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A379210 List of integers of the form (N^2 - 4)/15.

%I A379210 #18 Dec 24 2024 13:48:27
%S A379210 0,3,4,11,19,32,35,52,68,91,96,123,147,180,187,224,256,299,308,355,
%T A379210 395,448,459,516,564,627,640,707,763,836,851,928,992,1075,1092,1179,
%U A379210 1251,1344,1363,1460,1540,1643,1664,1771,1859,1972,1995,2112,2208,2331,2356,2483
%N A379210 List of integers of the form (N^2 - 4)/15.
%C A379210 Compare with A204221.
%C A379210 The sequence terms occur as exponents in the expansion of Sum_{n >= 1} x^(n*(n-1)) * Product_{k >= 2*n} 1 - x^k = 1 - x^3 - x^4 + x^11 + x^19 - x^32 - x^35 + + - - ....
%C A379210 |A379212(n)| is the characteristic function of the numbers in this sequence.
%H A379210 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,2,-2,0,0,-1,1).
%F A379210 a(n) = (1/32)*(30*n^2 - 30*n + 1) + (-1)^(n)*(5/32)*(2*n - 1) - (-1)^(n*(n+1)/2)*(1/16)*(6*n - 3 + (-1)^n).
%F A379210 a(n) is quasi-periodic in n: for n >= 0,
%F A379210 a(4*n+1) = 15*n^2 - 26*n + 11; a(4*n+2) = 15*n^2 - 16*n + 4;
%F A379210 a(4*n+3) = 15*n^2 - 14*n + 3; a(4*n+4) = 15*n^2 - 4*n.
%F A379210 a(1-n) = a(n).
%F A379210 15*a(n) + 4 = A379211(n)^2.
%F A379210 G.f: x*(3*x^6 + x^5 + 7*x^4 + 8*x^3 + 7*x^2 + x + 3)/((1 + x)^2*(1 - x)^3*(1 + x^2)^2).
%F A379210 E.g.f.: ((15*x^2 + 35*x - 2)*cosh(x) + 2*(cos(x) + 3*x*cos(x) + 2*sin(x) - 3*x*sin(x)) + (15*x^2 + 25*x + 3)*sinh(x))/16. - _Stefano Spezia_, Dec 23 2024
%p A379210 A379210 := proc(q) local n;
%p A379210 for n from 0 to q do
%p A379210  if type(sqrt(15*n+4), integer) then print(n);
%p A379210 fi; od; end:
%p A379210 A379210(2500);
%t A379210 LinearRecurrence[{1,0,0,2,-2,0,0,-1,1},{0, 3, 4, 11, 19, 32, 35, 52, 68},52] (* _James C. McMahon_, Dec 24 2024 *)
%Y A379210 Cf. A204220, A204221, A204542, A379211 (square roots of 15*a(n) + 4), A379212.
%K A379210 nonn,easy
%O A379210 1,2
%A A379210 _Peter Bala_, Dec 18 2024