A161595 The list of the A values in the common solutions to the 2 equations 15*k+1=A^2, 19*k+1=B^2.

1, 16, 271, 4591, 77776, 1317601, 22321441, 378146896, 6406175791, 108526841551, 1838550130576, 31146825378241, 527657481299521, 8939030356713616, 151435858582831951, 2565470565551429551, 43461563755791470416, 736281113282903567521, 12473317362053569177441

Offset: 1

Author

Paul Weisenhorn, Jun 14 2009

Keywords

• nonn
• easy

Comments

The 2 equations are equivalent to the Pell equation x^2- 285*y^2=1,

with x=(285*k+17)/2 and y=A*B/2, case C=15 in A160682.

Also: the first differences of A078366.

Positive values of x (or y) satisfying x^2 - 17xy + y^2 + 15 = 0. - Colin Barker, Feb 14 2014

Links

• J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014-2020.
• Index entries for linear recurrences with constant coefficients, signature (17,-1).

Crossrefs

Cf. A078366, A160682, A161599 (sequence of B), A161583 (sequence of k).

Cf. similar sequences listed in A238379.

Programs

Maple

t:=0: for a from 1 to 1000000 do b:=sqrt((19*a^2-4)/15):
if (trunc(b)=b) then t:=t+1: n:=(a^2-1)/15: print(t,a,b,n): end if: end do:

Mathematica

Rest[CoefficientList[Series[x (1-x)/(1-17x+x^2),{x,0,40}],x]] (* or *) LinearRecurrence[{17,-1},{1,16},20] (* Harvey P. Dale, Oct 12 2012 *)

PARI

Vec(x*(1-x)/(1-17*x+x^2) + O(x^100)) \\ Colin Barker, Feb 14 2014

Formula

a(t+2) = 17*a(t+1)-a(t).

a(t) = ((285+15*w)*((17+w)/2)^(t-1)+(285-15*w)*((17-w)/2)^(t-1))/570, where w=sqrt(285).

a(t) = ceiling of ((285+15*w)*((17+w)/2)^(t-1))/570.

G.f.: x*(1-x)/(1-17*x+x^2).

a(n) = 17*a(n-1)-a(n-2). - Colin Barker, Feb 14 2014

Extensions

Edited, extended by R. J. Mathar, Sep 02 2009