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.
For n=6 we have the following row of triangle A245618: 1 2 3 8 3 2 1. We have 1<+>2=1, 1<+>3=4, 4<+>8=12, 12<+>3=9, 9<+>2=7, 7<+>1=8. So, a(6)=8.
parityAdd[a_,b_]:=Abs[a+b (-1)^(a+b)];
t[n_,0]:=1;
t[n_,n_]:=1;
t[n_,k_]:=t[n,k]=parityAdd[t[n-1,k-1],t[n-1,k]];
Map[Fold[parityAdd,First[#],Rest[#]]&,Table[t[n,k],{n,0,40},{k,0,n}]] (* Peter J. C. Moses, Nov 05 2014 *)
For n=4, we have row 1,4,6,4,1. By definition of <+>, we find 1<+>4=3, 3<+>6=3, 3<+>4=1, 1<+>1=2. So a(4)=2.
a249779[n_Integer] := Module[{m0082, pls, lst},
m0082[j_] := Table[Binomial[j, k], {k, 0, j}];
pls[k_, m_] := Abs[k + (-1)^(k + m)*m];
lst = m0082[n];
For[i = 0, i < n, i++, lst[[2]] = pls[lst[[1]], lst[[2]]];
lst = Drop[lst, 1]];
lst[[1]]
]; a249779 /@ Range[35] (* Michael De Vlieger, Nov 23 2014 *)
parityAdd[a_,b_]:=Abs[a+b (-1)^(a+b)];
Map[Fold[parityAdd,First[#],Rest[#]]&[Binomial[#,Range[0,#]]]&,Range[0,35]] (* Peter J. C. Moses, Dec 01 2014 *)
Module[{n = 0, f}, NestList[If[(f = Fibonacci[n++]) < #, # - f, # + f] &, 0, 49]] (* Paolo Xausa, Nov 08 2024 *)
Flatten[Join[{0, 0}, Table[{1, Fibonacci[{k, k+2}] + 1}, {k, 2, 49, 3}]]] (* Paolo Xausa, Nov 08 2024 *)
LinearRecurrence[{1, 0, 4, -4, 0, 1, -1}, {0, 0, 1, 2, 4, 1, 6, 14, 1}, 50] (* Paolo Xausa, Nov 08 2024 *)
a(n) = if (n==0, 0, my(d=a(n-1)-fibonacci(n-1)); if (d>0, d, d+2*fibonacci(n-1))) \\ Michel Marcus, Dec 29 2018
a(n) = if (n<=1, 0, my(m=(n % 3)); if (m==0, fibonacci(n-1)+1, if (m==1, fibonacci(n)+1, 1))); \\ Michel Marcus, Dec 29 2018
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