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 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). 0
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 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) LINKS 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). EXAMPLE 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. PROG (PARI) {T(n, k) = fibonacci(n) * fibonacci(k) + fibonacci(n+1) * fibonacci(k-1)}; /* Michael Somos, Apr 03 2016 */ 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. Sequence in context: A027350 A029327 A079135 * A212813 A112219 A035458 Adjacent sequences:  A271312 A271313 A271314 * A271316 A271317 A271318 KEYWORD nonn,tabl AUTHOR Bob Selcoe, Apr 03 2016 STATUS approved

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Last modified October 21 06:01 EDT 2019. Contains 328291 sequences. (Running on oeis4.)