A255498 5th diagonal of triangle in A255494.

1, 280, 62475, 11179686, 1736613466, 243125885240, 31464032862802, 3828473678068060, 443307088929919375, 49283438913963499728, 5295767249826282145413, 552902424623732460251730, 56318224867097916236530640, 5615280578269206770801490160, 549533929275081475149009571700

Offset: 0

Author

N. J. A. Sloane, Mar 06 2015

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..500
• S. Falcon, On The Generating Functions of the Powers of the K-Fibonacci Numbers, Scholars Journal of Engineering and Technology (SJET), 2014; 2 (4C):669-675.
• Index entries for linear recurrences with constant coefficients, signature (376, -54802, 3630508, -75022815, -2846082932, 114238747024, 1577306027464, -52069433611135, -1016200021352656, -3020413156112394, 29965893152789468, 72435932210073135, -546365140007154292, 650692815293657132, 267744542455319216, -440297864251362544, -214251046924716480, -9998773345956800, 3992836965024000, -117063803520000, -2034547200000).

Crossrefs

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

Programs

Mathematica

P[n_]:= Fibonacci[n,2]; Q[n_]:= LucasL[n,2];
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]);
a[n_]:= a[n]= If[n<2, (280)^n, 70*a[n-1] +P[n+1]*A255497[n]];
Table[a[n], {n, 0, 30}] (* G. C. Greubel, Sep 22 2021 *)

Sage

@CachedFunction
def P(n): return lucas_number1(n, 2, -1)
def Q(n): return lucas_number2(n, 2, -1)
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) )
def a(n): return (280)^n if (n<2) else 70*a(n-1) + P(n+1)*A255497(n)
[a(n) for n in (0..30)] # G. C. Greubel, Sep 20 2021

Formula

From G. C. Greubel, Sep 22 2021: (Start)

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

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) +

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.

G.f.: (1 -96*x +11997*x^2 -596862*x^3 [phone]*x^4 -135141972*x^5 [phone]*x^6 -30606125134*x^7 - 254125754944*x^8 -657125309064*x^9 [phone]*x^10 -2048614425760*x^11 [phone]*x^12 [phone]*x^13 [phone]*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)

Extensions

Terms a(7) onward added by G. C. Greubel, Sep 22 2021