A255498 5th diagonal of triangle in A255494.

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

Offset

0,2

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).

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 +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)

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A218411 A272715 A282439 * A091034 A294165 A259079

Adjacent sequences: A255495 A255496 A255497 * A255499 A255500 A255501

Keyword

nonn

Author

N. J. A. Sloane, Mar 06 2015

Extensions

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

Status

approved