A060922 -id:A060922 - cp's OEIS frontend

A060924 Bisection of Lucas triangle A060922: odd-indexed members of column sequences of A060922 (not counting leading zeros).

3, 7, 6, 18, 38, 9, 47, 158, 120, 12, 123, 566, 753, 280, 15, 322, 1880, 3612, 2568, 545, 18, 843, 5964, 15040, 16220, 7043, 942, 21, 2207, 18342, 57366, 83780, 57560, 16536, 1498, 24, 5778, 55162, 206115

Offset: 0

Wolfdieter Lang, Apr 20 2001

Row sums give A060927. Column sequences (without leading zeros) are, for m=0..5: A005248(n+1), 2*A061171, A061172, 4*A061173, A061174, 2*A061175.

Companion triangle A060923 (even part).

			{3}; {7,6}; {18,38,9}; {47,158,120,12}; ...; pLo(2,x)= 2*(3+x-2*x^2).

Cf. A005248.

a(n, m) = A060922(2*n+1-m, m).
a(n, m) = ((2*n-m+1)*A060923(n, m-1) + 2*(2*(2*n+1)-3*m)*a(n-1, m-1) + 4*(2*n-m)*A060923(n-1, m-1))/(5*m), m >= n >= 1; a(n, 0) = A005248(n); otherwise 0.
G.f. for column m >= 0: x^m*pLo(m+1, x)/(1-3*x+x^2)^(m+1), where pLo(n, x) := Sum_{m=0..n+floor((n-1)/2)} A061187(n-1, m)*x^m are the row polynomials of the (signed) staircase A061187.
T(n,k) = 3*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k) + 2*T(n-2,k-1) - T(n-2,k-2) + 4*T(n-3,k-2), T(0,0) = 3, T(1,0) = 7, T(1,1) = 6, T(2,0) = 18, T(2,1) = 38, T(2,2) = 9, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Jan 21 2014

A060923 Bisection of Lucas triangle A060922: even-indexed members of column sequences of A060922 (not counting leading zeros).

Original entry on oeis.org

1, 4, 1, 11, 17, 1, 29, 80, 39, 1, 76, 303, 315, 70, 1, 199, 1039, 1687, 905, 110, 1, 521, 3364, 7470, 6666, 2120, 159, 1, 1364, 10493, 29634, 37580, 20965, 4311, 217, 1, 3571, 31885, 109421, 181074, 148545

Offset: 0

Wolfdieter Lang, Apr 20 2001

			Triangle begins:
  {1};
  {4,1};
  {11,17,1};
  {29,80,39,1};
  ...
pLe(2,x) = 1+11*x-11*x^2+4*x^3.

Row sums give A060926.
Column sequences (without leading zeros) are, for m=0..3: A002878, A060934-A060936.
Companion triangle A060924 (odd part).
Cf. A060922.

a(n, m) = A060922(2*n-m, m).
a(n, m) = ((2*(n-m)+1)*A060924(n-1, m-1) + 2*(4*n-3*m)*a(n-1, m-1) + 4*(2*n-m-1)*A060924(n-2, m-1))/(5*m), m >= n >= 1; a(n, 0)= A002878(n); else 0.
G.f. for column m >= 0: x^m*pLe(m+1, x)/(1-3*x+x^2)^(m+1), where pLe(n, x) := Sum_{m=0..n+floor(n/2)} A061186(n, m)*x^m are the row polynomials of the (signed) staircase A061186.
T(n,k) = 3*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k) + 2*T(n-2,k-1) - T(n-2,k-2) + 4*T(n-3,k-2), T(0,0) = 1, T(1,0) = 4, T(1,1) = 1, T(2,0) = 11, T(2,1) = 17, T(2,2) = 1, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Jan 21 2014

A060925 a(n) = 2a(n-1) + 3a(n-2), a(0) = 1, a(1) = 4.

Original entry on oeis.org

1, 4, 11, 34, 101, 304, 911, 2734, 8201, 24604, 73811, 221434, 664301, 1992904, 5978711, 17936134, 53808401, 161425204, 484275611, 1452826834, 4358480501, 13075441504, 39226324511, 117678973534, 353036920601

Offset: 0

Wolfdieter Lang, Apr 20 2001

Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=-1, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=charpoly(A,2). - Milan Janjic, Jan 26 2010

Harry J. Smith, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (2,3).

Cf. A000204, A003947, A060922.

