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 1-5 of 5 results.

A191361 Number of the diagonal of the Wythoff difference array that contains n.

Original entry on oeis.org

0, 1, -1, -2, 2, -3, 0, -4, -5, -1, -6, -7, 3, -8, -2, -9, -10, 1, -11, -3, -12, -13, -4, -14, -15, 0, -16, -5, -17, -18, -6, -19, -20, 4, -21, -7, -22, -23, -1, -24, -8, -25, -26, -9, -27, -28, 2, -29, -10, -30, -31, -2, -32, -11, -33, -34, -12, -35, -36, -3
Offset: 1

Views

Author

Clark Kimberling, May 31 2011

Keywords

Comments

Every integer occurs in A191361 (infinitely many times).
Represent the array as {g(i,j): i>=1, j>=1}. Then for m>=0, (diagonal #m) is the sequence (g(i,i+m)), i>=1;
for m<0, (diagonal #m) is the sequence (g(i+m,i)), i>=1.

Examples

			Diagonal #0 (the main diagonal) of A080164 is (1,7,26,...), so a(1)=0, a(7)=0, a(26)=0.

Crossrefs

Cf. A080164, A114327, A191360, A191362.

Programs

  • Mathematica
    r = GoldenRatio; f[n_] := Fibonacci[n];
g[i_, j_] := f[2 j - 1]*Floor[i*r] + (i - 1) f[2 j - 2];
TableForm[Table[g[i, j], {i, 1, 10}, {j, 1, 5}]]
(* A080164, Wythoff difference array *)
a = Flatten[Table[If[g[i, j] == n, j - i, {}], {n, 60}, {i, 50}, {j, 50}]]
(* a=A191361 *)

A191362 Number of the diagonal of the dispersion of the even positive integers that contains n.

Original entry on oeis.org

0, -1, 1, -2, 2, 0, 3, -3, 4, 1, 5, -1, 6, 2, 7, -4, 8, 3, 9, 0, 10, 4, 11, -2, 12, 5, 13, 1, 14, 6, 15, -5, 16, 7, 17, 2, 18, 8, 19, -1, 20, 9, 21, 3, 22, 10, 23, -3, 24, 11, 25, 4, 26, 12, 27, 0, 28, 13, 29, 5, 30, 14, 31, -6, 32, 15, 33, 6, 34, 16, 35, 1, 36, 17, 37, 7, 38, 18, 39, -2, 40, 19, 41, 8, 42, 20, 43, 2, 44, 21, 45, 9, 46, 22, 47, -4, 48, 23, 49, 10
Offset: 1

Views

Author

Clark Kimberling, May 31 2011

Keywords

Comments

Every integer occurs in A191362 (infinitely many times).
Represent the array as {f(i,j): i>=1, j>=1}. Then for m>=0, (diagonal #m) is the sequence (f(i,i+m)), i>=1;
for m<0, (diagonal #m) is the sequence (f(i+m,i)), i>=1.

Crossrefs

Cf. A191362, A114327, A191360, A191361.

Programs

  • Mathematica
    f[i_, j_] := (2 j - 1)*2^(i - 1);
t=TableForm[Table[f[i, j], {i, 1, 10}, {j, 1, 8}]]
(* t=A054582, the dispersion of the even positive integers *)
a = Flatten[Table[If[f[i, j] == n, j - i, {}], {n, 100}, {i, 10}, {j, 80}]]
(* a=A191362 *)

A360380 a(n) = number of the diagonal of the Wythoff array, A035513, that includes prime(n). See Comments.

Original entry on oeis.org

1, 2, 3, 0, 1, 5, -6, -7, -4, 3, -6, -3, -8, -16, 4, -20, -22, -23, -25, -8, -3, -9, -18, 9, -1, -38, -39, -6, -24, -14, -48, -3, -52, -18, -10, -34, 0, -62, -38, -5, -68, -24, -1, -44, -75, 7, -80, -30, -52, -87, 11, -91, -9, -19, -35, -100, -62, -103, -64
Offset: 1

Views

Author

Clark Kimberling, Feb 07 2023

Keywords

Comments

The indexing of diagonals is given in A191360. Conjecture: every integer occurs infinitely many times in this sequence; i.e., every diagonal includes infinitely many primes.

