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 1-4 of 4 results.

A265104 a(n) = A265100(n+1) - 6, n >= 1.

Original entry on oeis.org

8, 26, 35, 80, 89, 107, 116, 242, 251, 269, 278, 323, 332, 350, 359, 728, 737, 755, 764, 809, 818, 836, 845, 971, 980, 998, 1007, 1052, 1061, 1079, 1088, 2186, 2195, 2213, 2222, 2267, 2276, 2294, 2303, 2429, 2438, 2456, 2465, 2510, 2519, 2537
Offset: 1

Views

Author

L. Edson Jeffery, Dec 01 2015

Keywords

Comments

In the following, let "gap" and "gap number" be as defined in A265100, and let C(m) denote the m-th Catalan number (A000108).
Conjecture 1: The sequence contains all possible gap numbers.
Conjecture 2: For any gap G, the order |G| of G is the constant |G| = 6.
Conjecture 3: If g is a gap number, then 3*g + 2 is a gap number.
Conjecture 4: If C(m) =!= 0 (mod 3), then C(3*m+1) =!= 0 (mod 3) (=!= means "not congruent") or, what is the same thing, if m lies in a gap, then 3*m + 1 lies in a gap.

Crossrefs

Cf. A265100.

Programs

  • Mathematica
    a005836[1] := 0; a005836[n_] := If[OddQ[n], 3*a005836[Floor[(n + 1)/2]], a005836[n - 1] + 1]; a265100[n_] := 9*a005836[n] + 5; a265104[n_] := a265100[n+1] - 6; Table[a265104[n], {n, 46}]
(* Or: *)
a007814[x_] := IntegerExponent[x, 2]; a003602[x_] := (1 + x/2^a007814[x])/2; a005836[1] := 0; a005836[n_] := If[OddQ[n], 3*a005836[Floor[(n + 1)/2]], a005836[n - 1] + 1]; a265100[n_] := 9*a005836[n] + 5; a265104[n_] := (3^(a007814[n] + 2) - 3)/2 + a265100[2^(a007814[n])*(2*a003602[n] - 1)]; Table[a265104[n], {n, 46}]

Formula

a(n) = (3^(A007814(n) + 2) - 3)/2 + A265100(2^(A007814(n))*(2*A003602(n) - 1)), n >= 1.
Conjecture: a(n) = A265100(n) + A085296(n).

A111018 Indices of Catalan numbers that are divisible by 3.

Original entry on oeis.org

5, 6, 7, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 32, 33, 34, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 86, 87, 88, 95, 96, 97, 98, 99, 100
Offset: 1

Views

Author

Robert G. Wilson v, Sep 09 2005

Keywords

Comments

Conjecture: The sequence contains all numbers n such that n and n+1 are both in the sequence A074940, or, equivalently, such that neither n nor n+1 is in the sequence A005836. - L. Edson Jeffery, Dec 02 2015
The asymptotic density of this sequence is 1 (Burns, 2016). - Amiram Eldar, Jan 26 2021

Links

Crossrefs

Cf. A000108.
Cf. A085296, A265100, A265104.

Programs

A265159 Rectangular array A read by upward antidiagonals in which the entry in row n and column k is defined by A(n,k) = 5 + 9*A005836(2^(k - 1)*(2 n - 1)), n,k >= 1.

Original entry on oeis.org

5, 32, 14, 86, 95, 41, 113, 257, 284, 122, 248, 338, 770, 851, 365, 275, 743, 1013, 2309, 2552, 1094, 329, 824, 2228, 3038, 6926, 7655, 3281, 356, 986, 2471, 6683, 9113, 20777, 22964, 9842, 734, 1067, 2957, 7412, 20048, 27338, 62330, 68891, 29525
Offset: 1

Views

Author

L. Edson Jeffery, Dec 03 2015

Keywords

Comments

Conjecture 1: The array contains without duplication all possible "block numbers" defined in A265100.

