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Numbers n such that the octagonal number N(n) is equal to the sum of the octagonal numbers N(m) and N(m+1) for some m.
2

%I #7 Aug 25 2024 15:39:14

%S 1,43,521,49235,600849,56816763,693378841,65566494883,800158581281,

%T 75663678277835,923382309419049,87315819166126323,1065582384911000881,

%U 100762379654031498523,1229681148804985597241,116279698804933183168835,1419050980138568468214849

%N Numbers n such that the octagonal number N(n) is equal to the sum of the octagonal numbers N(m) and N(m+1) for some m.

%C Also nonnegative integers y in the solutions to 12*x^2-6*y^2+4*x+4*y+2 = 0, the corresponding values of x being A251895.

%H Colin Barker, <a href="/A251896/b251896.txt">Table of n, a(n) for n = 1..653</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,1154,-1154,-1,1).

%F a(n) = a(n-1)+1154*a(n-2)-1154*a(n-3)-a(n-4)+a(n-5).

%F G.f.: -x*(3*x^4+246*x^3-676*x^2+42*x+1) / ((x-1)*(x^2-34*x+1)*(x^2+34*x+1)).

%e 43 is in the sequence because N(43) = 5461 = 2640+2821 = N(30)+N(31).

%t LinearRecurrence[{1,1154,-1154,-1,1},{1,43,521,49235,600849},20] (* _Harvey P. Dale_, Aug 25 2024 *)

%o (PARI) Vec(-x*(3*x^4+246*x^3-676*x^2+42*x+1) / ((x-1)*(x^2-34*x+1)*(x^2+34*x+1)) + O(x^100))

%Y Cf. A000567, A251895.

%K nonn,easy

%O 1,2

%A _Colin Barker_, Dec 10 2014