A103625 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.

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

Offset

1,3

Comments

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

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (1,14,-14,-1,1).

Formula

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

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

E.g.f.: (15*cosh(sqrt(3)*x)*(2*cosh(2*x) + sinh(2*x))/2 - sqrt(3)*(4*cosh(x) + sinh(x))*(cosh(x) + 4*sinh(x))*sinh(sqrt(3)*x) - 3*(4 + exp(x)))/6. - Stefano Spezia, Jun 02 2024

Mathematica

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 *)

Prog

(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))
[A103625(n) for n in range(1, 41)] # G. C. Greubel, Mar 22 2024

Crossrefs

Cf. A053141, A103200, A103737, A125905.

Sequence in context: A373344 A051225 A306582 * A006989 A236399 A132529

Adjacent sequences: A103622 A103623 A103624 * A103626 A103627 A103628

Keyword

nonn,easy

Author

Pierre CAMI, Mar 29 2005

Extensions

Terms a(17) onward added by G. C. Greubel, Jul 15 2018

Edited by G. C. Greubel, Mar 22 2024

Status

approved