A214992 -id:A214992 - cp's OEIS frontend

A218984 Power floor sequence of 2+sqrt(6).

4, 17, 75, 333, 1481, 6589, 29317, 130445, 580413, 2582541, 11490989, 51129037, 227498125, 1012250573, 4503998541, 20040495309, 89169978317, 396760903885, 1765383572173, 7855056096461, 34950991530189, 155514078313677, 691958296315085, 3078861341887693

Offset: 0

Clark Kimberling, Nov 11 2012

See A214992 for a discussion of power floor sequence and the power floor function, p1(x) = limit of a(n,x)/x^n. The present sequence is a(n,r), where r = 2+sqrt(6), and the limit p1(r) = 3.77794213613376987528458445727451673384055973517...

			a(0) = [r] = 4, where r = 2+sqrt(6); a(1) = [4*r] = 17; a(2) = [17*r] = 75.

Clark Kimberling, Table of n, a(n) for n = 0..250
Index entries for linear recurrences with constant coefficients, signature (5,-2,-2).

Cf. A214992, A090017, A123347, A218985.

x = 2 + Sqrt[6]; z = 30; (* z = # terms in sequences *)
f[x_] := Floor[x]; c[x_] := Ceiling[x];
p1[0] = f[x]; p2[0] = f[x]; p3[0] = c[x]; p4[0] = c[x];
p1[n_] := f[x*p1[n - 1]]
p2[n_] := If[Mod[n, 2] == 1, c[x*p2[n - 1]], f[x*p2[n - 1]]]
p3[n_] := If[Mod[n, 2] == 1, f[x*p3[n - 1]], c[x*p3[n - 1]]]
p4[n_] := c[x*p4[n - 1]]
t1 = Table[p1[n], {n, 0, z}]  (* A218984 *)
t2 = Table[p2[n], {n, 0, z}]  (* A090017 *)
t3 = Table[p3[n], {n, 0, z}]  (* A123347 *)
t4 = Table[p4[n], {n, 0, z}]  (* A218985 *)

Vec((4 - 3*x - 2*x^2) / ((1 - x)*(1 - 4*x - 2*x^2)) + O(x^40)) \\ Colin Barker, Nov 13 2017

a(n) = [x*a(n-1)], where x=2+sqrt(6), a(0) = [x].
a(n) = 5*a(n-1) - 2*a(n-2) - 2*a(n-3).
G.f.: (4 - 3*x - 2*x^2)/(1 - 5*x + 2*x^2 + 2*x^3).
a(n) = (1/30)*(6 + (57-23*sqrt(6))*(2-sqrt(6))^n + (2+sqrt(6))^n*(57+23*sqrt(6))). - Colin Barker, Nov 13 2017

A218985 Power ceiling sequence of 2+sqrt(6).

Original entry on oeis.org

5, 23, 103, 459, 2043, 9091, 40451, 179987, 800851, 3563379, 15855219, 70547635, 313900979, 1396699187, 6214598707, 27651793203, 123036370227, 547449067315, 2435869009715, 10838374173491, 48225234713395, 214577687200563, 954761218229043, 4248200247317299

Offset: 0

Clark Kimberling, Nov 11 2012

See A214992 for a discussion of power ceiling sequence and the power ceiling function, p4(x) = limit of a(n,x)/x^n. The present sequence is a(n,r), where r = 2+sqrt(6), and the limit p4(r) = 5.2127890589687233047437696796862841514303439...

See A218984 for the power floor function, p1(x). For comparison of p4 and p1, limit(p4(r)/p1(r)) = 2*(1+sqrt(6))/5 = 1.379795897113271239278913629882356556786378...

			a(0) = ceiling(r) = 5, where r = 2+sqrt(6).
a(1) = ceiling(5*r) = 23; a(2) = ceiling(23*r) = 103.

Clark Kimberling, Table of n, a(n) for n = 0..250
Index entries for linear recurrences with constant coefficients, signature (5,-2,-2).

Cf. A214992, A090017, A123347, A218984.

