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.

A206611 Fibonacci sequence beginning 13, 7.

Original entry on oeis.org

13, 7, 20, 27, 47, 74, 121, 195, 316, 511, 827, 1338, 2165, 3503, 5668, 9171, 14839, 24010, 38849, 62859, 101708, 164567, 266275, 430842, 697117, 1127959, 1825076, 2953035, 4778111, 7731146, 12509257, 20240403, 32749660, 52990063, 85739723, 138729786
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Feb 10 2012

Keywords

Links

Crossrefs

Cf. A000032, A000045, A206609, A206610.

Programs

  • Magma
    I:=[13, 7]; [n le 2 select I[n] else Self(n-1)+Self(n-2): n in [1..40]]; // Vincenzo Librandi, Feb 16 2012
  • Mathematica
    LinearRecurrence[{1, 1}, {13, 7}, 80]

A206612 Fibonacci sequence beginning 13, 6.

Original entry on oeis.org

13, 6, 19, 25, 44, 69, 113, 182, 295, 477, 772, 1249, 2021, 3270, 5291, 8561, 13852, 22413, 36265, 58678, 94943, 153621, 248564, 402185, 650749, 1052934, 1703683, 2756617, 4460300, 7216917, 11677217, 18894134, 30571351, 49465485, 80036836, 129502321
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Feb 10 2012

Keywords

Links

Crossrefs

Cf. A000032, A000045, A206610, A206611.

Programs

  • Magma
    I:=[13, 6]; [n le 2 select I[n] else Self(n-1)+Self(n-2): n in [1..40]]; // Vincenzo Librandi, Feb 17 2012
  • Mathematica
    LinearRecurrence[{1, 1}, {13, 6}, 80]
  • PARI
    Vec((13 - 7*x)/(1 - x - x^2) + O(x^30)) \\ Andrew Howroyd, Aug 28 2018

Formula

From Andrew Howroyd, Aug 28 2018: (Start)
a(n) = a(n-1) + a(n-2) for n > 2.
a(n) = 13*Fibonacci(n) - 7*Fibonacci(n-1).
G.f.: x*(13 - 7*x)/(1 - x - x^2).
(End)

A271315 Array T(n,k) read by diagonals: T(n,k) = T(n,k-1) + T(n,k-2) where T(n,0) = F(n+1), T(n,1) = F(n); F(n) = Fibonacci(n) = A000045(n).

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 1, 2, 1, 3, 2, 3, 3, 2, 5, 3, 5, 4, 5, 3, 8, 5, 8, 7, 7, 8, 5, 13, 8, 13, 11, 12, 11, 13, 8, 21, 13, 21, 18, 19, 19, 18, 21, 13, 34, 21, 34, 29, 31, 30, 31, 29, 34, 21, 55, 34, 55, 47, 50, 49, 49, 50, 47, 55, 34, 89
Offset: 0

