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.

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A372501 The 2-Zeckendorf array of the second kind, read by upward antidiagonals.

Original entry on oeis.org

0, 2, 1, 5, 4, 3, 7, 9, 8, 6, 10, 12, 16, 14, 11, 13, 17, 21, 27, 24, 19, 15, 22, 29, 35, 45, 40, 32, 18, 25, 37, 48, 58, 74, 66, 53, 20, 30, 42, 61, 79, 95, 121, 108, 87, 23, 33, 50, 69, 100, 129, 155, 197, 176, 142, 26, 38, 55, 82, 113, 163, 210, 252, 320, 286, 231
Offset: 1

Views

Author

A.H.M. Smeets, May 03 2024

Keywords

Comments

The 2-Zeckendorf array of the second kind is based on the dual Zeckendorf representation of numbers (see A104326).
Column k contains the numbers whose dual Zeckendorf expansion ends "... 0 1^(k-1)" where ^ denotes repetition.
Rows satisfy this recurrence: T(n,k+1) = T(n,k) + T(n,k-1) + 2 for all n > 0 and k > 1.
As a sequence, the array is a permutation of the nonnegative integers.
As an array, T is an interspersion (hence also a dispersion). This holds as well for all Zeckendorf arrays of the second kind.
In general, for the m-Zeckendorf array of the second kind, the row recursion is given by T(n,k) = T(n,k-1) + T(n,k-m) + m, and the first column represent the "even" numbers.

Examples

			Array begins:
       k=1    2    3    4    5    6    7
      +---------------------------------
  n=1 |  0    1    3    6   11   19   32
  n=2 |  2    4    8   14   24   40   66
  n=3 |  5    9   16   27   45   74  121
  n=4 |  7   12   21   35   58   95  155
  n=5 | 10   17   29   48   79  129  210
  n=6 | 13   22   37   61  100  163  265
  n=7 | 15   25   42   69  113  184  299
The same in dual Zeckendorf form shows the pattern of digit suffixes, for example column k=3 is all numbers ending 011:
          k=1      2       3        4
      +------------------------------
  n=1 |     0      1      11      111
  n=2 |    10    101    1011    10111
  n=3 |   110   1101   11011   110111
  n=4 |  1010  10101  101011  1010111
  n=5 |  1110  11101  111011  1110111

Crossrefs

Cf. A104326.
Rows n=1..3: A001911, A019274, A014739.
Columns k=1..3: A090909, A276885, A188012.
Cf. k-th prepended column: A022342 (k=1), A023444 (k=2).

Formula

T(n,1) = A090909(n+1).
T(1,k) = A001911(k-1).
T(2,k) = A019274(k-2).
T(3,k) = A014739(k-1).
T(n,1) = floor((n-1)*phi^2) and T(n,k+1) = floor((T(n,k)+1)*phi) for k > 0, where phi = (1+sqrt(5))/2. This can be considered as an alternative way to define the array.

A053566 Expansion of (11*x-2)/(1-3*x)^2.

Original entry on oeis.org

-2, -1, 12, 81, 378, 1539, 5832, 21141, 74358, 255879, 866052, 2893401, 9565938, 31355019, 102036672, 330024861, 1061819118, 3400690959, 10847773692, 34480423521, 109252577898, 345191655699
Offset: 0

Views

Author

Barry E. Williams, Jan 17 2000

Keywords

References

  • A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 189, 194-196.

Links

Crossrefs

Cf. A023444.

Programs

  • GAP
    List([0..30], n-> 3^(n-1)*(5*n-6)) # G. C. Greubel, May 16 2019
  • Magma
    [3^(n-1)*(5*n-6) : n in [0..30]]; // G. C. Greubel, May 16 2019
  • Mathematica
    LinearRecurrence[{6,-9},{-2,-1},30] (* Harvey P. Dale, Jun 26 2012 *)
  • PARI
    Vec((11*x-2)/(1-3*x)^2 + O(x^30)) \\ Michel Marcus, Dec 03 2014
  • Sage
    [3^(n-1)*(5*n-6) for n in (0..30)] # G. C. Greubel, May 16 2019

Formula

a(n) = 3^(n-1)*(5*n-6).
a(n) = 6*a(n-1) - 9*a(n-2), with a(0) = -2, a(1) = -1.
E.g.f.: (5*x - 2)*exp(3*x). - G. C. Greubel, May 16 2019

A243860 a(n) = 2^(n+1) - (n-1)^2.

