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 A093953 #22 Aug 20 2025 05:13:53 %S A093953 1,10,63,344,1781,9030,45403,227524,1138641,5695250,28480343, %T A093953 142409904,712065901,3560362270,17801876883,89009515484 %N A093953 a(n) = rightmost term in M^n * [1,1,1], where M = a 3 X 3 matrix composed of the first 3 rows of A050166 (fill in the matrix with zeros): = [1 0 0 / 1 2 0 / 1 4 5]. %C A093953 A sequence relating to Catalan numbers. %C A093953 a(n)/a(n-1) tends to 5, a Catalan number. E.g. a(6)/a(5) = 45403/9030 = 4.9948... %C A093953 Generally, with M = an N X N matrix composed of rows of A050166 (along with zeros), M^n * [1,1,1...] generates terms [a, b, c, d...] such that sequences of which a,b,c,d...are members converge upon the Catalan numbers: 1, 2, 5, 14, 42, 132... %C A093953 Companion (M^n)[3,2] = 4*A016127(n), (M^n)[3,3] = 5^n = A000351(n), so a(n) = a(n-1) + 4*A016127(n-1) + 5^(n-1) for n>0. - Lambert Klasen (lambert.klasen(AT)gmx.net), Jan 30 2005 %H A093953 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (8,-17,10). %F A093953 Or simply with M=[1, 0, 0;1, 2, 0;1, 4, 5], a(n)=(M^n)[3, 1], (adds a leading 0 to sequence). - Lambert Klasen (lambert.klasen(AT)gmx.net), Jan 30 2005 %F A093953 G.f.: (1+2*x)/(1-8*x+17*x^2-10*x^3). - _Colin Barker_, Jan 31 2012 %e A093953 a(4) = 1781 since M^4 * [1,1,1] = [1, 31, 1781]. %o A093953 (PARI) M=[1,0,0;1,2,0;1,4,5];for(i=0,16,print1((M^i)[3,1]", ")) \\ Klasen %Y A093953 Cf. A050166, A016127, A000351. %K A093953 nonn,easy %O A093953 0,2 %A A093953 _Gary W. Adamson_, Apr 18 2004