Views

Author

Bob Selcoe, Apr 03 2016

Keywords

Comments

The array is built by treating rows as Fibonacci-type sequences with seed values being two consecutive Fibonacci numbers (A000045(n) = F(n)) in reverse order: For row n, a(0) = F(n+1), a(1) = F(n). As a result, columns are Fibonacci-type sequences with seed values b(0) = F(k-1), b(1) = F(k+1); so starting with T(n,1), Row n == Column k=n+1.
Therefore, an alternative title is: Array T(n,k) read by diagonals: T(n,k) = T(n-1,k) + T(n-2,k) where T(0,k) = F(k-1) and T(1,k) = F(k+1), k>=1.
Patterns exist for certain generalized (a,b)-Pascal triangle transforms of row sequences. Definitions, explanation and examples: (Start)
Define (a,b)-Pascal triangles as having conditions T(0,0) = 1, a = left boundary and b = right boundary.
Let R_n be Row n, and R_n(k) be terms k in sequence R_n.
Let Tr_(k) be the (a,b)-Pascal triangle transform of R_n; define Tr_n(k) as when a = R_n(1) and b = R_n(0). Then Tr_n(k) = R_n(n+2k-2), k>=1. (Trivially, Tr_n(0) = R_n(0)).
For example, n=4: R_4 = {5, 3, 8, 11, 19, 30, 49, 79, 128, 207, 335, 542...}; a=3, b=5.
(3,5)-Pascal triangle is:
1
3 5
3 8 5
3 11 13 5
3 14 24 18 5
etc.
Transform Tr_4(k) is:
Tr_4(0) = 5*1 = 5 = R_4(0).
Tr_4(1) = 5*3 + 3*5 = 30 = R_4(5).
Tr_4(2) = 5*3 + 3*8 + 8*5 = 79 = R_4(7).
Tr_4(3) = 5*3 + 3*11 + 8*13 + 11*5 = 207 = R_4(9).
Tr_4(4) = 5*3 + 3*14 + 8*24 + 11*18 + 19*5 = 542 = R_4(11).
etc.
Examples of sequences where such transforms apply:
Tr_0 = A001906 starting A001906(0)=0.
Tr_1 = A001519 starting A001519(2)=2.
Tr_2 = A002878 starting A002878(1)=4.
Tr_4 = A167375 starting A167375(3)=30.
(End)

Examples

			Array Starts:
  n/k   0   1   2    3    4    5    6    7     8     9     10
  0     1   0   1    1    2    3    5    8     13    21    34
  1     1   1   2    3    5    8    13   21    34    55    89
  2     2   1   3    4    7    11   18   29    47    76    123
  3     3   2   5    7    12   19   31   50    81    131   212
  4     5   3   8    11   19   30   49   79    128   207   335
  5     8   5   13   18   31   49   80   129   209   338   547
  6     13  8   21   29   50   79   129  208   337   545   882
  7     21  13  34   47   81   128  209  337   546   883   1429
  8     34  21  55   76   131  207  338  545   883   1428  2311
  9     55  34  89   123  212  335  547  882   1429  2311  3740
  10    89  55  144  199  343  542  885  1427  2312  3739  6051
Row 7 starts {21,13} because A000045(8)=21 and A000045(7)=13.
T(9,2)=89 + T(9,3)=123 = T(9,4)=212; alternatively, T(7,4)=81 + T(8,4)=131 = T(9,4)=212.

Crossrefs

Cf. A000045 (Fibonacci numbers).
Cf. additional sequences related to rows and columns: A000032 (Lucas numbers), A013655, A022121, A022138, A206610.
Cf. sequences related to falling diagonals: A061646, A079472.
Cf. sequences related to (a,b)-Pascal triangle transforms of rows: A001906, A001519, A002878, A167375.

Programs

  • PARI
    {T(n, k) = fibonacci(n) * fibonacci(k) + fibonacci(n+1) * fibonacci(k-1)}; /* Michael Somos, Apr 03 2016 */

Formula

T(n,k) = T(n,k-1) + T(n,k-2) = T(n-1,k) + T(n-2,k).
T(n,n) = T(n-1,n+1) = A061646(n).
T(n,n+1) = A079472(n+1). Omitting T(n,0), the array is symmetric about this falling diagonal.
Treating rows and columns as individual sequences, let R_n be Row n and C_k be Column k; let R_n(k) and C_k(n) be terms k and n, respectively, in these sequences:
C_0(n) = A000045(n+1).
R_0(k) = A000045(k-1); C_1(n) = A000045(n).
R_1(k) = A000045(k+1); C_2(n) = A000045(n+2).
R_2(k) = A000032(k); C_3(n) = A000032(n+1) .
R_3(k) = A013655(k); C_4(n) = A013655(n+1).
R_4(k) = A022121(k-1); C_5(n) = A022121(n).
R_5(k) = A022138(k-1); C_6(n) = A022138(n).
R_6(k) = A206610(k+1); C_7(n) = A206610(n+2).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.