[(5*3^n - (-1)^n)/4: n in [0..30]]; // G. C. Greubel, Apr 06 2021

A060925:= n-> (5*3^n - (-1)^n)/4; seq(A060925(n), n=0..30); # G. C. Greubel, Apr 06 2021

f[n_]:=3/(n+2);x=2;Table[x=f[x];Denominator[x],{n,0,5!}] (* Vladimir Joseph Stephan Orlovsky, Mar 11 2010 *)
LinearRecurrence[{2,3},{1,4},30] (* Harvey P. Dale, Mar 07 2014 *)

{a(n) = (5*3^n - (-1)^n)/4};
vector(30, n, a(n-1)) \\ Harry J. Smith, Jul 19 2009 \\ modified by G. C. Greubel, Apr 06 2021

[(5*3^n - (-1)^n)/4 for n in (0..30)] # G. C. Greubel, Apr 06 2021

Row sums of Lucas convolution triangle A060922.
Inverse binomial transform of A003947. - Philippe Deléham, Jul 23 2005
a(n) = Sum_{m=0..n} A060922(n, m) = Sum_{j=1..n} (a(j-1)*A000204(n-j+1)) + A000204(n+1).
a(n) = (5*3^n - (-1)^n)/4.
G.f.: (1+2*x)/(1 - 2*x - 3*x^2).
a(2n) = 3*a(2n-1) - 1; a(2n+1) = 3*a(2n) + 1. - Philippe Deléham, Jul 23 2005
Binomial transform is A003947. - Paul Barry, May 19 2003
E.g.f.: (-exp(-x) + 5*exp(3*x))/4. - G. C. Greubel, Apr 06 2021

Recurrence, now used as definition, from Philippe Deléham, Jul 23 2005

Entry revised by N. J. A. Sloane, Sep 10 2006

A004799 Self-convolution of Lucas numbers.

Original entry on oeis.org

1, 6, 17, 38, 80, 158, 303, 566, 1039, 1880, 3364, 5964, 10493, 18342, 31885, 55162, 95032, 163114, 279051, 475990, 809771, 1374316, 2327372, 3933528, 6636025, 11176518, 18794633, 31560206, 52925984, 88646390, 148303719, 247841654

Offset: 1

Clark Kimberling

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..1000
É. Czabarka, R. Flórez, and L. Junes, A Discrete Convolution on the Generalized Hosoya Triangle, Journal of Integer Sequences, 18 (2015), #15.1.6.
Index entries for linear recurrences with constant coefficients, signature (2, 1, -2, -1).

Cf. A000032, A000204, A060922.

[((5*n-4)*Lucas(n+1) + 2*Lucas(n))/5: n in [1..30]]; // G. C. Greubel, Dec 17 2017

a:= n-> (Matrix([[17, 6, 1, 0]]). Matrix(4, (i,j)-> if i=j-1 then 1 elif j=1 then [2, 1, -2, -1][i] else 0 fi)^n) [1,4]: seq (a(n), n=1..40); # Alois P. Heinz, Oct 28 2008

a[n_]:= ((5*n-4)*LucasL[n+1] + 2*LucasL[n])/5; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Nov 12 2015 *)

Vec(x*((1+2*x)/(1-x-x^2))^2 + O(x^50)) \\ Altug Alkan, Nov 12 2015

[((5*n-4)*lucas_number2(n+1,1,-1) + 2*lucas_number2(n,1,-1))/5 for n in (1..30)] # G. C. Greubel, Apr 07 2021

From Wolfdieter Lang, Apr 24 2001: (Start)
a(n) = A060922(n, 1) (second column of Lucas triangle).
a(n) = ((-4 + 5*n)*L(n+1) + 2*L(n))/5 with L(n) = A000032(n) = A000204(n), n >= 1.
G.f.: x*((1+2*x)/(1-x-x^2))^2. (End)

More terms from Alois P. Heinz, Oct 28 2008

A060929 Second convolution of Lucas numbers A000032(n+1), n >= 0.

Original entry on oeis.org

1, 9, 39, 120, 315, 753, 1687, 3612, 7470, 15040, 29634, 57366, 109421, 206115, 384105, 709152, 1298613, 2360943, 4264835, 7659870, 13686456, 24340184, 43102644, 76031100, 133636825, 234116493, 408900987

Offset: 0

Wolfdieter Lang, Apr 20 2001

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,0,-5,0,3,1).

Cf. A004799, A060922.

