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 A140995 #54 Jan 28 2024 18:10:14
%S A140995 1,1,1,1,2,1,1,2,4,1,1,2,4,8,1,1,2,4,8,16,1,1,2,4,8,17,31,1,1,2,4,8,
%T A140995 17,35,60,1,1,2,4,8,17,35,72,116,1,1,2,4,8,17,35,72,148,224,1,1,2,4,8,
%U A140995 17,35,72,149,303,432,1,1,2,4,8,17,35,72,149,308,618,833,1,1,2,4,8,17,35,72,149,308,636,1257,1606,1
%N A140995 Triangle G(n, k) read by rows, for 0 <= k <= n, where G(n, n) = G(n+1, 0) = 1, G(n+2, 1) = 2, G(n+3, 2) = 4, G(n+4, 3) = 8, and G(n+5, m) = G(n+1, m-3) + G(n+1, m-4) + G(n+2, m-3) + G(n+3, m-2) + G(n+4, m-1) for n >= 0 and m = 4..(n+4).
%C A140995 From _Petros Hadjicostas_, Jun 13 2019: (Start)
%C A140995 This is a mirror image of the triangular array A140996. The current array has index of asymmetry s = 3 and index of obliqueness (obliquity) e = 1. Array A140996 has the same index of asymmetry, but has index of obliqueness e = 0. (In other related sequences, the author uses the letter y for the index of asymmetry and the letter z for the index of obliqueness, but in a picture that he posted in those sequences, the letters s and e are used instead. See, for example, the documentation for sequences A140998, A141065, A141066, and A141067.)
%C A140995 Pascal's triangle A007318 has s = 0 and is symmetric, arrays A140998 and A140993 have s = 1 (with e = 0 and e = 1, respectively), and arrays A140997 and A140994 have s = 2 (with e = 0 and e = 1, respectively).
%C A140995 If A(x,y) = Sum_{n,k >= 0} G(n, k)*x^n*y^k is the bivariate g.f. for this array (with G(n, k) = 0 for 0 <= n < k) and B(x, y) = Sum_{n, k} A140996(n, k)*x^n*y^k, then A(x, y) = B(x*y, y^(-1)). This can be proved using formal manipulation of double series expansions and the fact G(n, k) = A140996(n, n-k) for 0 <= k <= n.
%C A140995 If we let b(k) = lim_{n -> infinity} G(n, k) for k >= 0, then b(0) = 1, b(1) = 2, b(2) = 4, b(3) = 8, and b(k) = b(k-1) + b(k-2) + 2*b(k-3) + b(k-4) for k >= 4. (The existence of the limit can be proved by induction on k.) Thus, the limiting sequence is 1, 2, 4, 8, 17, 35, 72, 149, 308, 636, 1314, 2715, 5609, 11588, 23941, 49462, 102188, 211120, 436173, ... (sequence A309462). (End)
%H A140995 Juri-Stepan Gerasimov, <a href="/A140998/a140998.jpg">Stepan's triangles and Pascal's triangle are connected by the recurrence relation ...</a>
%F A140995 From _Petros Hadjicostas_, Jun 13 2019: (Start)
%F A140995 G(n, k) = A140996(n, n-k) for 0 <= k <= n.
%F A140995 Bivariate g.f.: Sum_{n,k >= 0} G(n, k)*x^n*y^k = (x^5*y^4 - x^4*y^4 - x^3*y^3 + x^3*y^2 - x^2*y^2 + x^2*y - x*y + 1)/((1- x*y) * (1- x) * (1 - x*y - x^2*y^2 -x^3*y^3 - x^4*y^4 - x^4*y^3)).
%F A140995 Substituting y = 1 in the above bivariate function and simplifying, we get the g.f. of row sums: 1/(1 - 2*x). Hence, the row sums are powers of 2; i.e., A000079.
%F A140995 (End)
%e A140995 Triangle begins:
%e A140995 1
%e A140995 1 1
%e A140995 1 2 1
%e A140995 1 2 4 1
%e A140995 1 2 4 8 1
%e A140995 1 2 4 8 16 1
%e A140995 1 2 4 8 17 31 1
%e A140995 1 2 4 8 17 35 60 1
%e A140995 1 2 4 8 17 35 72 116 1
%e A140995 1 2 4 8 17 35 72 148 224 1
%e A140995 1 2 4 8 17 35 72 149 303 432 1
%e A140995 1 2 4 8 17 35 72 149 308 618 833 1
%e A140995 ...
%Y A140995 Cf. A000079, A007318, A140993, A140994, A140996, A140997, A140998, A141020, A141021, A141031, A141065, A141066, A141067, A141068, A141069, A141070, A141072, A141073, A309462.
%K A140995 nonn,tabl
%O A140995 0,5
%A A140995 _Juri-Stepan Gerasimov_, Jul 08 2008
%E A140995 Entries checked by _R. J. Mathar_, Apr 14 2010
%E A140995 Name edited by and more terms from _Petros Hadjicostas_, Jun 13 2019