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Numbers n such that the octagonal numbers N(n), N(n+1) and N(n+2) sum to another octagonal number.
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%I #12 May 30 2017 08:45:22

%S 278,752958,2034494038,5497202139518,14853438146485398,

%T 40133984374601407678,108442010926734857062358,

%U 293010273390053209181085438,791713650257912844472435792918,2139209989986607115711312331380798,5780144601230162168739121446955125078

%N Numbers n such that the octagonal numbers N(n), N(n+1) and N(n+2) sum to another octagonal number.

%C Also nonnegative integers x in the solutions to 18*x^2-6*y^2+24*x+4*y+18 = 0, the corresponding values of y being A251625.

%C It seems that the least significant digit of each term is 8.

%H Colin Barker, <a href="/A251624/b251624.txt">Table of n, a(n) for n = 1..291</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2703,-2703,1).

%F a(n) = 2703*a(n-1)-2703*a(n-2)+a(n-3).

%F G.f.: 2*x*(x^2-762*x-139) / ((x-1)*(x^2-2702*x+1)).

%F a(n) = 2*(-3 - (2*sqrt(3)+3)*(1351+780*sqrt(3))^(-n) + (2*sqrt(3)-3)*(1351+780*sqrt(3))^n) / 9. - _Colin Barker_, May 30 2017

%e 278 is in the sequence because N(278)+N(279)+N(280) = 231296+232965+234640 = 698901 = N(483).

%t LinearRecurrence[{2703,-2703,1},{278,752958,2034494038},20] (* _Harvey P. Dale_, Feb 14 2015 *)

%o (PARI) Vec(2*x*(x^2-762*x-139)/((x-1)*(x^2-2702*x+1)) + O(x^100))

%Y Cf. A000567, A251625.

%K nonn,easy

%O 1,1

%A _Colin Barker_, Dec 06 2014