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Row sums of triangle A131779.
3

%I #12 Aug 27 2021 12:23:05

%S 1,4,5,10,15,26,41,68,109,178,287,466,753,1220,1973,3194,5167,8362,

%T 13529,21892,35421,57314,92735,150050,242785,392836,635621,1028458,

%U 1664079,2692538,4356617,7049156,11405773,18454930,29860703,48315634,78176337,126491972

%N Row sums of triangle A131779.

%C a(n)/a(n-1) tends to phi; e.g., a(10)/a(9) = 178/109 = 1.633...

%H Andrew Howroyd, <a href="/A131780/b131780.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-1,-1)

%F From _Andrew Howroyd_, Sep 01 2018: (Start)

%F a(n) = 2*Fibonacci(n+1) - (1 - (-1)^n)/2.

%F a(n) = a(n-1) + 2*a(n-2) - a(n-3) - a(n-4) for n > 4.

%F G.f.: x*(1 + 3*x - x^2 - 2*x^3)/((1 - x)*(1 + x)*(1 - x - x^2)).

%F (End)

%e a(4) = 10 = sum of row 4 terms of triangle A131779: (3 + 1 + 5 + 1).

%t 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 (PARI) a(n) = 2*fibonacci(n+1) - (1 - (-1)^n)/2; \\ _Andrew Howroyd_, Sep 01 2018

%o (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 Row sums of A131779.

%K nonn,easy

%O 1,2

%A _Gary W. Adamson_, Jul 14 2007

%E Terms a(11) and beyond from _Andrew Howroyd_, Sep 01 2018