This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A276600 #26 Jun 16 2025 00:00:06
%S A276600 0,2,3,7,15,20,42,88,117,245,513,682,1428,2990,3975,8323,17427,23168,
%T A276600 48510,101572,135033,282737,592005,787030,1647912,3450458,4587147,
%U A276600 9604735,20110743,26735852,55980498,117214000,155827965,326278253,683173257,908231938
%N A276600 Values of m such that m^2 + 6 is a triangular number (A000217).
%C A276600 2*a(n+2) gives the y members of all positive solutions (x(n), y(n)), proper and improper, of the Pell equation x^2 - 2*y^2 = 7^2, n >= 0. The corresponding x members are x(n) = A106525(n). - _Wolfdieter Lang_, Sep 29 2016
%H A276600 Colin Barker, <a href="/A276600/b276600.txt">Table of n, a(n) for n = 1..1000</a>
%H A276600 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,6,0,0,-1).
%F A276600 a(n) = 6*a(n-3) - a(n-6) for n>6.
%F A276600 G.f.: x^2*(2 + 3*x + 7*x^2 + 3*x^3 + 2*x^4)/(1 - 6*x^3 + x^6).
%F A276600 From _Wolfdieter Lang_, Sep 29 2016: (Start)
%F A276600 Trisection:
%F A276600 a(2+3*n) = 15*S(n-1,6) - 2*S(n-2,6) = A275794(n),
%F A276600 a(3+3*n) = 20*S(n-1,6) - 3*S(n-2,6) = A275796(n),
%F A276600 a(4+3*n) = 7*(6*S(n-1,6) - S(n-2,6)) = 7*A001109(n+1) for n >= 0, with the Chebyshev polynomials S(n, 6) = A001109(n+1), n >= -1, with S(-2, 6) = -1.
%F A276600 (End)
%e A276600 7 is in the sequence because 7^2 + 6 = 55, which is a triangular number.
%t A276600 LinearRecurrence[{0,0,6,0,0,-1}, {0,2,3,7,15,20}, 41] (* _G. C. Greubel_, Sep 15 2021 *)
%o A276600 (PARI) concat(0, Vec(x^2*(2+3*x+7*x^2+3*x^3+2*x^4)/(1-6*x^3+x^6) + O(x^40)))
%o A276600 (Magma) I:=[0,2,3,7,15,20]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..41]]; // _G. C. Greubel_, Sep 15 2021
%o A276600 (Sage)
%o A276600 def A276600_list(prec):
%o A276600 P.<x> = PowerSeriesRing(ZZ, prec)
%o A276600 return P( x^2*(2+3*x+7*x^2+3*x^3+2*x^4)/(1-6*x^3+x^6) ).list()
%o A276600 a=A276600_list(41); a[1:] # _G. C. Greubel_, Sep 15 2021
%Y A276600 Cf. A000217, A230044.
%Y A276600 Cf. A001109 (k=0), A106328 (k=1), A077241 (k=2), A276598 (k=3), A276599 (k=5), A276601 (k=9), A276602 (k=10), where k is the value added to n^2.
%Y A276600 Cf. A275794, A275796, A106525.
%K A276600 nonn,easy
%O A276600 1,2
%A A276600 _Colin Barker_, Sep 07 2016