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%I #21 Mar 23 2024 20:22:52
%S 0,0,2,4,34,62,480,870,6692,12124,93214,168872,1298310,2352090,
%T 18083132,32760394,251865544,456293432,3508034490,6355347660,
%U 48860617322,88518573814,680540608024,1232904685742,9478707895020,17172147026580,132021369922262,239177153686384
%N a(n) = 3 + 7*a(n-2) + sqrt(1 + 48*a(n-2) + 48*a(n-2)^2), with a(1) = 0, a(2) = 0, a(3) = 2.
%C Define j(n) = sqrt(48*a(n)^2 + 48*a(n) + 1), then j(n) is prime for n=3, 4, 5, 6, 7, 25, 28, 32, 35, 48, 65, 66, 88, 96, 113, 119, 151, 155, 182, 220, 231, 316, 488, 531, 599, 722, 1049, 1176, ...
%C For n > 1, first member of the Diophantine pair (m,k) that satisfies 12*(m^2 + m) = k^2 + k; a(n)=m. - _Herbert Kociemba_, May 12 2008
%C Former name: Define a(1)=0, a(2)=0, a(3)=2, a(4)=4, a(5)=34, a(6)=62, a(7)=480, a(8)=870 such that from i=1 to 8: 48*a(i)^2 + 48*a(i) + 1 = j(i)^2 with j(1)=1, j(2)=1, j(3)=17, j(4)=31, j(5)=239, j(6)=433, j(7)=3329, j(8)=6031. Then a(n) = a(n-8) + 28*sqrt(48*(a(n-4)^2) + 48*a(n-4) + 1). - _G. C. Greubel_, Mar 22 2024
%H G. C. Greubel, <a href="/A103625/b103625.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,14,-14,-1,1).
%F G.f.: 2*x^3*(1+x+x^2)/((1-x)*(1-4*x+x^2)*(1+4*x+x^2)). - _Ralf Stephan_, May 18 2007
%F a(n) = (1/8)*(-16*[n=0] - 4 + 5*(-1)^n*(3*A125905(n) + 11*A125905(n-1)) + (5*A125905(n) + 19*A125905(n-1))), where A125905(n) = ChebyshevU(n, -2). - _G. C. Greubel_, Mar 22 2024
%t a[1]=0; a[2]=0; a[3]=2; a[n_]:=a[n]= 3+7a[n-2]+Sqrt[1+48a[n-2]+48a[n-2]^2]; Table[a[n],{n,1,20}] (* _Herbert Kociemba_, May 12 2008 *)
%t Rest@CoefficientList[Series[2*x^3*(1+x+x^2)/(1-x-14*x^2+14*x^3+x^4-x^5), {x,0,30}], x] (* _G. C. Greubel_, Jul 15 2018 *)
%t LinearRecurrence[{1,14,-14,-1,1},{0,0,2,4,34},30] (* _Harvey P. Dale_, Jun 04 2021 *)
%o (PARI) my(x='x+O('x^30)); concat([0,0], Vec(2*x^3*(1+x+x^2)/(1-x-14*x^2+14*x^3 + x^4-x^5))) \\ _G. C. Greubel_, Jul 15 2018
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 30); [0,0] cat Coefficients(R!(2*(x^2+x+1)/(1-x-14*x^2+14*x^3+x^4-x^5))); // _G. C. Greubel_, Jul 15 2018
%o (SageMath)
%o @CachedFunction
%o def b(n): return chebyshev_U(n, -2) # A125905
%o def A103625(n): return (1/8)*(-16*int(n==0) -4 +5*(-1)^n*(3*b(n) +11*b(n-1)) +5*b(n) +19*b(n-1))
%o [A103625(n) for n in range(1,41)] # _G. C. Greubel_, Mar 22 2024
%Y Cf. A053141, A103200, A103737, A125905.
%K nonn
%O 1,3
%A _Pierre CAMI_, Mar 29 2005
%E Terms a(17) onward added by _G. C. Greubel_, Jul 15 2018
%E Edited by _G. C. Greubel_, Mar 22 2024
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