I:=[1,9,39,120,315,753]; [n le 6 select I[n] else 3*Self(n-1) - 5*Self(n-3) + 3*Self(n-5) + Self(n-6): n in [1..30]]; // G. C. Greubel, Dec 21 2017

CoefficientList[Series[((1 + 2*x)/(1 - x - x^2))^3, {x, 0, 50}], x] (* or *) LinearRecurrence[{3,0,-5,0,3,1}, {1,9,39,120,315,753}, 30] (* G. C. Greubel, Dec 21 2017 *)

x='x+O('x^30); Vec(((1+2*x)/(1-x-x^2))^3) \\ G. C. Greubel, Dec 21 2017

G.f.: ((1+2*x)/(1-x-x^2))^3.
a(n) = A060922(n+2, 2) (third column of Lucas triangle).
a(n) = (n+1)*((5*n+4)*L(n+2) + (5*n-2)*L(n+1))/10, n >= 1, with the Lucas numbers L(n)=A000032(n)=A000204(n), n >= 1.

A060930 Third convolution of Lucas numbers A000032(n+1), n >= 0.

Original entry on oeis.org

1, 12, 70, 280, 905, 2568, 6666, 16220, 37580, 83780, 181074, 381488, 786715, 1593160, 3176210, 6246732, 12139859, 23344760, 44471340, 84005640, 157483176, 293201912, 542468100, 997906400, 1826073525

Offset: 0

Wolfdieter Lang, Apr 20 2001

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-2,-8,5,8,-2,-4,-1).

Cf. A000032, A000204, A004799, A060922, A060929.

R:=PowerSeriesRing(Integers(), 40);
Coefficients(R!( ((1+2*x)/(1-x-x^2))^4 )); // G. C. Greubel, Apr 08 2021

Table[((25*n^3+90*n^2+95*n+6)*LucasL[n+4] -12*(5*n^2+10*n-3)*LucasL[n+2])/150, {n, 0, 40}] (* G. C. Greubel, Apr 08 2021 *)

def A060930_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( ((1+2*x)/(1-x-x^2))^4 ).list()
A060930_list(40) # G. C. Greubel, Apr 08 2021

G.f.: ((1+2*x)/(1-x-x^2))^4.
a(n) = A060922(n+3, 3) (fourth column of Lucas triangle).
a(n) = (2*(25*n^3 + 60*n^2 + 35*n +24)*L(n+2) + (25*n^3 + 90*n^2 + 95*n + 6)*L(n+1))/(3!*5^2), with the Lucas numbers L(n) = A000032(n).

A060931 Fourth convolution of Lucas numbers A000032(n+1), n >= 0.

Original entry on oeis.org

1, 15, 110, 545, 2120, 7043, 20965, 57560, 148545, 365045, 862224, 1970905, 4382820, 9520315, 20265665, 42385132, 87284120, 177293730, 355738710, 705980760, 1387213926, 2701362950, 5217448800, 10001654350

Offset: 0

Wolfdieter Lang, Apr 20 2001

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-5,-10,15,11,-15,-10,5,5,1).

Cf. A000032, A000204, A004799, A060922, A060929, A060930.

R:=PowerSeriesRing(Integers(), 40);
Coefficients(R!( ((1+2*x)/(1-x-x^2))^5 )); // G. C. Greubel, Apr 08 2021

Table[((n+1)/120)*((5*n^3+5*n^2-10*n+72)*LucasL[n+5] + 4*(5*n^2+10*n-24)*LucasL[n+ 4]), {n, 0, 40}] (* G. C. Greubel, Apr 08 2021 *)

def A060931_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( ((1+2*x)/(1-x-x^2))^5 ).list()
A060931_list(40) # G. C. Greubel, Apr 08 2021

a(n) = A060921(n+4, 4) (fifth column of Lucas triangle).
a(n) = (n+1)*( (15*n^3 +55*n^2 +50*n +24)*L(n+2) + 2*(5*n^3 +15*n^2 +10*n +24)*L(n+1))/5!, with the Lucas numbers L(n)=A000032(n).
G.f.: ((1+2*x)/(1-x-x^2))^5.

A060932 Fifth convolution of Lucas numbers A000032(n+1), n >= 0.

Original entry on oeis.org

1, 18, 159, 942, 4311, 16536, 55898, 171924, 491487, 1325546, 3409347, 8430246, 20164223, 46880424, 106350942, 236147828, 514553154, 1102562952, 2327442276, 4847463408, 9974081130, 20297335340

Offset: 0

Wolfdieter Lang, Apr 20 2001

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-9,-10,30,6,-41,-6,30,10,-9,-6,-1).

Cf. A000032, A000204, A004799, A060922, A060929, A060930, A060931.

