A049310 -id:A049310 - cp's OEIS frontend

A193663 Q-residue of A049310 (triangle of coefficients of Fibonacci polynomials), where Q is the triangle given by t(n,k)=k+1 for 0<=k<=n. (See Comments.)

0, 1, 1, 9, 17, 80, 198, 748, 2107, 7236, 21680, 71279, 219879, 708436, 2215513, 7071210, 22256567, 70723367, 223272153, 708017329, 2238347440, 7091170416, 22433032016

Offset: 0

Clark Kimberling, Aug 02 2011

nonn

The definition of Q-residue is given at A193649.

Cf. A049310, A193649.

q[n_, k_] := k + 1;
r[0] = 1; r[k_] := Sum[q[k - 1, i] r[k - 1 - i], {i, 0, k - 1}];
f[n_, x_] := Fibonacci[n, x]; (* A049310 *)
p[n_, k_] := Coefficient[f[n, x], x, k];
v[n_] := Sum[p[n, k] r[n - k], {k, 0, n}]
Table[v[n], {n, 0, 22}]    (* A193663 *)
TableForm[Table[q[i, k], {i, 0, 4}, {k, 0, i}]]
Table[r[k], {k, 0, 8}]
TableForm[Table[p[n, k], {n, 0, 6}, {k, 0, n}]]

Conjecture: G.f.: x*(1-x+x^2) / ( 1-2*x-6*x^2+7*x^3+x^4 ). - R. J. Mathar, Feb 19 2015

A220665 Array of coefficients of powers of x^2 for (S(2*n+1,x)/x)^3, with Chebyshev's S polynomials A049310.

Original entry on oeis.org

1, -8, 12, -6, 1, 27, -108, 171, -136, 57, -12, 1, -64, 480, -1488, 2488, -2472, 1524, -588, 138, -18, 1, 125, -1500, 7575, -21200, 36690, -41700, 32211, -17184, 6330, -1580, 255, -24, 1, -216, 3780, -28098, 117323, -308688, 546864, -680474, 611019, -402264, 195444, -69894, 18153, -3328, 408, -30, 1

Offset: 0

Wolfdieter Lang, Dec 17 2012

The row lengths sequence of this array is 3*n+1 = A016777(n).
For the coefficient array of S(n,x)^3 see A219240. The present array is the odd part of the bisection of that one divided by x^3.
The row polynomials in powers of x^2 are (S(2*n+1,x)/x)^3 = sum(a(n,m)*x^(2*m), m=0..3*n), n >= 0. The o.g.f. for these row polynomials is GS3odd(x,z) = ((z+1)^2 +2*z*(x^2-3))/ (((z+1)^2-z*x^2)*((z+1)^2-z*x^2*(x^2-3)^2)). This is obtained from the odd part of the bisection of the o.g.f. for A219240.

			The array a(n,m) begins:
n\m  0    1     2     3      4     5     6    7    8  9
0:   1
1:  -8   12    -6     1
2:  27 -108   171  -136     57   -12     1
3: -64  480 -1488  2488  -2472  1524  -588  138  -18  1
...
Row n=4: [125 -1500, 7575, -21200, 36690, -41700, 32211, -17184, 6330, -1580, 255, -24, 1],
Row n=5: [-216, 3780, -28098, 117323, -308688, 546864, -680474, 611019, -402264, 195444, -69894, 18153, -3328, 408, -30, 1],
Row n=6: [343, -8232, 84378, -489608, 1809129, -4562292, 8219967, -10918992, 10927077, -8356272, 4923132, -2240256, 784840, -209580, 41853, -6048, 597, -36, 1],
Row n=1: (S(3,x)/x)^3 = -8 + 12*x^2 - 6*x^4 + 1*x^6, with Chebyshev's S polynomial.

Cf. A219240, A220666 (even part of the bisection).

a(n,m) = [x^m](S(2*n+1,x)/x)^3, n>=0, 0 <= m <= 3*n.

a(n,m) = [x^m]([z^n]GS3odd(x,z)) with GS3odd(x,z) the o.g.f. for the row polynomials in powers of x^2, given in a comment above.

