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 A178239 #15 Jan 01 2023 19:28:49
%S A178239 1,1,1,1,1,1,1,1,2,1,1,1,3,1,1,1,1,4,1,3,1,1,1,5,1,5,2,1,1,1,6,1,7,3,
%T A178239 3,1,1,1,7,1,9,4,7,1,1,1,1,8,1,11,5,13,1,4,1,1,1,9,1,13,6,21,1,7,3,1,
%U A178239 1,1,10,1,15,7,31,1,10,5,5,1
%N A178239 Triangle read by rows, antidiagonals of an array generated from a(n) = a(2n), a(2n+1) = r*a(n) + a(n+1).
%C A178239 Partial sums of array terms in groups of 1, next 2, next 4, ... 8 = powers of (r+2).
%C A178239 Row sums = A178240: (1, 2, 3, 5, 7, 11, 16, 23, ...).
%C A178239 Row 1 of the array = A002487.
%C A178239 Row 2 = .............A116528.
%C A178239 Row 3 = .............A342633.
%C A178239 Row 4 = .............A342634.
%C A178239 ...
%C A178239 Row 10 = ............A178243.
%C A178239 Polcoeff row r of the array as f(x) satisfies f(x)/f(x^2) = (1 + x + r*x^2).
%C A178239 Let q(x) = (1 + x + r*x^2). Then polcoeff row 4 = q(x) * q(x^2) * q(x^4) * q(x^8) * ...
%F A178239 Antidiagonals of an array generated from a(n) = a(2n); a(2n+1) = r*a(n) + a(n+1).
%F A178239 Given a triangle M with columns stepped down twice from the previous column, for columns > 0, with (1, 1, r, 0, 0, 0, ...) in each column, r-th row of the array = lim_{n->oo} M^n.
%e A178239 First few rows of the array =
%e A178239 n=1 n=2 n=3 n=4 n=5 n=6 n=7 n=8 n=9 n=10 n=11 n=12 n=13 n=14 n=15
%e A178239 r=0: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e A178239 r=1: 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, ...
%e A178239 r=2: 1, 1, 3, 1, 5, 3, 7, 1, 7, 5, 13, 3, 13, 7, 15, ...
%e A178239 r=3: 1, 1, 4, 1, 7, 4, 13, 1, 10, 7, 25, 4, 25, 13, 40, ...
%e A178239 r=4: 1, 1, 5, 1, 9, 5, 21, 1, 13, 9, 41, 5, 41, 21, 85, ...
%e A178239 r=5: 1, 1, 6, 1, 11, 6, 31, 1, 16, 11, 61, 6, 61, 31, 156, ...
%e A178239 ...
%e A178239 Example: In row 3: (1, 1, 4, 1, 7, 4, 13, ...) = A342633, r = 3.
%e A178239 A342633(7) = 13 = 3*4 + 1. In blocks of 1, 2, 4, 8, ... terms, partial sums are powers of (r+2) = 5: (1, 5, 25, ...).
%e A178239 First few rows of the triangle =
%e A178239 1;
%e A178239 1, 1;
%e A178239 1, 1, 1;
%e A178239 1, 1, 2, 1;
%e A178239 1, 1, 3, 1, 1;
%e A178239 1, 1, 4, 1, 3, 1;
%e A178239 1, 1, 5, 1, 5, 2, 1;
%e A178239 1, 1, 6, 1, 7, 3, 3, 1;
%e A178239 1, 1, 7, 1, 9, 4, 7, 1, 1;
%e A178239 1, 1, 8, 1, 11, 5, 13, 1, 4, 1;
%e A178239 1, 1, 9, 1, 13, 6, 21, 1, 7, 3, 1;
%e A178239 1, 1, 10, 1, 15, 7, 31, 1, 10, 5, 5, 1;
%e A178239 1, 1, 11, 1, 17, 8, 43, 1, 13, 7, 13, 2, 1;
%e A178239 1, 1, 12, 1, 19, 9, 57, 1, 16, 9, 21, 3, 5, 1;
%e A178239 1, 1, 13, 1, 21, 11, 73, 1, 19, 11, 31, 4, 13, 2, 1;
%e A178239 ...
%Y A178239 Cf. A178240, A359250 (column polynomials).
%Y A178239 Array rows r=1..10: A002487, A116528, A342633, A342634, A342635, A342603, A342636, A342637, A342638, A178243.
%K A178239 nonn,easy,tabl
%O A178239 0,9
%A A178239 _Gary W. Adamson_, May 23 2010