Mathematica
```
(See A218984.)
```

Vec((5 - 2*x - 2*x^2) / ((1 - x)*(1 - 4*x - 2*x^2)) + O(x^40)) \\ Colin Barker, Nov 13 2017

a(n) = [x*a(n-1)], where x=2+sqrt(6), a(0) = [x].
a(n) = 5*a(n-1) - 2*a(n-2) - 2*a(n-3).
G.f.: (5 - 2*x - 2*x^2)/(1 - 5*x + 2*x^2 + 2*x^3).
a(n) = (1/15)*(-3 + (39-16*sqrt(6))*(2-sqrt(6))^n + (2+sqrt(6))^n*(39+16*sqrt(6))). - Colin Barker, Nov 13 2017

A218986 Power floor sequence of 2+sqrt(7).

Original entry on oeis.org

4, 18, 83, 385, 1788, 8306, 38587, 179265, 832820, 3869074, 17974755, 83506241, 387949228, 1802315634, 8373110219, 38899387777, 180716881764, 839565690386, 3900413406835, 18120350698497, 84182643014492, 391091624153458, 1816914425657307, 8440932575089601

Offset: 0

Clark Kimberling, Nov 11 2012

See A214992 for a discussion of power floor sequence and the power floor function, p1(x) = limit of a(n,x)/x^n. The present sequence is a(n,r), where r = 2+sqrt(7), and the limit p1(r) = 3.83798607113023840500712572585708...

See A218987 for the power floor function, p4. For comparison with p1, limit(p4(r)/p1(r) = 4-sqrt(7).

			a(0) = [r] = 4, where r = 2+sqrt(7);
a(1) = [4*r] = 18; a(2) = [18*r] = 83.

Clark Kimberling, Table of n, a(n) for n = 0..250
Index entries for linear recurrences with constant coefficients, signature (5,-1,-3).

Cf. A214992, A015530, A126473, A218987.

x = 2 + Sqrt[7]; z = 30; (* z = # terms in sequences *)
f[x_] := Floor[x]; c[x_] := Ceiling[x];
p1[0] = f[x]; p2[0] = f[x]; p3[0] = c[x]; p4[0] = c[x];
p1[n_] := f[x*p1[n - 1]]
p2[n_] := If[Mod[n, 2] == 1, c[x*p2[n - 1]], f[x*p2[n - 1]]]
p3[n_] := If[Mod[n, 2] == 1, f[x*p3[n - 1]], c[x*p3[n - 1]]]
p4[n_] := c[x*p4[n - 1]]
t1 = Table[p1[n], {n, 0, z}]  (* A218986 *)
t2 = Table[p2[n], {n, 0, z}]  (* A015530 *)
t3 = Table[p3[n], {n, 0, z}]  (* A126473 *)
t4 = Table[p4[n], {n, 0, z}]  (* A218987 *)
LinearRecurrence[{5,-1,-3},{4,18,83},30] (* Harvey P. Dale, Jun 18 2014 *)

a(n) = round((14+(161-61*sqrt(7))*(2-sqrt(7))^n+(2+sqrt(7))^n*(161+61*sqrt(7)))/84) \\ Colin Barker, Sep 02 2016

Vec((4-2*x-3*x^2)/((1-x)*(1-4*x-3*x^2)) + O(x^30)) \\ Colin Barker, Sep 02 2016

a(n) = [x*a(n-1)], where x=2+sqrt(7), a(0) = [x].
a(n) = 5*a(n-1) - a(n-2) - 3*a(n-3).
G.f.:  (4 - 2*x - 3*x^2)/(1 - 5*x + x^2 + 3*x^3).
a(n) = (14+(161-61*sqrt(7))*(2-sqrt(7))^n+(2+sqrt(7))^n*(161+61*sqrt(7)))/84. - Colin Barker, Sep 02 2016

A218987 Power ceiling sequence of 2+sqrt(7).

Original entry on oeis.org

5, 24, 112, 521, 2421, 11248, 52256, 242769, 1127845, 5239688, 24342288, 113088217, 525379733, 2440783584, 11339273536, 52679444897, 244735600197, 1136980735480, 5282129742512, 24539461176489, 114004233933493, 529635319263440, 2460553978854240

Offset: 0

Clark Kimberling, Nov 11 2012

See A214992 for a discussion of power ceiling sequence and the power ceiling function, p4(x) = limit of a(n,x)/x^n. The present sequence is a(n,r), where r = 2+sqrt(7), and the limit p4(r) = 5.19758760498048832156707270895307875397561324042...