A286717 a(n) is the number of zeros of the Chebyshev S(n, x) polynomial (A049310) in the open interval (-phi, +phi), with the golden section phi = (1 + sqrt(5))/2.

Original entry on oeis.org

0, 1, 2, 3, 2, 3, 4, 5, 6, 5, 6, 7, 8, 9, 8, 9, 10, 11, 12, 11, 12, 13, 14, 15, 14, 15, 16, 17, 18, 17, 18, 19, 20, 21, 20, 21, 22, 23, 24, 23, 24, 25, 26, 27, 26, 27, 28, 29, 30, 29, 30, 31, 32, 33, 32, 33, 34, 35, 36, 35, 36, 37, 38, 39, 38, 39, 40, 41, 42

Offset: 0

Wolfdieter Lang, May 13 2017

See a May 06 2017 comment on A049310 where these problems are considered which originated in a conjecture by Michel Lagneau (see A008611) on Fibonacci polynomials.

			a(4) = 2: S(4, x) = 1+x^4-3*x^2, and only two of the four zeros -phi, -1/phi, +1/phi, phi are in the open interval (-phi, +phi), the other two are at the borders.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Cf. A008611(n-1) (1), A285869 (sqrt(2)), A285872 (sqrt(3)).

m:=80; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(1+x+x^2-x^3+x^4)/((1-x)^2*(1+x+x^2+x^3+x^4)))); // G. C. Greubel, Mar 08 2018

CoefficientList[Series[x*(1+x+x^2-x^3+x^4)/((1-x)^2*(1+x+x^2+x^3+x^4)), {x, 0, 50}], x] (* G. C. Greubel, Mar 08 2018 *)
LinearRecurrence[{1,0,0,0,1,-1},{0,1,2,3,2,3},80] (* Harvey P. Dale, Aug 20 2020 *)

concat(0, Vec(x*(1 + x + x^2 - x^3 + x^4) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4)) + O(x^100))) \\ Colin Barker, May 18 2017

a(n) = 2*b(n) if n is even and 1 + 2*b(n) if n is odd with b(n) = floor(n/2) - floor((n+1)/6) = A286716(n). See the g.f. for {b(n)}_{n>=0} there.
From Colin Barker, May 18 2017: (Start)
G.f.: x*(1 + x + x^2 - x^3 + x^4) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4)).
a(n) = a(n-1) + a(n-5) - a(n-6) for n>5.
(End)

A302707 Number of factors of Chebyshev polynomial S(2*n+1, x) (A049310) over the integers. Factorization is into the minimal integer polynomials C (A187360).

Original entry on oeis.org

1, 2, 4, 3, 4, 6, 4, 4, 7, 6, 4, 8, 4, 6, 10, 5, 4, 10, 4, 8, 10, 6, 4, 10, 7, 6, 10, 8, 4, 14, 4, 6, 10, 6, 10, 13, 4, 6, 10, 10, 4, 14, 4, 8, 16, 6, 4, 12, 7, 10, 10, 8, 4, 14, 10, 10, 10, 6, 4, 18, 4, 6, 16, 7, 10, 14, 4, 8, 10, 14, 4, 16, 4, 6, 16, 8, 10, 14, 4, 12, 13

Offset: 0

Wolfdieter Lang, Apr 12 2018

For the factorization of the Chebyshev S polynomials (coefficients in A049310) with odd index into the minimal polynomials of {2*cos(Pi/k)}_{k>=1} (coefficients in A187360) see an Apr 12 2018 comment in A049310.
Note that factors -C(k, -x) may appear also and they come always together with C(k, x) (the minus signs are not counted as factors here). C(2, x) = x is always a factor.
For the number of factors of S(2*n, x) see 2*(tau(2*n+1) - 1) = 2*A095374(n).

			a(2) = 4 because S(5, x) = 3*x-4*x^3+x^5 = x*(-1 + x)*(1 + x)*(-3 + x^2) = C(2, x)*C(3, x)*(-C(3, -x))*C(6, x).
a(5) = 6 because S(11, x) = -6*x + 35*x^3 - 56*x^5 + 36*x^7 - 10*x^9 + x^11 = x*(-1 + x)*(1 + x)*(-2 + x^2)*(-3 + x^2)*(1 - 4*x^2 + x^4) = C(2, x)*C(3, x)*(-C(3, -x))*C(4, x)*C(6, x)*C(12, x).

