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A103625
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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.
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1
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0, 0, 2, 4, 34, 62, 480, 870, 6692, 12124, 93214, 168872, 1298310, 2352090, 18083132, 32760394, 251865544, 456293432, 3508034490, 6355347660, 48860617322, 88518573814, 680540608024, 1232904685742, 9478707895020, 17172147026580, 132021369922262, 239177153686384
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OFFSET
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1,3
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COMMENTS
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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, ...
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
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
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LINKS
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FORMULA
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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
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MATHEMATICA
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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 *)
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 *)
LinearRecurrence[{1, 14, -14, -1, 1}, {0, 0, 2, 4, 34}, 30] (* Harvey P. Dale, Jun 04 2021 *)
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PROG
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(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
(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
(SageMath)
@CachedFunction
def b(n): return chebyshev_U(n, -2) # A125905
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))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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