%I #26 Apr 25 2022 08:12:16

%S 0,1,1,11,12,32,44,85,129,223,352,584,936,1529,2465,4003,6468,10480,

%T 16948,27437,44385,71831,116216,188056,304272,492337,796609,1288955,

%U 2085564,3374528,5460092,8834629,14294721,23129359,37424080,60553448,97977528,158530985

%N Sums of NE-SW diagonals of triangle A172171.

%H Colin Barker, <a href="/A172173/b172173.txt">Table of n, a(n) for n = 0..1000</a>

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

%F For n=even: a(n) = a(n-1) + a(n-2); for n=odd: a(n) = a(n-1) + a(n-2) + 9 ; with a(0) = 0 and a(1) = 1.

%F From _Colin Barker_, Feb 18 2013: (Start)

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

%F G.f.: x*(1+8*x^2) / ((1-x)*(1+x)*(1-x-x^2)).

%F (End)

%F a(n) = (2^(-1-n)*(-45*((-2)^n+2^n) + (45-7*sqrt(5))*(1+sqrt(5))^n + (1-sqrt(5))^n*(45+7*sqrt(5)))) / 5. - _Colin Barker_, Jul 13 2017

%F a(n) = Fibonacci(n+1) + 8*Fibonacci(n-1) - 9*((1+(-1)^n)/2). - _G. C. Greubel_, Apr 25 2022

%t CoefficientList[Series[x*(1+8*x^2)/((1-x^2)*(1-x-x^2)), {x,0,50}], x] (* _G. C. Greubel_, Jul 13 2017 *)

%o (PARI) concat(0, Vec(x*(1+8*x^2)/((1-x)*(1+x)*(1-x-x^2)) + O(x^50))) \\ _Colin Barker_, Jul 13 2017

%o (Magma) [Lucas(n) +7*Fibonacci(n-1) -9*((n+1) mod 2): n in [0..50]]; // _G. C. Greubel_, Apr 25 2022

%o (Sage) [fibonacci(n+1) +8*fibonacci(n-1) -9*((n+1)%2) for n in (0..50)] # _G. C. Greubel_, Apr 25 2022

%Y Cf. A000032, A000045, A172171, A172172.

%K nonn,easy

%O 0,4

%A _Mark Dols_, Jan 28 2010

%E Offset corrected by _Colin Barker_, Feb 18 2013