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A103625 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). 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

j(n) = sqrt(48*a(n)^2 + 48*a(n) + 1); 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

LINKS

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

FORMULA

G.f.: 2*(x^2+x+1)/(1-x-14*x^2+14*x^3+x^4-x^5). - Ralf Stephan, May 18 2007

MATHEMATICA

a[1]=0; a[2]=0; a[3]=2; a[n_]:=(6+14a[n-2]+2Sqrt[1+48a[n-2]+48a[n-2]^2])/2; Table[a[i], {i, 1, 20}] (* Herbert Kociemba, May 12 2008 *)

Join[{0, 0}, CoefficientList[Series[2*(x^2+x+1)/(1-x-14*x^2+14*x^3+x^4-x^5), {x, 0, 30}], x]] (* G. C. Greubel, Jul 15 2018 *)

PROG

(PARI) x='x+O('x^30); concat([0, 0], Vec(2*(x^2+x+1)/(1-x-14*x^2+14*x^3 + x^4-x^5))) \\ G. C. Greubel, Jul 15 2018

(MAGMA) m:=30; R<x>:=PowerSeriesRing(Integers(), m); [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

CROSSREFS

Cf. A053141, A103200, A103737.

Sequence in context: A178811 A099433 A051225 * A006989 A236399 A132529

Adjacent sequences:  A103622 A103623 A103624 * A103626 A103627 A103628

KEYWORD

uned,nonn

AUTHOR

Pierre CAMI, Mar 29 2005

EXTENSIONS

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

STATUS

approved

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Last modified March 20 20:14 EDT 2019. Contains 321352 sequences. (Running on oeis4.)