A255494 -id:A255494 - cp's OEIS frontend

A255495 2nd diagonal of triangle in A255494.

1, 13, 130, 1106, 8575, 62475, 435576, 2939208, 19342285, 124800361, 792586270, 4969028750, 30822650251, 189500937303, 1156406300340, 7012380492516, 42294614785465, 253926386816725, 1518506730836026, 9050029200532298, 53778595325886295, 318762380704793571, 1885254096749834160

Offset: 0

N. J. A. Sloane, Mar 06 2015

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
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 (14,-56,14,189,84,-20).

Cf. A000129, A094706, A255494.

I:=[1,13,130,1106,8575,62475]; [n le 6 select I[n] else 14*Self(n-1) - 56*Self(n-2) +14*Self(n-3) +189*Self(n-4) + 84*Self(n-5) -20*Self(n-6): n in [1..31]]; // G. C. Greubel, Sep 20 2021

a[n_]:= (1/2)*(Fibonacci[n+3, 2]*Fibonacci[n+4, 2] + 2^(n+4)*Fibonacci[n+4, 2] - 2*5^(n+3));
Table[a[n], {n,0,30}] (* G. C. Greubel, Sep 20 2021 *)

def P(n): return lucas_number1(n, 2, -1)
def A255495(n): return (1/2)*(P(n+3)*P(n+4) + 2^(n+4)*P(n+4) - 2*5^(n+3))
[A255495(n) for n in (0..30)] # G. C. Greubel, Sep 20 2021

G.f.: (1 -x +4*x^2)/((1+x)*(1-5*x)*(1-6*x+x^2)*(1-4*x-4*x^2)). - R. J. Mathar, Jun 14 2015
From G. C. Greubel, Sep 20 2021: (Start)
a(n) = (1/2)*(P(n+3)*P(n+4) + 2^(n+4)*P(n+4) - 2*5^(n+3)), where P(n) = A000129(n).
a(n) = 5*a(n-1) + P(n+1)*(P(n+3) - 2^(n+2)) = 5*a(n) + P(n+1)*A094706(n+1). (End)

Terms a(13) onward from G. C. Greubel, Sep 20 2021

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

A094706 Convolution of Pell(n) and 2^n.

Original entry on oeis.org

0, 1, 4, 13, 38, 105, 280, 729, 1866, 4717, 11812, 29365, 72590, 178641, 438064, 1071153, 2613138, 6362965, 15470140, 37565389, 91125206, 220864377, 534951112, 1294960905, 3133261530, 7578261181, 18323338324, 44292046693, 107041649438

Offset: 0

Paul Barry, May 21 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
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 (4,-3,-2).

Cf. A000079, A000129 (Pell numbers), A101164, A255494.

I:=[0, 1, 4]; [n le 3 select I[n] else 4*Self(n-1)-3*Self(n-2)-2*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 24 2012

LinearRecurrence[{4,-3, -2},{0,1,4},40] (* Vincenzo Librandi, Jun 24 2012 *)

[lucas_number1(n+2, 2, -1) - 2^(n+1) for n in (0..30)] # G. C. Greubel, Sep 16 2021

G.f.: x/((1-2*x-x^2)*(1-2*x)).
a(n) = Sum_{k=0..n} ((1+sqrt(2))^n - (1-sqrt(2))^n)/(2*sqrt(2))*2^(n-k).
a(n) = (1 + 3*sqrt(2)/4)*(1 + sqrt(2))^n + (1 - 3*sqrt(2)/4)*(1-sqrt(2))^n - 2^(n+1).
a(n) = 4*a(n-1) - 3*a(n-2) - 2*a(n-3).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k+1)*2^(n-2k-1);
a(n) = Sum_{k=0..n} binomial(k, n-k+1)*2^k*(1/2)^(n-k+1). - Paul Barry, Oct 07 2004
a(n) = sum of n-th row in A101164 = A000129(n) - A000079(n). - Reinhard Zumkeller, Dec 03 2004
a(n) = A000129(n+2) - 2^(n+1). - R. J. Mathar, Jan 29 2012
a(n) = 2*a(n-1) + A000129(n), with a(0) = 0, a(1) = 1. - G. C. Greubel, Sep 20 2021

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A255495 2nd diagonal of triangle in A255494.

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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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A094706 Convolution of Pell(n) and 2^n.

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