A255495 -id:A255495 - cp's OEIS frontend

A255494 Triangle read by rows: coefficients of numerator of generating functions for powers of Pell numbers.

1, 1, 1, 1, 4, 1, 1, 13, 13, 1, 1, 38, 130, 38, 1, 1, 105, 1106, 1106, 105, 1, 1, 280, 8575, 26544, 8575, 280, 1, 1, 729, 62475, 567203, 567203, 62475, 729, 1, 1, 1866, 435576, 11179686, 32897774, 11179686, 435576, 1866, 1, 1, 4717, 2939208, 207768576, 1736613466, 1736613466, 207768576, 2939208, 4717, 1

Offset: 0

N. J. A. Sloane, Mar 06 2015

Note that Table 8 by Falcon should be labeled with the powers n (not r) and that the labels are off by 1. - R. J. Mathar, Jun 14 2015

			Triangle begins:
  1;
  1,    1; # see A079291
  1,    4,      1; # see A110272
  1,   13,     13,        1;
  1,   38,    130,       38,        1;
  1,  105,   1106,     1106,      105,        1;
  1,  280,   8575,    26544,     8575,      280,      1;
  1,  729,  62475,   567203,   567203,    62475,    729,    1;
  1, 1866, 435576, 11179686, 32897774, 11179686, 435576, 1866, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
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.

Cf. A000129, A079291, A094706, A110272.

Diagonals: A094706, A255495, A255496, A255497, A255498.

P:= func< n | Round(((1 + Sqrt(2))^n - (1 - Sqrt(2))^n)/(2*Sqrt(2))) >;
function T(n,k)
  if k eq 0 or k eq n then return 1;
  else return P(n-k+1)*T(n-1,k-1) + P(k+1)*T(n-1,k);
  end if; return T;
end function;
[T(n,k): k in [0..n], n in [0..12]];

T[n_, k_]:= T[n,k]= If[k==0 || k==n, 1, Fibonacci[n-k+1, 2]*T[n-1, k-1] + Fibonacci[k+1, 2]*T[n-1, k]]; Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Sep 19 2021 *)

@CachedFunction
def P(n): return lucas_number1(n, 2, -1)
def T(n,k): return 1 if (k==0 or k==n) else P(n-k+1)*T(n-1, k-1) + P(k+1)*T(n-1, k)
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Sep 19 2021

From G. C. Greubel, Sep 19 2021: (Start)
T(n, k) = P(n-k+1)*T(n-1, k-1) + P(k+1)*T(n-1, k), where T(n, 0) = T(n, n) = 1 and P(n) = A000129(n).
T(n, k) = T(n, n-k).
T(n, 1) = A094706(n).
T(n, 2) = A255495(n-2).
T(n, 3) = A255496(n-3).
T(n, 4) = A255497(n-4).
T(n, 5) = A255498(n-5). (End)

A255496 3rd diagonal of triangle in A255494.

Original entry on oeis.org

1, 38, 1106, 26544, 567203, 11179686, 207768576, 3692419776, 63361188037, 1057109514902, 17235551954894, 275697361933728, 4339725043253447, 67384965236252310, 1034147721558836220, 15711425790758327952, 236612932874975360809, 3536182524466029241958, 52494462902614684280330

Offset: 0

N. J. A. Sloane, Mar 06 2015

nonn

G. C. Greubel, Table of n, a(n) for n = 0..850
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 (44,-649,2770,11885,-65240,-215431,-67286,139956,-23560,-2400).

