A394245 - OEIS

%I #30 May 19 2026 00:19:10

%S 1,10,100,110,121,231,441,672,1024,1696,2809,4505,7225,11730,19044,

%T 30774,49729,80503,130321,210824,341056,551880,893025,1444905,2337841,

%U 3782746,6120676,9903422,16024009,25927431,41951529,67878960,109830400,177709360,287539849,465249209,752788969,1218038178,1970827236

%N a(n) = a(n-1) + a(n-3) + a(n-4), a(0) = 1, a(1) = 10, a(2) = 100, a(3) = 110.

%H Tomás Guardia, Douglas Jiménez, and Alexander McCurdy, <a href="https://doi.org/10.2478/rmm-2024-0002">Fiboquadratic numbers and Rithmomachia</a>, Recreational Mathematics Magazine, Vol. 11, No. 18 (2024), pp. 17-29.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,1).

%F a(n) = (10*F(n/2) + F((n/2)-1))^2 if n is even and (10*F((n-1)/2) + F(((n-1)/2)-1))*(10*F(((n-1)/2)+1) + F((n-1)/2)) if n is odd where F(n) = Fibonacci(n).

%F G.f.: (1 + 9*x + 90*x^2 + 9*x^3)/(1 - x - x^3 - x^4). \\ _Hoang Xuan Thanh_, May 15 2026

%t LinearRecurrence[{1, 0, 1, 1}, {1, 10, 100, 110}, 40] (* _Amiram Eldar_, May 15 2026 *)

%o (PARI) Vec((1 + 9*x + 90*x^2 + 9*x^3)/(1 - x - x^3 - x^4) + O(x^40)) \\ _Hoang Xuan Thanh_, May 15 2026

%Y Cf. A000045, A006498, A006499.

%K nonn,easy

%O 0,2

%A _Alexander McCurdy_, May 14 2026