Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #9 Apr 03 2016 20:32:02
%S 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,
%T 11,13,8,21,13,21,18,19,19,18,21,13,34,21,34,29,31,30,31,29,34,21,55,
%U 34,55,47,50,49,49,50,47,55,34,89
%N 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).
%C 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.
%C 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.
%C Patterns exist for certain generalized (a,b)-Pascal triangle transforms of row sequences. Definitions, explanation and examples (start):
%C Define (a,b)-Pascal triangles as having conditions T(0,0) = 1, a = left boundary and b = right boundary.
%C Let R_n be Row n, and R_n(k) be terms k in sequence R_n.
%C 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)).
%C For example, n=4: R_4 = {5, 3, 8, 11, 19, 30, 49, 79, 128, 207, 335, 542...}; a=3, b=5.
%C (3,5)-Pascal triangle is:
%C 1
%C 3 5
%C 3 8 5
%C 3 11 13 5
%C 3 14 24 18 5
%C etc.
%C Transform Tr_4(k) is:
%C Tr_4(0) = 5*1 = 5 = R_4(0).
%C Tr_4(1) = 5*3 + 3*5 = 30 = R_4(5).
%C Tr_4(2) = 5*3 + 3*8 + 8*5 = 79 = R_4(7).
%C Tr_4(3) = 5*3 + 3*11 + 8*13 + 11*5 = 207 = R_4(9).
%C Tr_4(4) = 5*3 + 3*14 + 8*24 + 11*18 + 19*5 = 542 = R_4(11).
%C etc.
%C Examples of sequences where such transforms apply:
%C Tr_0 = A001906 starting A001906(0)=0.
%C Tr_1 = A001519 starting A001519(2)=2.
%C Tr_2 = A002878 starting A002878(1)=4.
%C Tr_4 = A167375 starting A167375(3)=30.
%C (end)
%F T(n,k) = T(n,k-1) + T(n,k-2) = T(n-1,k) + T(n-2,k).
%F T(n,n) = T(n-1,n+1) = A061646(n).
%F T(n,n+1) = A079472(n+1). Omitting T(n,0), the array is symmetric about this falling diagonal.
%F 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:
%F C_0(n) = A000045(n+1).
%F R_0(k) = A000045(k-1); C_1(n) = A000045(n).
%F R_1(k) = A000045(k+1); C_2(n) = A000045(n+2).
%F R_2(k) = A000032(k); C_3(n) = A000032(n+1) .
%F R_3(k) = A013655(k); C_4(n) = A013655(n+1).
%F R_4(k) = A022121(k-1); C_5(n) = A022121(n).
%F R_5(k) = A022138(k-1); C_6(n) = A022138(n).
%F R_6(k) = A206610(k+1); C_7(n) = A206610(n+2).
%e Array Starts:
%e n/k 0 1 2 3 4 5 6 7 8 9 10
%e 0 1 0 1 1 2 3 5 8 13 21 34
%e 1 1 1 2 3 5 8 13 21 34 55 89
%e 2 2 1 3 4 7 11 18 29 47 76 123
%e 3 3 2 5 7 12 19 31 50 81 131 212
%e 4 5 3 8 11 19 30 49 79 128 207 335
%e 5 8 5 13 18 31 49 80 129 209 338 547
%e 6 13 8 21 29 50 79 129 208 337 545 882
%e 7 21 13 34 47 81 128 209 337 546 883 1429
%e 8 34 21 55 76 131 207 338 545 883 1428 2311
%e 9 55 34 89 123 212 335 547 882 1429 2311 3740
%e 10 89 55 144 199 343 542 885 1427 2312 3739 6051
%e Row 7 starts {21,13} because A000045(8)=21 and A000045(7)=13.
%e 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.
%o (PARI) {T(n, k) = fibonacci(n) * fibonacci(k) + fibonacci(n+1) * fibonacci(k-1)}; /* _Michael Somos_, Apr 03 2016 */
%Y Cf. A000045 (Fibonacci numbers)
%Y Cf. additional sequences related to rows and columns: A000032 (Lucas numbers), A013655, A022121, A022138, A206610.
%Y Cf. sequences related to falling diagonals: A061646, A079472.
%Y Cf. sequences related to (a,b)-Pascal triangle transforms of rows: A001906, A001519, A002878, A167375.
%K nonn,tabl
%O 0,6
%A _Bob Selcoe_, Apr 03 2016