, 14, 9, 59, 49, 39, 31, 24, 17, 12, 78, 66, 55, 45, 36, 28, 21, 15, 99, 86, 73, 62, 51, 41, 33, 25, 19, 123, 108, 94, 81, 69, 57, 47, 38, 30, 23, 150, 133, 117, 103, 89, 76, 64, 53, 44, 35
Offset: 1

Views

Author

Clark Kimberling, Mar 18 2017

Keywords

Comments

Row n is the ordered sequence of numbers k such that A118276(k) = n. As a sequence, A283938 is a permutation of the positive integers. As an array, A283938 is the joint-rank array (defined at A182801) of the numbers {i+j*r}, for i>=1, j>=1, where r = tau^2 = (3 + sqrt(5))/2. This is a transposable interspersion; i.e., every row intersperses all other rows, and every column intersperses all other columns.

Examples

			Northwest corner:
1   4  10   18  29  43  59   78  99   123
2   6  13   22  34  49  66   86  108  133
3   8  16   26  39  55  73   94  117  143
5  11  20   31  45  62  81  103  127  154
7  14  24   36  51  69  89  112  137  165
9  17  28   41  57  76  97  121  147  176
From _Indranil Ghosh_, Mar 19 2017: (Start)
Triangle formed when the array is read by antidiagonals:
    1;
    4,   2;
   10,   6,  3;
   18,  13,  8,  5;
   29,  22, 16, 11,  7;
   43,  34, 26, 20, 14,  9;
   59,  49, 39, 31, 24, 17, 12;
   78,  66, 55, 45, 36, 28, 21, 15;
   99,  86, 73, 62, 51, 41, 33, 25, 19;
  123, 108, 94, 81, 69, 57, 47, 38, 30, 23;
  ...
(End)

Links

Crossrefs

Cf. A118276, A283961, A283968, A283969.

Programs

  • Mathematica
    r = GoldenRatio^2; z = 100;
s[0] = 1; s[n_] := s[n] = s[n - 1] + 1 + Floor[n*r];
u = Table[n + 1 + Sum[Floor[(n - k)/r], {k, 0, n}], {n, 0, z}] (* A283968, row 1 of A283938 *)
v = Table[s[n], {n, 0, z}] (* A283969, col 1 of A283938 *)
w[i_, j_] := v[[i]] + u[[j]] + (i - 1)*(j - 1) - 1;
Grid[Table[w[i, j], {i, 1, 10}, {j, 1, 10}]] (* A283938, array *)
Flatten[Table[w[k, n - k + 1], {n, 1, 20}, {k, 1, n}]] (* A283938, sequence *)
  • PARI
    \\ This code produces the triangle mentioned in the example section
r = (3 +sqrt(5))/2;
z = 100;
s(n) = if(n<1, 1, s(n - 1) + 1 + floor(n*r));
p(n) = n + 1 + sum(k=0, n, floor((n - k)/r));
u = v = vector(z + 1);
for(n=1, 101, (v[n] = s(n - 1)));
for(n=1, 101, (u[n] = p(n - 1)));
w(i,j) = v[i] + u[j] + (i - 1) * (j - 1) - 1;
tabl(nn) = {for(n=1, nn, for(k=1, n, print1(w(n - k + 1, k),", ");); print(););};
tabl(10) \\ Indranil Ghosh, Mar 19 2017
Showing 1-3 of 3 results.

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