R:=PowerSeriesRing(Integers(), 40);
Coefficients(R!( ((1+2*x)/(1-x-x^2))^6 )); // G. C. Greubel, Apr 08 2021

Table[((744+2990*n+2895*n^2+1925*n^3+825*n^4+125*n^5)*LucasL[n+2] +3*(256+390*n + 505*n^2+425*n^3+175*n^4+25*n^5)*LucasL[n+1])/(5^2*5!), {n,0,40}] (* G. C. Greubel, Apr 08 2021 *)

def A060932_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( ((1+2*x)/(1-x-x^2))^6 ).list()
A060932_list(40) # G. C. Greubel, Apr 08 2021

a(n) = A060922(n+5, 5) (sixth column of Lucas triangle).
G.f.: ((1+2*x)/(1-x-x^2))^6.
a(n) = ( 25*(125*n^5 +825*n^4 +1925*n^3 +2895*n^2 +2990*n +744)*L(n+2) +(1875*n^5 +13125*n^4 +31875*n^3 +37875*n^2 +29250*n +19200)*L(n+1))/(5!*5^4), with the Lucas numbers L(n)=A000032(n).

A061188 Triangle of coefficients of polynomials (rising powers) useful for convolutions of A000032(n+1), n >= 0 (Lucas numbers).

Original entry on oeis.org

0, 1, 5, 20, 45, 25, 240, 350, 600, 250, 3000, 9250, 13125, 8750, 1875, 93000, 373750, 361875, 240625, 103125, 15625, 3690000, 11077500, 12818750, 8531250, 4156250, 1181250, 125000, 116550000, 312037500

Offset: 0

Wolfdieter Lang, Apr 20 2001

The row polynomials pL1(n,x) := Sum_{m=0..n} a(n,m)*x^m and pL2(n,x) := Sum_{m=0..n} A061189(n,m)*x^m appear in the k-fold convolution of the Lucas numbers L(n+1) = A000204(n+1) = A000032(n+1), n >= 0, as follows: L(k; n) := A060922(n+k,k) = (pL1(k,n)*L(n+2)+pL2(k,n)*L(n+1))/(k!*5^k).

			Triangle begins:
  {0};
  {1,5};
  {20,45,25};
  {240,350,600,250};
  ...;
pL1(2,n) = 5*(4+9*n+5*n^2) = 5*(1+n)*(4+5*n).

Cf. A061189(n, m) (companion triangle), A060922(n, m) (Lucas convolution triangle).

A061189 Triangle of coefficients of polynomials (rising powers) useful for convolutions of A000204(n+1), n >= 0 (Lucas numbers).

Original entry on oeis.org

1, 2, 0, -10, 15, 25, 30, 475, 450, 125, 6000, 8500, 6250, 5000, 1250, 96000, 146250, 189375, 159375, 65625, 9375, 180000, 5355000, 8881250, 5578125, 2515625, 721875, 78125, 44100000, 254700000, 341775000

Offset: 0

Wolfdieter Lang, Apr 20 2001

The row polynomials pL2(n,x) := Sum_{m=0..n} a(n,m)*x^m and pL1(n,x) := Sum_{m=0..n} A061188(n,m)*x^m appear in the k-fold convolution of the Lucas numbers L(n+1) = A000204(n+1) = A000032(n+1), n >= 0, as follows: L(k; n) := A060922(n+k,k) = (pL1(k,n)*L(n+2)+pL2(k,n)*L(n+1))/(k!*5^k).

			Triangle begins:
  {1};
  {2,0};
  {-10,15,25};
  {30,475,450,125};
  ...;
pL2(2,n) = 5*(-2+3*n+5*n^2) = 5*(1+n)*(-2+5*n).
L(2; n) := A060922(n+2,2) = A060929(n) = (1+n)*((4+5*n)*L(n+2)+(-2+5*n)*L(n+1))/(2*5).

Cf. A061188(n, m) (companion triangle), A060922(n, m) (Lucas convolution triangle).

cp's OEIS Frontend

A060924 Bisection of Lucas triangle A060922: odd-indexed members of column sequences of A060922 (not counting leading zeros).

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A060923 Bisection of Lucas triangle A060922: even-indexed members of column sequences of A060922 (not counting leading zeros).

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A060925 a(n) = 2*a(n-1) + 3*a(n-2), a(0) = 1, a(1) = 4.

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A004799 Self-convolution of Lucas numbers.

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A060929 Second convolution of Lucas numbers A000032(n+1), n >= 0.

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A060930 Third convolution of Lucas numbers A000032(n+1), n >= 0.

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A060931 Fourth convolution of Lucas numbers A000032(n+1), n >= 0.

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A060932 Fifth convolution of Lucas numbers A000032(n+1), n >= 0.

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A060925 a(n) = 2a(n-1) + 3a(n-2), a(0) = 1, a(1) = 4.