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).

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

Author

Bob Selcoe, Apr 03 2016

Keywords

• nonn
• tabl

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).