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A000005, A001227, A001620, A049310, A095374, A187360.

a[n_] := DivisorSigma[0, 2*(n+1)] + DivisorSigma[0, (n+1)/2^IntegerExponent[n+1, 2]] - 2; Array[a, 100, 0] (* Amiram Eldar, Feb 03 2025 *)

A001227(n) = numdiv(n>>valuation(n,2));
A302707(n) = (A001227(1+n) + numdiv(2*(n+1)) - 2); \\ Antti Karttunen, Sep 30 2018

a(n) = tau_{odd}(n+1) + tau(2*(n+1)) - 2, n >= 0, with tau_{odd} = A001227 and tau = A000005.
G.f.: Sum_{k>=1} (x^(k-1)/(1-x^(2*k)) + x^(k-1)*(2+x^k)/(1-x^(2*k))) - 2/(1-x).
Sum_{k=1..n} a(k) ~ 2*n * (log(n) + 2*gamma + log(2)/2 - 2), where gamma is Euler's constant (A001620). - Amiram Eldar, Feb 03 2025

Typo in the first formula corrected by Antti Karttunen, Sep 30 2018

A309213 A007318 + A065941 - A049310.

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 2, 6, 5, 1, 1, 5, 12, 6, 1, 2, 3, 14, 17, 8, 1, 3, 7, 14, 24, 26, 9, 1, 2, 12, 27, 30, 45, 33, 11, 1, 1, 9, 45, 62, 70, 66, 45, 12, 1, 2, 5, 44, 111, 147, 120, 104, 54, 14, 1, 3, 11, 39, 128, 273, 273, 217, 140, 69, 15, 1, 2, 18, 65, 139, 366, 546, 518, 329, 200, 80, 17, 1

Offset: 0

Gary W. Adamson, Jul 04 2007

Row sums = 1, 3, 7, 14, 25, 45, 85, ... (This is probably a new sequence and should be added to the OEIS.) - N. J. A. Sloane, Aug 09 2019

			First few rows of the triangle are:
1,
2, 1,
3, 3, 1,
2, 6, 5, 1,
1, 5, 12, 6, 1,
2, 3, 14, 17, 8, 1,
3, 7, 14, 24, 26, 9, 1,
...

Jinyuan Wang, Table of n, a(n) for n = 0..5150

Cf. A007318, A065941, A049310, A117591, A131375, A168561.

T007318(n, k) = binomial(n, k);
T065941(n, k) = binomial(n - (k+1)\2, k\2);
T049310(n, k) = if ((n+k)%2, 0, (-1)^((n+k)/2 + k) * binomial((n+k)/2, k));
T(n, k) = T007318(n, k) + T065941(n, k) - T049310(n, k); \\ Michel Marcus, Apr 28 2014

A007318 + A065941 - A168561 as infinite lower triangular matrices.

The old definition of A131376 did not match the data, as Michel Marcus pointed out. The definition there has been corrected, keeping the old data. The present sequence uses the old definition with corrected data from Michel Marcus. - N. J. A. Sloane, Aug 09 2019

More terms from Jinyuan Wang, Aug 29 2019

A128932 Define the Fibonacci polynomials by F[1] = 1, F[2] = x; for n > 2, F[n] = xF[n-1] + F[n-2] (cf. A049310, A053119). Swamy's inequality implies that F[n] <= F[n]^2 <= G[n] = (x^2 + 1)^2(x^2 + 2)^(n-3) for n >= 3 and x >= 1. The sequence gives a triangle of coefficients of G[n] - F[n] read by rows.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, -2, 5, -1, 4, 0, 1, 3, 0, 9, 0, 12, 0, 6, 0, 1, 8, -3, 28, -4, 38, -1, 25, 0, 8, 0, 1, 15, 0, 58, 0, 99, 0, 87, 0, 41, 0, 10, 0, 1, 32, -4, 144, -10, 272, -6, 280, -1, 170, 0, 61, 0, 12, 0, 1, 63, 0, 310, 0, 673, 0, 825, 0, 619, 0, 292, 0, 85, 0, 14, 0, 1

