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 A159683 The general form of the recurrences are the a(j), b(j) and n(j) solutions of the 2 equations problem: 3*n(j)+1=a(j)*a(j) and 5*n(j)+1=b(j)*b(j) with positive integer numbers. 3
 0, 16, 1008, 62496, 3873760, 240110640, 14882985936, 922505017408, 57180428093376, 3544264036771920, 219687189851765680, 13617061506772700256, 844038126230055650208, 52316746764756677612656, 3242794261288683956334480, 201000927453133648615125120 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS Colin Barker, Table of n, a(n) for n = 1..559 Index entries for linear recurrences with constant coefficients, signature (63,-63,1). FORMULA The a(j) recurrence is a(1)=1; a(2)=7; a(t+2)=8*a(t+1)-a(t) resulting in terms 1, 7, 55. 433, 3409. The b(j) recurrence is b(1)=1; b(2)=9; b(t+2)=8*b(t+1)-b(t) resulting in terms 1, 9, 71, 559, 4401. The n(j) recurrence is n(0)=n(1)=0; n(2)=16; n(t+3)=63*(n(t+2)-n(t+1))+n(t) resulting in terms 0, 0, 16, 1008, 62496 as listed above. From Colin Barker, Sep 25 2015: (Start) a(n) = 63*a(n-1)-63*a(n-2)+a(n-3) for n>3. G.f.: -16*x^2 / ((x-1)*(x^2-62*x+1)). (End) a(n) = (-8+(4+sqrt(15))*(31+8*sqrt(15))^(-n) -(-4+sqrt(15))*(31+8*sqrt(15))^n)/30. - Colin Barker, Mar 03 2016 MAPLE for a from 1 by 2 to 100000 do b:=sqrt((5*a*a-2)/3): if (trunc(b)=b) then n:=(a*a-1)/3: La:=[op(La), a]:Lb:=[op(Lb), b]:Ln:=[op(Ln), n]: end if: end do: MATHEMATICA CoefficientList[Series[16*x^2/((1-x)*(x^2-62*x+1)), {x, 0, 50}], x] (* G. C. Greubel, Jun 02 2018 *) PROG (PARI) concat(0, Vec(-16*x^2/((x-1)*(x^2-62*x+1)) + O(x^20))) \\ Colin Barker, Sep 25 2015 (MAGMA) m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(16*x^2/((1-x)*(x^2-62*x+1)))); // G. C. Greubel, Jun 02 2018 CROSSREFS Cf. A057080, A070997, A157456, A245031. Sequence in context: A211090 A160012 A070307 * A197104 A067490 A130036 Adjacent sequences:  A159680 A159681 A159682 * A159684 A159685 A159686 KEYWORD nonn,easy AUTHOR Paul Weisenhorn, Apr 19 2009 STATUS approved

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Last modified September 23 16:09 EDT 2020. Contains 337311 sequences. (Running on oeis4.)