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 A141073 #50 Jul 16 2025 14:25:40
%S A141073 1,1,4,2,8,4,17,8,35,17,72,35,149,72,308,149,636,308,1314,636,2715,
%T A141073 1314,5609,2715,11588,5609,23941,11588,49462,23941,102188,49462,
%U A141073 211120,102188,436173,211120,901131,436173,1861732,901131,3846329,1861732,7946496,3846329
%N A141073 List of central integer pairs in Pascal-like triangles with index of asymmetry y = 3 and index of obliqueness z = 0 or z = 1.
%C A141073 For the Pascal-like triangle G(n, k) with index of asymmetry y = 3 and index of obliqueness z = 0, which is read by rows, we have G(n, 0) = G(n+1, n+1) = 1, G(n+2, n+1) = 2, G(n+3, n+1) = 4, G(n+4, n+1) = 8, and G(n+5, k) = G(n+1, k-1) + G(n+1, k) + G(n+2, k) + G(n+3, k) + G(n+4, k) for n >= 0 and k = 1..(n+1). (This is array A140996.)
%C A141073 For the Pascal-like triangle G(n, k) with index of asymmetry y = 3 and index of obliqueness z = 1, which is read by rows, we have 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, k) = G(n+1, k-3) + G(n+1, k-4) + G(n+2, k-3) + G(n+3, k-2) + G(n+4, k-1) for n > = 0 and k = 4..(n+4). (This is array A140995.)
%C A141073 Arrays A140995 and A140996 are mirror images of each other. For discussion about their properties and their connection to Stepan's triangles, see their documentation. See also the documentation of the sequences in the CROSSREFS. - _Petros Hadjicostas_, Jun 13 2019
%H A141073 Juri-Stepan Gerasimov, <a href="/A140998/a140998.jpg">Stepan's triangles and Pascal's triangle are connected by the recurrence relation ...</a>
%H A141073 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-4,1,2,-1).
%F A141073 From _Petros Hadjicostas_, Jun 13 2019: (Start)
%F A141073 a(2*n - 1) = A140996(2*n - 1, n - 1) = A140995(2*n - 1, n) and a(2*n) = A140996(2*n - 1, n) = A140995(2*n - 1, n - 1) for n >= 1.
%F A141073 a(2*n) = a(2*n - 3) for n >= 3.
%F A141073 a(n) = 2*a(n-2) + A129847(floor(n/2) - (4 + (-1)^n)) for n >= 9.
%F A141073 G.f.: x*(x^8 + 3*x^6 + x^5 + 3*x^4 + x^3 + 3*x^2 + x + 1)/(1 - x^2 - x^4 - 2*x^6 -x^8). (End)
%e A141073 Pascal-like triangle with y = 3 and z = 0 (i.e., A140996) begins as follows:
%e A141073 1, so no central pair.
%e A141073 1 1, so a(1) = 1 and a(2) = 1.
%e A141073 1 2 1, so no central pair.
%e A141073 1 4 2 1, so a(3) = 4 and a(4) = 2.
%e A141073 1 8 4 2 1, so no central pair.
%e A141073 1 16 8 4 2 1, so a(5) = 8 and a(6) = 4.
%e A141073 1 31 17 8 4 2 1, so no central pair.
%e A141073 1 60 35 17 8 4 2 1, so a(7) = 17 and a(8) = 8.
%e A141073 1 116 72 35 17 8 4 2 1, so no central pair.
%e A141073 1 224 148 72 35 17 8 4 2 1, so a(9) = 35 and a(10) = 17.
%e A141073 1 432 303 149 72 35 17 8 4 2 1, so no central pair.
%e A141073 1 833 618 308 149 72 35 17 8 4 2 1, so a(11) = 72 and a(12) = 35.
%e A141073 ... [edited by _Petros Hadjicostas_, Jun 13 2019]
%t A141073 Rest[CoefficientList[Series[x*(x^8 + 3*x^6 + x^5 + 3*x^4 + x^3 + 3*x^2 + x + 1)/(1 - x^2 - x^4 - 2*x^6 -x^8),{x,0,44}],x]] (* _James C. McMahon_, Jul 16 2025 *)
%Y A141073 Cf. A129847, A140993, A140994, A140995, A140996, A140997, A140998, A141065, A141066, A141067, A141068, A141069, A141070, A141072.
%K A141073 nonn,easy
%O A141073 1,3
%A A141073 _Juri-Stepan Gerasimov_, Jul 16 2008
%E A141073 Partially edited by _N. J. A. Sloane_, Jul 18 2008
%E A141073 More terms from _Petros Hadjicostas_, Jun 13 2019