A213570 - OEIS

%I #27 Sep 08 2022 08:46:02

%S 1,9,37,110,272,598,1213,2323,4265,7588,13184,22500,37881,63125,

%T 104381,171602,280896,458330,746085,1212415,1967761,3190824,5170752,

%U 8375400,13561777,21954753,35536213,57512918,93073520,150613438

%N Antidiagonal sums of the convolution array A213566.

%H Clark Kimberling, <a href="/A213570/b213570.txt">Table of n, a(n) for n = 1..500</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (5,-9,6,1,-3,1).

%F a(n) = 5*a(n-1) - 9*a(n-2) + 6*a(n-3) + a(n-4) - 3*a(n-5) + a(n-6).

%F G.f.: f(x)/g(x), where f(x) = x*(1 + 4*x + x^2) and g(x) = (1 - x - x^2)*(1 - x)^4.

%F a(n) = Fibonacci(n+9) + Lucas(n+8) - n*(n^2 + 9*n + 39) - 81. - _Ehren Metcalfe_, Jul 10 2019

%F a(n) = Sum_{k=1..n} k^3 * Fibonacci(n+1-k). - _Greg Dresden_, Feb 27 2022

%t (* First program *)

%t b[n_]:= Fibonacci[n]; c[n_]:= n^2;

%t t[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]

%t TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]

%t Flatten[Table[t[n-k+1, k], {n, 12}, {k, n, 1, -1}]]

%t r[n_]:= Table[t[n, k], {k, 1, 60}]  (* A213566 *)

%t d = Table[t[n, n], {n, 1, 40}] (* A213567 *)

%t s[n_]:= Sum[t[i, n+1-i], {i, 1, n}]

%t s1 = Table[s[n], {n, 1, 50}] (* A213570 *)

%t (* Second program *)

%t Table[Fibonacci[n+9] + LucasL[n+8] -(n^3+9*n^2+39*n+81), {n,35}] (* _G. C. Greubel_, Jul 26 2019 *)

%o (PARI) vector(35,n, f=fibonacci; 2*f(n+9)+f(n+7) -(n^3+9*n^2+39*n+81)) \\ _G. C. Greubel_, Jul 26 2019

%o (Magma) [Fibonacci(n+9) +Lucas(n+8) -(n^3+9*n^2+39*n+81): n in [1..35]]; // _G. C. Greubel_, Jul 26 2019

%o (Sage) [fibonacci(n+9) +lucas_number2(n+8,1,-1) -(n^3+9*n^2+39*n+81) for n in (1..35)] # _G. C. Greubel_, Jul 26 2019

%o (GAP) List([1..35], n-> Fibonacci(n+9)+Lucas(1,-1,n+8)[2] -(n^3+9*n^2 +39*n+81)); # _G. C. Greubel_, Jul 26 2019

%Y Cf. A213566, A213500.

%K nonn,easy

%O 1,2

%A _Clark Kimberling_, Jun 19 2012