See A218986 for the power floor function, p1(x); for comparison of p1 and p4, limit(p4(r)/p1(r) = 4 - sqrt(7).

			a(0) = ceiling(r) = 5, where r = 2+sqrt(7);
a(1) = ceiling(5*r) = 24; a(2) = ceiling(24*r) = 112.

Clark Kimberling, Table of n, a(n) for n = 0..250
Index entries for linear recurrences with constant coefficients, signature (5,-1,-3).

Cf. A214992, A015530, A126473, A218986.

Mathematica
```
(See A218986.)
```

a(n) = round((-14+(217-83*sqrt(7))*(2-sqrt(7))^n+(2+sqrt(7))^n*(217+83*sqrt(7)))/84) \\ Colin Barker, Sep 02 2016

Vec((5-x-3*x^2)/((1-x)*(1-4*x-3*x^2)) + O(x^30)) \\ Colin Barker, Sep 02 2016

a(n) = ceiling(x*a(n-1)), where x=2+sqrt(7), a(0) = ceiling(x).
a(n) = 5*a(n-1) - a(n-2) - 3*a(n-3).
G.f.: (5 - x - 3*x^2)/(1 - 5*x + x^2 + 3*x^3).
a(n) = (-14+(217-83*sqrt(7))*(2-sqrt(7))^n+(2+sqrt(7))^n*(217+83*sqrt(7)))/84. - Colin Barker, Sep 02 2016
E.g.f.: exp(x)*(exp(x)*(217*cosh(sqrt(7)*x) + 83*sqrt(7)*sinh(sqrt(7)*x)) - 7)/42. - Stefano Spezia, Aug 05 2025

A218991 Power floor sequence of 3+sqrt(10).

Original entry on oeis.org

6, 36, 221, 1361, 8386, 51676, 318441, 1962321, 12092366, 74516516, 459191461, 2829665281, 17437183146, 107452764156, 662153768081, 4080375372641, 25144406003926, 154946811396196, 954825274381101, 5883898457682801

Offset: 0

Clark Kimberling, Nov 12 2012

See A214992 for a discussion of power floor sequence and the power floor function, p1(x) = lim_{n->oo} a(n,x)/x^n. The present sequence is a(n,r), where r = 3+sqrt(10), and the limit p1(r) = 5.815421188487681054332319082...

See A218992 for the power floor function, p4. For comparison with p1, we have lim_{r->oo} p4(r)/p1(r) = (3+sqrt(10))/5 = 1.23245553....

			a(0) = floor(r) = 6, where r = 3+sqrt(10);
a(1) = floor(6*r) = 36;
a(2) = floor(36*r) = 221.

Clark Kimberling, Table of n, a(n) for n = 0..250
Index entries for linear recurrences with constant coefficients, signature (7,-5,-1).

Cf. A214992, A005668, A015451, A218992.

Cf. A176398 (3+sqrt(10)).

[IsZero(n) select Floor(r) else Floor(r*Self(n)) where r is 3+Sqrt(10): n in [0..20]]; // Bruno Berselli, Nov 22 2012

x = 3 + Sqrt[10]; z = 30; (* z = # terms in sequences *)
f[x_] := Floor[x]; c[x_] := Ceiling[x];
p1[0] = f[x]; p2[0] = f[x]; p3[0] = c[x]; p4[0] = c[x];
p1[n_] := f[x*p1[n - 1]]
p2[n_] := If[Mod[n, 2] == 1, c[x*p2[n - 1]], f[x*p2[n - 1]]]
p3[n_] := If[Mod[n, 2] == 1, f[x*p3[n - 1]], c[x*p3[n - 1]]]
p4[n_] := c[x*p4[n - 1]]
t1 = Table[p1[n], {n, 0, z}]  (* A218991 *)
t2 = Table[p2[n], {n, 0, z}]  (* A005668 *)
t3 = Table[p3[n], {n, 0, z}]  (* A015451 *)
t4 = Table[p4[n], {n, 0, z}]  (* A218992 *)

