A255498 - OEIS

%I #13 Sep 23 2021 02:25:51

%S 1,280,62475,11179686,1736613466,243125885240,31464032862802,

%T 3828473678068060,443307088929919375,49283438913963499728,

%U 5295767249826282145413,552902424623732460251730,56318224867097916236530640,5615280578269206770801490160,549533929275081475149009571700

%N 5th diagonal of triangle in A255494.

%H G. C. Greubel, <a href="/A255498/b255498.txt">Table of n, a(n) for n = 0..500</a>

%H S. Falcon, <a href="http://saspublisher.com/wp-content/uploads/2014/06/SJET24C669-675.pdf">On The Generating Functions of the Powers of the K-Fibonacci Numbers</a>, Scholars Journal of Engineering and Technology (SJET), 2014; 2 (4C):669-675.

%H <a href="/index/Rec#order_21">Index entries for linear recurrences with constant coefficients</a>, signature (376, -54802, 3630508, -75022815, -2846082932, 114238747024, 1577306027464, -52069433611135, -1016200021352656, -3020413156112394, 29965893152789468, 72435932210073135, -546365140007154292, 650692815293657132, 267744542455319216, -440297864251362544, -214251046924716480, -9998773345956800, 3992836965024000, -117063803520000, -2034547200000).

%F From _G. C. Greubel_, Sep 22 2021: (Start)

%F a(n) = 70*a(n-1) + A000129(n+1)*A255497(n), a(0) = 1, a(1) = 280.

%F a(n) = (1/222720)*(435*2^(n+7) + 2320*(-12)^(n+7) - 222720*(70)^(n+6) - 29*2^(n+6)*Q(4*n+22) + 1160*(12)^(n+7)*Q(2*n+13) + 87*(-2)^(n+8)*Q(2*n+11) +

%F   P(5*n+25) - 2784*5^(n+6)*P(3*n+18) + 29*(-1)^n*P(3*n+15) + 7680*(29)^(n+7)*P(n + 7) + 2784*(-5)^(n+7)*P(n+6) - 174*P(n+5)), where P = A000129, Q(n) = A002203.

%F G.f.: (1 -96*x +11997*x^2 -596862*x^3 +15287055*x^4 -135141972*x^5 +366556867*x^6 -30606125134*x^7 - 254125754944*x^8 -657125309064*x^9 +376990806976*x^10 -2048614425760*x^11 +1171618742400*x^12 +77172576000*x^13 +29064960000*x^14)/((1-2*x)*(1+12*x)*(1-70*x)*(1 -2*x -x^2)*(1 +10*x -25*x^2)*(1 +12*x +4*x^2)*(1 +14*x -x^2)*(1 -58*x -841*x^2)*(1 -68*x +4*x^2)*(1 -70*x -25*x^2)*(1 -72*x +144*x^2)*(1 -82*x -x^2)). (End)

%t P[n_]:= Fibonacci[n,2]; Q[n_]:= LucasL[n,2];

%t A255497[n_]:= (1/7680)*(7680*(29)^(n+5) -192*(-5)^(n+6) -30 +Q[4*n+18] -96*5^(n+6)*Q[2*n+11] +12*(-1)^n*Q[2*n+9] +3*2^(n+10)*P[3*n+15] -640*(12)^(n+6)*P[n+6] -15*(-2)^(n+10)*P[n+5]);

%t a[n_]:= a[n]= If[n<2, (280)^n, 70*a[n-1] +P[n+1]*A255497[n]];

%t Table[a[n], {n, 0, 30}] (* _G. C. Greubel_, Sep 22 2021 *)

%o (Sage)

%o @CachedFunction

%o def P(n): return lucas_number1(n, 2, -1)

%o def Q(n): return lucas_number2(n, 2, -1)

%o def A255497(n): return (1/7680)*( 7680*(29)^(n+5) -192*(-5)^(n+6) -30 + Q(4*n+18) -96*5^(n+6)*Q(2*n+11) +12*(-1)^n*Q(2*n+9) +3*2^(n+10)*P(3*n+15) -640*(12)^(n+6)*P(n+6) -15*(-2)^(n+10)*P(n+5) )

%o def a(n): return (280)^n if (n<2) else 70*a(n-1) + P(n+1)*A255497(n)

%o [a(n) for n in (0..30)] # _G. C. Greubel_, Sep 20 2021

%Y Cf. A000129, A002203, A255494, A255495, A255496, A255497.

%K nonn

%O 0,2

%A _N. J. A. Sloane_, Mar 06 2015

%E Terms a(7) onward added by _G. C. Greubel_, Sep 22 2021