Examples

			a(n) = A191360(prime(n)).

Crossrefs

Cf. A000040, A035513, A191360, A332938, A360376, A360377, A360378, A360379.

Programs

  • Mathematica
    w[n_, k_] := Fibonacci[k + 1] Floor[n*GoldenRatio] + (n - 1) Fibonacci[k];
t = Table[w[n - k + 1, k], {n, 300}, {k, n, 1, -1}];
Map[1 + #[[1]] - 2 #[[2]] &, Most[Reap[NestWhile[# + 1 &, 1,
Length[Sow[FirstPosition[t, Prime[#]]]] > 1 &]][[2]][[1]]]]
(* Peter J. C. Moses, Feb 07 2023 *)

A360485 a(n) = index of the antidiagonal of the Wythoff array (A035513) that includes n.

Original entry on oeis.org

1, 2, 3, 2, 4, 3, 3, 5, 4, 4, 4, 5, 6, 6, 5, 5, 7, 5, 8, 6, 7, 9, 7, 6, 10, 6, 11, 8, 6, 12, 9, 7, 13, 8, 14, 10, 8, 15, 7, 16, 11, 7, 17, 12, 9, 18, 7, 19, 13, 10, 20, 8, 21, 14, 9, 22, 15, 11, 23, 9, 24, 16, 8, 25, 17, 12, 26, 8, 27, 18, 13, 28, 10, 29, 19
Offset: 1

Views

Author

Clark Kimberling, Feb 09 2023

Keywords

Comments

Each m appears exactly m times.

Crossrefs

Cf. A035513, A191360, A360379.

Programs

  • Mathematica
    W[n_, k_] := Fibonacci[k + 1] Floor[n*GoldenRatio] + (n - 1) Fibonacci[k];
t = Table[W[n - k + 1, k], {n, 300}, {k, n, 1, -1}];
Map[#[[1]] &, Most[Reap[NestWhileList[# + 1 &, 1,
Length[Sow[FirstPosition[t, #]]] > 1 &]][[2]][[1]]]]
(* Peter J. C. Moses, Feb 08 2023 *)

A360533 a(n) = index of the diagonal of the natural number array, A000027, that includes prime(n). See Comments.

Original entry on oeis.org

1, -1, 0, 3, 4, 0, 3, -1, 4, 7, 3, 8, 0, -4, 7, -5, 4, 0, 11, 3, -1, 12, 4, -8, 3, -5, -9, 12, 8, 0, 3, -5, 16, 12, -8, -12, 11, -1, -9, 16, 4, 0, 19, 15, 7, 3, 20, -4, -12, -16, 19, 7, 3, -17, 16, 4, -8, -12, 23, 15, 11, -9, 12, 4, 0, -8, 15, 3, -17, -21
Offset: 1

Views

Author

Clark Kimberling, Feb 10 2023

Keywords

Comments

The natural number array, A000027 = (w(n,k)) = (n + (n + k - 2) (n + k - 1)/2), has corner:
1 2 4 7 ...
3 5 8 12 ...
6 9 13 18 ...
10 14 19 25 ...
The indexing of diagonals is given in A191360. Conjecture: Every odd-indexed diagonal contains infinitely many primes.

Examples

			Prime(1) = 2  is in the diagonal (w(n,n+1)), so a(1) = 1.
Prime(13) = 43 is in the diagonal (w(n,n-4)), so a(7) = -4.

Crossrefs

Cf. A000027, A000040, A185787, A191360.

Programs

  • Mathematica
    Map[1 + #[[1]] - 2 #[[2]] &[{#[[2]], #[[1]] - ((#[[2]] - 1) + (#[[2]] - 1)^2)/
2} &[{#, Floor[(1 + Sqrt[8 # - 7])/2]}] &[Prime[#]]] &, Range[1000]]
(* Peter J. C. Moses, Feb 07 2023 *)
Showing 1-5 of 5 results.

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