Offset: 3

N. J. A. Sloane, Apr 28 2007

From Petros Hadjicostas, Jun 10 2020: (Start)
Swamy's (1966) inequality states that F[n]^2 <= G[n] for all real x and all integers n >= 3. Because F[n] >= 1 for all real x >= 1, we get F[n] <= G[n] for all integers n >= 3 and all real x >= 1.
Row n >= 3 of this irregular table gives the coefficients of the polynomial G[n] - F[n] (with exponents in increasing order). The degree of G[n] - F[n] is 2*n - 2, so row n >= 3 contains 2*n - 1 terms.
Guilfoyle (1967) notes that F[n] = det(A_n), where A_n is the (n-1) X (n-1) matrix [[x, -1, 0, 0, ..., 0, 0, 0], [1, x, -1, 0, ..., 0, 0, 0], [0, 1, x, -1, ..., 0, 0, 0], ..., [0, 0, 0, 0, ..., 1, x, -1], [0, 0, 0, 0, ..., 0, 1, x]], and Swamy's original inequality follows from Hadamard's inequality.
Koshy (2019) writes Swamy's original inequality in the form x^(n-3)*F[n]^2 <= F[3]^2*F[4]^(n-3) for x >= 1, and gives a counterpart inequality for Lucas polynomials. Notice, however, that the original form of Swamy's inequality is true for all real x. (End)

			Triangle T(n,k) (with rows n >= 3 and columns k = 0..2*n-2) begins:
   0,  0,  1,  0,  1;
   2, -2,  5, -1,  4,  0,  1;
   3,  0,  9,  0, 12,  0,  6, 0,  1;
   8, -3, 28, -4, 38, -1, 25, 0,  8, 0,  1;
  15,  0, 58,  0, 99,  0, 87, 0, 41, 0, 10, 0, 1;
  ...

Thomas Koshy, Fibonacci and Lucas numbers with Applications, Vol. 2, Wiley, 2019; see p. 33. [Vol. 1 was published in 2001.]
D. S. Mitrinovic, Analytic Inequalities, Springer-Verlag, 1970; p. 232, Sect. 3.3.38.

Richard Guilfoyle, Comment to the solution of Problem E1846, Amer. Math. Monthly, 74(5), 1967, 593. [It is pointed out that the inequality is a special case of Hadamard's inequality.]
M. N. S. Swamy, Problem E1846 proposed for solution, Amer. Math. Monthly, 73(1) (1966), 81.
M. N. S. Swamy and R. E. Giudici, Solution to Problem E1846, Amer. Math. Monthly, 74(5), 1967, 592-593.
M. N. S. Swamy and Joel Pitcain, Comment to Problem E1846, Amer. Math. Monthly, 75(3) (1968), 295. [It is pointed out that I^{n-1}*F[n](x) = U_{n-1}(I*x/2), where U_{n-1}(cos(t)) = sin(n*t)/sin(t) and I = sqrt(-1); Cf. A049310 and A053119, but with different notation.]
Wikipedia, Fibonacci polynomials.
Wikipedia, Hadamard's inequality.

