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

Author

Pierre CAMI, Mar 29 2005

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A053141, A103200, A103737, A125905.

Programs

Magma

R:=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

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 *)

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

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

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

Extensions

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

Edited by G. C. Greubel, Mar 22 2024