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.
%I A271315 #14 Aug 15 2025 04:27:58
%S A271315 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 A271315 11,13,8,21,13,21,18,19,19,18,21,13,34,21,34,29,31,30,31,29,34,21,55,
%U A271315 34,55,47,50,49,49,50,47,55,34,89
%N 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).
%C A271315 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 A271315 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 A271315 Patterns exist for certain generalized (a,b)-Pascal triangle transforms of row sequences. Definitions, explanation and examples: (Start)
%C A271315 Define (a,b)-Pascal triangles as having conditions T(0,0) = 1, a = left boundary and b = right boundary.
%C A271315 Let R_n be Row n, and R_n(k) be terms k in sequence R_n.
%C A271315 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 A271315 For example, n=4: R_4 = {5, 3, 8, 11, 19, 30, 49, 79, 128, 207, 335, 542...}; a=3, b=5.
%C A271315 (3,5)-Pascal triangle is:
%C A271315 1
%C A271315 3 5
%C A271315 3 8 5
%C A271315 3 11 13 5
%C A271315 3 14 24 18 5
%C A271315 etc.
%C A271315 Transform Tr_4(k) is:
%C A271315 Tr_4(0) = 5*1 = 5 = R_4(0).
%C A271315 Tr_4(1) = 5*3 + 3*5 = 30 = R_4(5).
%C A271315 Tr_4(2) = 5*3 + 3*8 + 8*5 = 79 = R_4(7).
%C A271315 Tr_4(3) = 5*3 + 3*11 + 8*13 + 11*5 = 207 = R_4(9).
%C A271315 Tr_4(4) = 5*3 + 3*14 + 8*24 + 11*18 + 19*5 = 542 = R_4(11).
%C A271315 etc.
%C A271315 Examples of sequences where such transforms apply:
%C A271315 Tr_0 = A001906 starting A001906(0)=0.
%C A271315 Tr_1 = A001519 starting A001519(2)=2.
%C A271315 Tr_2 = A002878 starting A002878(1)=4.
%C A271315 Tr_4 = A167375 starting A167375(3)=30.
%C A271315 (End)
%F A271315 T(n,k) = T(n,k-1) + T(n,k-2) = T(n-1,k) + T(n-2,k).
%F A271315 T(n,n) = T(n-1,n+1) = A061646(n).
%F A271315 T(n,n+1) = A079472(n+1). Omitting T(n,0), the array is symmetric about this falling diagonal.
%F A271315 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 A271315 C_0(n) = A000045(n+1).
%F A271315 R_0(k) = A000045(k-1); C_1(n) = A000045(n).
%F A271315 R_1(k) = A000045(k+1); C_2(n) = A000045(n+2).
%F A271315 R_2(k) = A000032(k); C_3(n) = A000032(n+1) .
%F A271315 R_3(k) = A013655(k); C_4(n) = A013655(n+1).
%F A271315 R_4(k) = A022121(k-1); C_5(n) = A022121(n).
%F A271315 R_5(k) = A022138(k-1); C_6(n) = A022138(n).
%F A271315 R_6(k) = A206610(k+1); C_7(n) = A206610(n+2).
%e A271315 Array Starts:
%e A271315 n/k 0 1 2 3 4 5 6 7 8 9 10
%e A271315 0 1 0 1 1 2 3 5 8 13 21 34
%e A271315 1 1 1 2 3 5 8 13 21 34 55 89
%e A271315 2 2 1 3 4 7 11 18 29 47 76 123
%e A271315 3 3 2 5 7 12 19 31 50 81 131 212
%e A271315 4 5 3 8 11 19 30 49 79 128 207 335
%e A271315 5 8 5 13 18 31 49 80 129 209 338 547
%e A271315 6 13 8 21 29 50 79 129 208 337 545 882
%e A271315 7 21 13 34 47 81 128 209 337 546 883 1429
%e A271315 8 34 21 55 76 131 207 338 545 883 1428 2311
%e A271315 9 55 34 89 123 212 335 547 882 1429 2311 3740
%e A271315 10 89 55 144 199 343 542 885 1427 2312 3739 6051
%e A271315 Row 7 starts {21,13} because A000045(8)=21 and A000045(7)=13.
%e A271315 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 A271315 (PARI) {T(n, k) = fibonacci(n) * fibonacci(k) + fibonacci(n+1) * fibonacci(k-1)}; /* _Michael Somos_, Apr 03 2016 */
%Y A271315 Cf. A000045 (Fibonacci numbers).
%Y A271315 Cf. additional sequences related to rows and columns: A000032 (Lucas numbers), A013655, A022121, A022138, A206610.
%Y A271315 Cf. sequences related to falling diagonals: A061646, A079472.
%Y A271315 Cf. sequences related to (a,b)-Pascal triangle transforms of rows: A001906, A001519, A002878, A167375.
%K A271315 nonn,tabl
%O A271315 0,6
%A A271315 _Bob Selcoe_, Apr 03 2016