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 A253146 #46 Mar 14 2025 20:20:37
%S A253146 1,2,3,4,1,5,6,2,3,7,8,4,1,5,9,10,6,2,3,7,11,12,8,4,1,5,9,13,14,10,6,
%T A253146 2,3,7,11,15,16,12,8,4,1,5,9,13,17,18,14,10,6,2,3,7,11,15,19,20,16,12,
%U A253146 8,4,1,5,9,13,17,21,22,18,14,10,6,2,3,7,11,15,19,23
%N A253146 A fractal tree, read by rows: for n > 2, T(n,1) = T(n-1,1)+2, T(n,n) = T(n-1,1)+3, and for k=2..n-1, T(n,k) = T(n-2,k-1).
%C A253146 Eric Angelini's original posting to the Sequence Fans mailing list gave a similar but different lovely sequence, which is now A253028. - _N. J. A. Sloane_, Jan 04 2015, and _Felix Fröhlich_, May 23 2016
%C A253146 It appears that:
%C A253146 1) partial sums of terms, situated on the outer leftmost leftwise triangle diagonal are equal to A002061(k), k>=1;
%C A253146 2) partial sums of terms, situated on the second (from the left) leftwise triangle diagonal represent recurrence a(k+1) = ((k-1)*a(k))/(k-3)-(2*(k+3))/(k-3), k>=3
%C A253146 3) partial sums of terms, situated on the outer rightmost rightwise triangle diagonal are equal to A000290(k)=k^2, k>=1. - _Alexander R. Povolotsky_, Dec 28 2014
%H A253146 Reinhard Zumkeller, <a href="/A253146/b253146.txt">Rows n = 1..125 of triangle, flattened</a>
%H A253146 Éric Angelini, <a href="https://web.archive.org/web/20201230072549/http://list.seqfan.eu/pipermail/seqfan/2014-December/014193.html">A fractal tree</a>, SeqFan list, Dec 27 2014.
%e A253146 . 1: 1
%e A253146 . 2: 2 3
%e A253146 . 3: 4 1 5
%e A253146 . 4: 6 2 3 7
%e A253146 . 5: 8 4 1 5 9
%e A253146 . 6: 10 6 2 3 7 11
%e A253146 . 7: 12 8 4 1 5 9 13
%e A253146 . 8: 14 10 6 2 3 7 11 15
%e A253146 . 9: 16 12 8 4 1 5 9 13 17
%e A253146 . 10: 18 14 10 6 2 3 7 11 15 19
%e A253146 . 11: 20 16 12 8 4 1 5 9 13 17 21
%e A253146 . 12: 22 18 14 10 6 2 3 7 11 15 19 23 .
%e A253146 Removing the first and last entries from each row gives the same tree back again.
%t A253146 T[n_, 1] := 2n - 2;
%t A253146 T[n_, n_] := 2n - 1;
%t A253146 T[n_, k_] := T[n, k] = T[n-2, k-1];
%t A253146 Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Sep 20 2021 *)
%o A253146 (Haskell)
%o A253146 a253146 n k = a253146_tabl !! (n-1) !! (k-1)
%o A253146 a253146_row n = a253146_tabl !! (n-1)
%o A253146 a253146_tabl = [1] : [2,3] : f [1] [2,3] where
%o A253146 f us vs@(v:_) = ws : f vs ws where
%o A253146 ws = [v + 2] ++ us ++ [v + 3]
%Y A253146 Cf. A253028. Row sums appear to be A035608.
%Y A253146 Cf. A000290, A002061.
%K A253146 nonn,tabl
%O A253146 1,2
%A A253146 _Eric Angelini_ and _Reinhard Zumkeller_, Dec 27 2014