Cf. A000045, A003946, A049310, A053119, A166920.

lista(nn) = {my(f=vector(nn)); my(g=vector(nn)); my(h=vector(nn)); f[1]=1; f[2]=x; g[1]=0; g[2]=0; for(n=3, nn, g[n] = (x^2+1)^2*(x^2+2)^(n-3)); for(n=3, nn, f[n] = x*f[n-1]+f[n-2]); for(n=1, nn, h[n] = g[n]-f[n]); for(n=3, nn, for(k=0, 2*n-2, print1(polcoef(h[n], k, x), ",")); print(););} \\ Petros Hadjicostas, Jun 10 2020

From Petros Hadjicostas, Jun 10 2020: (Start)
T(n,0) = 2^(n-3) - (1 - (-1)^n)/2 = A166920(n-3) for n >= 3.
Sum_{k=0}^{2*n-2} T(n,k) = 4*3^(n-3) - Fib(n) = A003946(n-2) - A000045(n) for n >= 3. (End)

Name edited by Petros Hadjicostas, Jun 10 2020

A131350 2*A007318 - A049310 as infinite lower triangular matrices.

Original entry on oeis.org

1, 2, 1, 1, 4, 1, 2, 4, 6, 1, 1, 8, 9, 8, 1, 2, 7, 20, 16, 10, 1, 1, 12, 24, 40, 25, 12, 1, 2, 10, 42, 60, 70, 36, 14, 1, 1, 16, 46, 112, 125, 112, 49, 16, 1, 2, 13, 72, 148, 252, 231, 168, 64, 18, 1

Offset: 0

Gary W. Adamson, Jul 02 2007

Row sums = A099036 starting (1, 3, 6, 13, 27, 56, 115,...).

			First few rows of the triangle are:
1;
2, 1;
1, 4, 1;
2, 4, 6, 1;
1, 8, 9, 8, 1;
2, 7, 20, 16, 10, 1;
1, 12, 24, 40, 25, 12, 1;
...

Cf. A007318, A049310, A131350.

A135830 A000012(signed) * A049310 * A000012.

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 1, 3, 0, 1, 4, 1, 4, 0, 1, 4, 7, 1, 5, 0, 1, 9, 5, 11, 1, 6, 0, 1, 12, 16, 6, 16, 1, 7, 0, 1, 22, 17, 27, 7, 22, 1, 8, 0, 1, 33, 38, 23, 43, 8, 29, 1, 9, 0, 1

Offset: 1

Gary W. Adamson, Nov 30 2007

Row sums A010049: (1, 1, 3, 5, 10, 18, 33, ...).

A135831 = A000012 * A000012(signed) * A049310.

			First few rows of the triangle:
   1;
   0,  1;
   2,  0,  1;
   1,  3,  0,  1;
   4,  1,  4,  0,  1;
   4,  7,  1,  5,  0,  1;
   9,  5, 11,  1,  6,  0,  1;
  12, 16,  6, 16,  1,  7,  0,  1;
  ...

Cf. A049310, A010049, A135831.

A000012(signed) * A049310 * A000012, as infinite lower triangular matrices, where A000012(signed) = (1; -1,1; 1,-1,1; ...).

A137865 Triangle read by rows, antidiagonals of an array formed by A000012 * A049310(transform).

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 0, 1, 1, 2, 3, 4, 3, 1, 1, 1, 2, 3, 4, 3, 1, 0, 1, 1, 2, 3, 5, 7, 7, 4, 1, 1, 1, 2, 3, 5, 7, 7, 4, 1, 0, 1, 1, 2, 3, 5, 8, 12, 14, 11, 5, 1, 1, 1, 2, 3, 5, 12, 14, 11, 5, 1, 0

Offset: 0

Gary W. Adamson, Feb 18 2008

Rows of the array tend to the Fibonacci sequence.

Row sums of the triangle = A052551: (1, 1, 3, 3, 7, 7, 15, 15, 31, 31, 63, 63, ...).

			First few rows of the array:
  1, 0, 1, 0, 1, 0,  1, ...
  1, 1, 1, 2, 1, 3,  1, ...
  1, 1, 2, 2, 4, 3,  7, ...
  1, 1, 2, 3, 4, 7,  7, ...
  1, 1, 2, 3, 5, 7, 12, ...
  1, 1, 2, 3, 5, 8, 12, ...
  ...
First few rows of the triangle:
  1;
  1, 0;
  1, 1, 1;
  1, 1, 1, 0;
  1, 1, 2, 2, 1;
  1, 1, 2, 2, 1, 0;
  1, 1, 2, 3, 4, 3,  1;
  1, 1, 2, 3, 4, 3,  1,  0;
  1, 1, 2, 3, 5, 7,  7,  4,  1;
  1, 1, 2, 3, 5, 7,  7,  4,  1, 0;
  1, 1, 2, 3, 5, 8, 12, 14, 11, 5, 1;
  1, 1, 2, 3, 5, 8, 12, 14, 11, 5, 1, 0;
  ...