Original entry on oeis.org

1, 4, 7, 12, 23, 48, 103, 220, 463, 960, 1967, 3996, 8071, 16240, 32599, 65340, 130847, 261888, 523999, 1048252, 2096791, 4193904, 8388167, 16776732, 33553903, 67108288, 134217103, 268434780, 536870183, 1073741040, 2147482807, 4294966396, 8589933631, 17179868160, 34359737279, 68719475580
Offset: 0

Views

Author

Juri-Stepan Gerasimov, Jun 12 2014

Keywords

Comments

Sequences of the form (k-1)^m - m^(k+1):
k\m | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
-----------------------------------------------------------------------
0 | 1 | -2 | -1 | -4 | -3 | -6 | -5 |
1 | 1 | -1 | -4 | -9 | -16 | -25 | -36 |
2 | 1 | 0 | -7 | -26 | -63 | -124 | -215 |
3 | 1 | 1 | -12 | -73 | -240 | -593 | -1232 |
4 | 1 | 2 | -23 | -216 | -943 | -2882 | -7047 |
5 | 1 | 3 | -43 | -665 | -3840 | -14601 | -42560 |
6 | 1 | 4 | -103 | -2062 | -15759 | -75000 | -264311 |
7 | 1 | 5 | -220 | -6345 | -64240 | -382849 | -1632960 |
8 | 1 | 6 | -463 | -19340 | -259743 | -1936318 | -9960047 |
9 | 1 | 7 | -960 | -58537 | -1044480 | -9732857 | -60204032 |
10 | 1 | 8 | -1967 | -176418 | -4187743 | -48769076 | -362265615 |
11 | 1 | 9 | -3996 | -530441 | -16767216 | -244040625 | -2175782336 |

Examples

			1 = 2^(0+1) - (0-1)^2, 4 = 2^(1+1) - (1-1)^2, 7 = 2^(2+1) - (2-1)^2.

Links

Crossrefs

Sequences of the form (k-1)^m - m^(k+1): A000012 (m = 0), A023444 (m = 1), (-1)*(this sequence) for m = 2, A114285 (k = 0),(A000007-A000290) for k = 1, A024001 (k = 2), A024014 (k = 3), A024028 (k = 4), A024042 (k = 5), A024056 (k = 6), A024070 (k = 7), A024084 (k = 8), A024098 (k = 9), A024112 (k = 10), A024126 (k = 11).

Programs

  • Magma
    [2^(n+1) - (n-1)^2: n in [0..35]];
  • Maple
    A243860:=n->2^(n + 1) - (n - 1)^2; seq(A243860(n), n=0..30); # Wesley Ivan Hurt, Jun 12 2014
  • Mathematica
    Table[2^(n + 1) - (n - 1)^2, {n, 0, 30}] (* Wesley Ivan Hurt, Jun 12 2014 *)
LinearRecurrence[{5,-9,7,-2},{1,4,7,12},40] (* Harvey P. Dale, Nov 29 2015 *)
  • PARI
    Vec((6*x^3-4*x^2-x+1)/((x-1)^3*(2*x-1)) + O(x^100)) \\ Colin Barker, Jun 12 2014

Formula

a(n) = 5*a(n-1)-9*a(n-2)+7*a(n-3)-2*a(n-4). - Colin Barker, Jun 12 2014
G.f.: (6*x^3-4*x^2-x+1) / ((x-1)^3*(2*x-1)). - Colin Barker, Jun 12 2014
Previous Showing 11-13 of 13 results.

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