a(n) = floor(r*a(n-1)), where r=3+sqrt(10), a(0) = floor(r).
a(n) = 7*a(n-1) - 5*a(n-2) - a(n-3).
G.f.: (6 - 6*x - x^2)/(1 - 7*x + 5*x^2 + x^3).
a(n) = ((5+sqrt(10))*(3-sqrt(10))^(n+2) + (5-sqrt(10))*(3+sqrt(10))^(n+2)+2)/12. - Bruno Berselli, Nov 22 2012

A218992 Power ceiling sequence of 3+sqrt(10).

Original entry on oeis.org

7, 44, 272, 1677, 10335, 63688, 392464, 2418473, 14903303, 91838292, 565933056, 3487436629, 21490552831, 132430753616, 816075074528, 5028881200785, 30989362279239, 190965054876220, 1176779691536560, 7251643204095581

Offset: 0

Clark Kimberling, Nov 12 2012

See A214992 for a discussion of power ceiling sequence and the power ceiling function, p4(x) = limit of a(n,x)/x^n. The present sequence is a(n,r), where r = 3+sqrt(10), and the limit p4(r) = 7.16724801485749657...

See A218991 for the power floor function, p1(x); for comparison of p1 and p4, we have limit(p4(r)/p1(r) = (3+sqrt(10))/5 = 1.23245553...

			a(0) = ceiling(r) = 7, where r = 3+sqrt(10);
a(1) = ceiling(7*r) = 44;
a(2) = ceiling(44*r) = 272.

Clark Kimberling, Table of n, a(n) for n = 0..250
Index entries for linear recurrences with constant coefficients, signature (7,-5,-1).

Cf. A214992, A005668, A015451, A218991.

Cf. A176398 (3+sqrt(10)).

[IsZero(n) select Ceiling(r) else Ceiling(r*Self(n)) where r is 3+Sqrt(10): n in [0..20]]; // Bruno Berselli, Nov 22 2012

(See A218991.)
LinearRecurrence[{7,-5,-1},{7,44,272},20] (* Harvey P. Dale, Sep 22 2016 *)

a(n) = ceiling(r*a(n-1)), where r=3+sqrt(10), a(0) = ceiling(r).
a(n) = 7*a(n-1) - 5*a(n-2) - a(n-3).
G.f.: (7 - 5*x - x^2)/(1 - 7*x + 5*x^2 + x^3).
a(n) = ((5+sqrt(10))*(3-sqrt(10))^(n+3)+(5-sqrt(10))*(3+sqrt(10))^(n+3)-10)/60. [Bruno Berselli, Nov 22 2012]

A214995 Power ceiling-floor sequence of (golden ratio)^6.

Original entry on oeis.org

18, 322, 5779, 103699, 1860804, 33390772, 599173093, 10751724901, 192931875126, 3462022027366, 62123464617463, 1114760341086967, 20003562674947944, 358949367807976024, 6441085057868620489, 115580581673827192777, 2074009385071020849498

Offset: 0

Clark Kimberling, Nov 09 2012

See A214992 for a discussion of power ceiling-floor sequence and the power ceiling-floor function, p3(x) = limit of a(n,x)/x^n. The present sequence is a(n,r), where r = (golden ratio)^6, and the limit p3(r) = 17.94722275971790615684809...

			a(0) = ceiling(r) = [17.9] = 18 , where r=(1+sqrt(5))^6;
a(1) = floor(18*r) = 322; a(2) = ceiling(322*r ) = 5779.

Clark Kimberling, Table of n, a(n) for n = 0..250
Index entries for linear recurrences with constant coefficients, signature (17,17,-1).