Examples

			Array A begins:
.      5    14    41    122    365    1094    3281     9842    29525
.     32    95   284    851   2552    7655   22964    68891   206672
.     86   257   770   2309   6926   20777   62330   186989   560966
.    113   338  1013   3038   9113   27338   82013   246038   738113
.    248   743  2228   6683  20048   60143  180428   541283  1623848
.    275   824  2471   7412  22235   66704  200111   600332  1800995
.    329   986  2957   8870  26609   79826  239477   718430  2155289
.    356  1067  3200   9599  28796   86387  259160   777479  2332436
.    734  2201  6602  19805  59414  178241  534722  1604165  4812494

Crossrefs

Cf. A005836, A055246, A265100, A265104, A265161.

Programs

  • Mathematica
    (* Array: *)
a005836[1] := 0; a005836[n_] := If[OddQ[n], 3*a005836[Floor[(n + 1)/2]], a005836[n - 1] + 1]; a265159[n_, k_] := 5 + 9*a005836[2^(k - 1)*(2 n - 1)]; Grid[Table[a265159[n, k], {n, 9}, {k, 9}]]
(* Array antidiagonals flattened: *)
a005836[1] := 0; a005836[n_] := If[OddQ[n], 3*a005836[Floor[(n + 1)/2]], a005836[n - 1] + 1]; a265159[n_, k_] := 5 + 9*a005836[2^(k - 1)*(2 n - 1)]; Flatten[Table[a265159[n - k + 1, k], {n, 9}, {k, n}]]

Formula

Conjecture 2: A(n,k) = (A055246(n)*3^k + 1)/2, so the array and A265100 are related to Cantor's ternary set.
G.f. for row n (conjectured): f(n,x) = x*(A265100(n)-(A265100(n)+1)*x)/((1-x)*(1-3*x)).

A265161 Array A read by upward antidiagonals in which the entry in row n and column k is defined by A(n,k) = (3/2)*(3^k - 1) + A265159(n,k), n,k >= 1.

Original entry on oeis.org

8, 35, 26, 89, 107, 80, 116, 269, 323, 242, 251, 350, 809, 971, 728, 278, 755, 1052, 2429, 2915, 2186, 332, 836, 2267, 3158, 7289, 8747, 6560, 359, 998, 2510, 6803, 9476, 21869, 26243, 19682, 737, 1079, 2996, 7532, 20411, 28430, 65609, 78731, 59048
Offset: 1

Views

Author

L. Edson Jeffery, Dec 03 2015

Keywords

Comments

Conjecture 1: The array contains without duplication all possible "gap numbers" as defined in A265100.

Examples

			Array A begins:
.      8    26    80    242    728    2186    6560    19682    59048
.     35   107   323    971   2915    8747   26243    78731   236195
.     89   269   809   2429   7289   21869   65609   196829   590489
.    116   350  1052   3158   9476   28430   85292   255878   767636
.    251   755  2267   6803  20411   61235  183707   551123  1653371
.    278   836  2510   7532  22598   67796  203390   610172  1830518
.    332   998  2996   8990  26972   80918  242756   728270  2184812
.    359  1079  3239   9719  29159   87479  262439   787319  2361959
.    737  2213  6641  19925  59777  179333  538001  1614005  4842017

Crossrefs

Cf. A191107, A265100, A265104, A265159.

Programs

  • Mathematica
    (* Array: *)
a005836[1] := 0; a005836[n_] := If[OddQ[n], 3*a005836[Floor[(n + 1)/2]], a005836[n - 1] + 1]; a265159[n_, k_] := 5 + 9*a005836[2^(k - 1)*(2 n - 1)]; a265161[n_, k_] := (3/2)*(3^k - 1) + a265159[n, k]; Grid[Table[a265161[n, k], {n, 9}, {k, 9}]]
(* Array antidiagonal flattened: *)
a005836[1] := 0; a005836[n_] := If[OddQ[n], 3*a005836[Floor[(n + 1)/2]], a005836[n - 1] + 1]; a265159[n_, k_] := 5 + 9*a005836[2^(k - 1)*(2 n - 1)]; a265161[n_, k_] := (3/2)*(3^k - 1) + a265159[n, k]; Flatten[Table[a265161[n - k + 1, k], {n, 9}, {k, n}]]

Formula

Conjecture 2: A(n,k) = A191107(n)*3^k - 1.
Showing 1-4 of 4 results.

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