Cf. A049310, A052551.

Triangle read by rows, antidiagonals of an array formed by taking A000012 * A049310(transform); given A049310 unsigned.

A220666 Array of coefficients of powers of x^2 for S(2*n,x)^3 with Chebyshev's S polynomials A049310.

Original entry on oeis.org

1, -1, 3, -3, 1, 1, -9, 30, -45, 30, -9, 1, -1, 18, -123, 399, -651, 588, -308, 93, -15, 1, 1, -30, 345, -1921, 5598, -9540, 10212, -7137, 3303, -1003, 192, -21, 1, -1, 45, -780, 6609, -29847, 80718, -141482, 168927, -141636, 84766, -36366, 11091, -2346, 327, -27, 1

Offset: 0

Wolfdieter Lang, Dec 17 2012

The row lengths sequence of this array is 3*n+1 = A016777(n).
For the coefficient array of S(n,x)^3 see A219240. The present array is the even part of the bisection of that one.
The row polynomials in powers of x^2 are (S(2*n,x))^3 =
  sum(a(n,m)*x^(2*m), m=0..3*n), n >= 0. The o.g.f. for these row polynomials is GS3even(x,z) = ((z+1)^3 + (1+z)*z*x^2*(3*x^2 - 7))/(((z+1)^2-z*x^2)*((z+1)^2-z*x^2*(x^2-3)^2)). This is obtained from the even part of the bisection of the o.g.f. for A219240.

			The array a(n,m) begins:
n\m  0    1     2     3      4     5     6    7    8  9
0:   1
1:  -1    3    -3     1
2:   1   -9    30   -45     30    -9     1
3:   1   18  -123   399   -651   588  -308   93  -15  1
...
Row n=4: [1, -30, 345, -1921, 5598, -9540, 10212, -7137, 3303, -1003, 192, -21, 1],
Row n=5: [-1, 45, -780, 6609, -29847, 80718, -141482, 168927, -141636, 84766, -36366, 11091, -2346, 327, -27, 1],
Row n=6: [1, -63, 1533, -18333, 118029, -460815, 1184872, -2118207, 2729922, -2598297, 1854177, -999687, 407472, -124680, 28164, -4553, 498, -33, 1].
Row n=2: S(4,x)^3 = 1 - 9*x^2 + 30*x^4 - 45*x^6 + 30*x^8  - 9*x^10 + 1*x^12.

Cf. A219240, A220665 (odd part of the bisection).

a(n,m) = [x^m] S(2*n,x)^3, n>=0, 0 <= m <= 3*n.

a(n,m) = [x^m]([z^n]GS3even(x,z)) with GS3even(x,z) the o.g.f. for the row polynomials in powers of x^2, given in a comment above.

cp's OEIS Frontend

A193663 Q-residue of A049310 (triangle of coefficients of Fibonacci polynomials), where Q is the triangle given by t(n,k)=k+1 for 0<=k<=n. (See Comments.)

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A220665 Array of coefficients of powers of x^2 for (S(2*n+1,x)/x)^3, with Chebyshev's S polynomials A049310.

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A286717 a(n) is the number of zeros of the Chebyshev S(n, x) polynomial (A049310) in the open interval (-phi, +phi), with the golden section phi = (1 + sqrt(5))/2.

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A302707 Number of factors of Chebyshev polynomial S(2*n+1, x) (A049310) over the integers. Factorization is into the minimal integer polynomials C (A187360).

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A309213 A007318 + A065941 - A049310.

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A131350 2*A007318 - A049310 as infinite lower triangular matrices.

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A135830 A000012(signed) * A049310 * A000012.

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