Cf. A214992, A007805, A156085, A049660.

x = GoldenRatio^6; z = 30; (* z = # terms in sequences *)
z1 = 100; (* z1 = # digits in approximations *)
f[x_] := Floor[x]; c[x_] := Ceiling[x];
p1[0] = f[x]; p2[0] = f[x]; p3[0] = c[x]; p4[0] = c[x];
p1[n_] := f[x*p1[n - 1]]
p2[n_] := If[Mod[n, 2] == 1, c[x*p2[n - 1]], f[x*p2[n - 1]]]
p3[n_] := If[Mod[n, 2] == 1, f[x*p3[n - 1]], c[x*p3[n - 1]]]
p4[n_] := c[x*p4[n - 1]]
Table[p1[n], {n, 0, z}]  (* A007805 *)
Table[p2[n], {n, 0, z}]  (* A156085 *)
Table[p3[n], {n, 0, z}]  (* A214995 *)
Table[p4[n], {n, 0, z}]  (* A049660 *)
Table[p4[n] - p1[n], {n, 0, z}]  (* A049660 *)
Table[p3[n] - p2[n], {n, 0, z}]  (* A099279 *)
LinearRecurrence[{17,17,-1},{18,322,5779},30] (* Harvey P. Dale, Feb 25 2013 *)

Vec((18+16*x-x^2)/((1+x)*(1-18*x+x^2)) + O(x^20)) \\ Colin Barker, Mar 04 2016

a(n) = floor(x*a(n-1)) if n is odd, a(n) = ceiling(x*a(n-1)) if n is even, where x=((1+sqrt(5))/2)^6 and a(0) = ceiling(x).
a(n) = 17*a(n-1) + 17*a(n-2) - a(n-3).
G.f.: (18 + 16*x - x^2)/((1 + x)*(1 - 18*x + x^2)).
a(n) = (4*(-1)^n+(718-321*sqrt(5))*(9+4*sqrt(5))^(-n)+(9+4*sqrt(5))^n*(718+321*sqrt(5)))/80. - Colin Barker, Mar 04 2016
E.g.f.: exp(-x)*(2 + exp(10*x)*(718*cosh(4*sqrt(5)*x) + 321*sqrt(5)*sinh(4*sqrt(5)*x)))/40. - Stefano Spezia, Oct 28 2024

A214998 Power ceiling-floor sequence of 2 + sqrt(3).

Original entry on oeis.org

4, 14, 53, 197, 736, 2746, 10249, 38249, 142748, 532742, 1988221, 7420141, 27692344, 103349234, 385704593, 1439469137, 5372171956, 20049218686, 74824702789, 279249592469, 1042173667088, 3889445075882, 14515606636441, 54172981469881, 202176319243084

Offset: 0

Clark Kimberling, Nov 10 2012

See A214992 for a discussion of power ceiling-floor sequence and power ceiling-floor function, p3(x) = limit of a(n,x)/x^n. The present sequence is a(n,r), where r = 2 + sqrt(3), and the limit p3(r) = (23 + 13*sqrt(3))/12.

			a(0) = ceiling(r) =  4, where r = 2+sqrt(3);
a(1) = floor(4*r) = 14; a(2) = ceiling(14*r) = 53.

Clark Kimberling, Table of n, a(n) for n = 0..250
R. C. Alperin, A family of nonlinear recurrences and their linear solutions, Fib. Q., 57:4 (2019), 318-321.
Index entries for linear recurrences with constant coefficients, signature (3,3,-1).

Cf. A214992, A001835, A109437, A001353, A001654.

x = 2 + Sqrt[3]; z = 30; (* z = # terms in sequences *)
z1 = 100; (* z1 = # digits in approximations *)
f[x_] := Floor[x]; c[x_] := Ceiling[x];
p1[0] = f[x]; p2[0] = f[x]; p3[0] = c[x]; p4[0] = c[x];
p1[n_] := f[x*p1[n - 1]]
p2[n_] := If[Mod[n, 2] == 1, c[x*p2[n - 1]], f[x*p2[n - 1]]]
p3[n_] := If[Mod[n, 2] == 1, f[x*p3[n - 1]], c[x*p3[n - 1]]]
p4[n_] := c[x*p4[n - 1]]
Table[p1[n], {n, 0, z}]  (* A001835 *)
Table[p2[n], {n, 0, z}]  (* A109437 *)
Table[p3[n], {n, 0, z}]  (* A214998 *)
Table[p4[n], {n, 0, z}]  (* A001353 *)