Cf. A000129, A002203, A255494, A255495.

a[n_]:= (12)^(n+4) -(-2)^(n+1) -2^n*LucasL[2*n+9, 2] -5^(n+4)*Fibonacci[n+5, 2] +(1/10)*Fibonacci[n+4, 2]*(Fibonacci[n+4, 2]^2 +(-1)^n);
Table[a[n], {n, 0, 30}] (* G. C. Greubel, Sep 20 2021 *)

def P(n): return lucas_number1(n, 2, -1)
def Q(n): return lucas_number2(n, 2, -1)
def a(n): return (12)^(n+4) - (-2)^(n+1) - 2^n*Q(2*n+9) - 5^(n+4)*P(n+5) + (1/10)*P(n+4)*(P(n+4)^2 + (-1)^n)
[a(n) for n in (0..30)] # G. C. Greubel, Sep 20 2021

From G. C. Greubel, Sep 20 2021: (Start)
a(n) = 12*a(n-1) + P(n+1)*A255495(n), where P(n) = A000129(n).
a(n) = (12)^(n+4) - (-2)^(n+1) - 2^n*Q(2*n+9) - 5^(n+4)*P(n+5) + (1/10)*P(n+4)*(P(n+4)^2 + (-1)^n), where P(n) = A000129(n), Q(n) = A002203(n).
G.f.: (1 -6*x +83*x^2 -228*x^3 -84*x^4 -200*x^5)/((1+2*x)*(1-12*x)*(1 +2*x -x^2)*(1 -10*x -25*x^2)*(1 -12*x +4*x^2)*(1 -14*x -x^2)). (End)

3 more terms. - R. J. Mathar, Jun 14 2015

Terms a(12) onward added by G. C. Greubel, Sep 20 2021

A255497 4th diagonal of triangle in A255494.

Original entry on oeis.org

1, 105, 8575, 567203, 32897774, 1736613466, 85474679858, 3985272984490, 177983686766655, 7675333342669951, 321533970710475033, 13145650587005246037, 526435406695455725140, 20710119055883150135480, 802278112017623387734420, 30663507276425403310594244, 1158197029073059563909854477

Offset: 0

N. J. A. Sloane, Mar 06 2015

nonn

G. C. Greubel, Table of n, a(n) for n = 0..600
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 (131, -6288, 119160,-72338,-20691314,43119120,1745477304, 5765440363,-4766158745, -22941732072,25990149864,-2819340784,-2805312400,-199344000,8352000).

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

A255496[n_]:= (12)^(n+4) -(-2)^(n+1) -2^n*LucasL[2*n+9, 2] -5^(n+4)*Fibonacci[n+5, 2] +(1/10)*Fibonacci[n+4, 2]*(Fibonacci[n+4, 2]^2 +(-1)^n);
a[n_]:= a[n]= If[n<2, (105)^n, 29*a[n-1] + Fibonacci[n+1,2]*A255496[n]];
Table[a[n], {n,0,30}] (* G. C. Greubel, Sep 20 2021 *)

def P(n): return lucas_number1(n, 2, -1)
def Q(n): return lucas_number2(n, 2, -1)
def a(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) )
[a(n) for n in (0..30)] # G. C. Greubel, Sep 20 2021

From G. C. Greubel, Sep 20 2021: (Start)
a(n) = 29*a(n-1) + P(n+1)*A255496(n).
a(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) ), where P(n) = A000129(n) and Q(n) = A002203(n).
G.f.: (1 -26*x +1108*x^2 -15042*x^3 +74319*x^4 +67340*x^5 +1376444*x^6 +2010720*x^7 -323920*x^8 +288000*x^9)/((1-x)*(1+5*x)*(1-29*x)*(1 +4*x -4*x^2)*(1 +6*x +x^2)*(1 -24*x -144*x^2)*(1 -28*x -4*x^2)*(1 -30*x +25*x^2)*(1 -34*x +x^2)).
(End)

a(8)-a(10) from R. J. Mathar, Jun 14 2015

Terms a(11) onward added by G. C. Greubel, Sep 20 2021

A255498 5th diagonal of triangle in A255494.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Mar 06 2015

nonn

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).

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

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 *)

@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

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)

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

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A255494 Triangle read by rows: coefficients of numerator of generating functions for powers of Pell numbers.

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A255496 3rd diagonal of triangle in A255494.

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A255497 4th diagonal of triangle in A255494.

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A255498 5th diagonal of triangle in A255494.

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