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 A131780 #16 Jul 11 2025 08:47:12
%S A131780 1,4,5,10,15,26,41,68,109,178,287,466,753,1220,1973,3194,5167,8362,
%T A131780 13529,21892,35421,57314,92735,150050,242785,392836,635621,1028458,
%U A131780 1664079,2692538,4356617,7049156,11405773,18454930,29860703,48315634,78176337,126491972
%N A131780 Row sums of triangle A131779.
%C A131780 a(n)/a(n-1) tends to phi; e.g., a(10)/a(9) = 178/109 = 1.633...
%H A131780 Andrew Howroyd, <a href="/A131780/b131780.txt">Table of n, a(n) for n = 1..1000</a>
%H A131780 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-1,-1)
%F A131780 From _Andrew Howroyd_, Sep 01 2018: (Start)
%F A131780 a(n) = 2*Fibonacci(n+1) - (1 - (-1)^n)/2.
%F A131780 a(n) = a(n-1) + 2*a(n-2) - a(n-3) - a(n-4) for n > 4.
%F A131780 G.f.: x*(1 + 3*x - x^2 - 2*x^3)/((1 - x)*(1 + x)*(1 - x - x^2)). (End)
%F A131780 E.g.f.: 2*exp(x/2)*(5*cosh(sqrt(5)*x/2) + sqrt(5)*sinh(sqrt(5)*x/2))/5 - sinh(x) - 2. - _Stefano Spezia_, Jul 11 2025
%e A131780 a(4) = 10 = sum of row 4 terms of triangle A131779: (3 + 1 + 5 + 1).
%t A131780 LinearRecurrence[{1,2,-1,-1},{1,4,5,10},40] (* or *) CoefficientList[ Series[(1+3*x-x^2-2*x^3)/(1-x-2*x^2+x^3+x^4),{x,0,40}],x] (* _Harvey P. Dale_, Aug 27 2021 *)
%o A131780 (PARI) a(n) = 2*fibonacci(n+1) - (1 - (-1)^n)/2; \\ _Andrew Howroyd_, Sep 01 2018
%o A131780 (PARI) Vec((1 + 3*x - x^2 - 2*x^3)/((1 - x)*(1 + x)*(1 - x - x^2)) + O(x^40)) \\ _Andrew Howroyd_, Sep 01 2018
%Y A131780 Row sums of A131779.
%K A131780 nonn,easy
%O A131780 1,2
%A A131780 _Gary W. Adamson_, Jul 14 2007
%E A131780 Terms a(11) and beyond from _Andrew Howroyd_, Sep 01 2018