Vec((4 + 2*x - x^2) / ((1 + x)*(1 - 4*x + x^2)) + O(x^30)) \\ Colin Barker, Nov 13 2017

a(n) = floor(x*a(n-1)) if n is odd, a(n) = ceiling(x*a(n-1)) if n is even, where x = 2+sqrt(3) and a(0) = ceiling(x).
a(n) = 3*a(n-1) + 3*a(n-2) - a(n-3).
G.f.: (4 + 2*x - x^2)/(1 - 3*x - 3*x^2 + x^3).
a(n) = (-1)^n + 4*a(n-1) - a(n-2) with a(0) = 4 and a(1) = 14. - Peter Bala, Nov 12 2017
a(n) = (1/12)*(2*(-1)^n + (23-13*sqrt(3))*(2-sqrt(3))^n + (2+sqrt(3))^n*(23+13*sqrt(3))). - Colin Barker, Nov 13 2017

A218990 Power ceiling-floor sequence of 3+sqrt(8).

Original entry on oeis.org

6, 34, 199, 1159, 6756, 39376, 229501, 1337629, 7796274, 45440014, 264843811, 1543622851, 8996893296, 52437736924, 305629528249, 1781339432569, 10382407067166, 60513102970426, 352696210755391, 2055664161561919, 11981288758616124, 69832068390134824

Offset: 0

Clark Kimberling, Nov 11 2012

See A214992 for a discussion of power ceiling-floor sequence and power ceiling-floor function, p3(x) = limit of a(n,x)/x^n. The present sequence is a(n,r), where r = 3+sqrt(8), and the limit p3(r) = 5.854315472394508538153482993682502287049948...

			a(0) = ceiling(r) = 6, where r = 3+sqrt(8);
a(1) = floor(6*r) = 34; a(2) = ceiling(35*r) = 199.

R. C. Alperin, A family of nonlinear recurrences and their linear solutions, Fib. Q., 57:4 (2019), 318-321.~

Clark Kimberling, Table of n, a(n) for n = 0..250
Index entries for linear recurrences with constant coefficients, signature (5,5,-1).

Cf. A214992, A001653, A084158, A001109.

x = 3 + Sqrt[8]; z = 30; (* z = # terms in sequences *)
f[x_] := Floor[x]; c[x_] := Ceiling[x];
p1[0] = f[x]; p2[0] = f[x]; p3[0] = c[x]; p4[0] = c[x];
p1[n_] := f[x*p1[n - 1]]
p2[n_] := If[Mod[n, 2] == 1, c[x*p2[n - 1]], f[x*p2[n - 1]]]
p3[n_] := If[Mod[n, 2] == 1, f[x*p3[n - 1]], c[x*p3[n - 1]]]
p4[n_] := c[x*p4[n - 1]]
t1 = Table[p1[n], {n, 0, z}]  (* A001653 *)
t2 = Table[p2[n], {n, 0, z}]  (* A084158 *)
t3 = Table[p3[n], {n, 0, z}]  (* A218990 *)
t4 = Table[p4[n], {n, 0, z}]  (* A001109 *)
LinearRecurrence[{5,5,-1},{6,34,199},30] (* Harvey P. Dale, Mar 21 2024 *)

Vec((6 + 4*x - x^2) / ((1 + x)*(1 - 6*x + x^2)) + O(x^50)) \\ Colin Barker, Nov 13 2017

a(n) = floor(x*a(n-1)) if n is odd, a(n) = ceiling(x*a(n-1)) if n is even, where x=3+sqrt(8) and a(0) = ceiling(x).
a(n) = 5*a(n-1) + 5*a(n-2) - a(n-3).
G.f.: (6 + 4*x - x^2)/(1 - 5*x - 5*x^2 + x^3).
a(n) = (1/16)*(2*(-1)^n + (47-33*sqrt(2))*(3-2*sqrt(2))^n + (3+2*sqrt(2))^n*(47+33*sqrt(2))). - Colin Barker, Nov 13 2017

cp's OEIS Frontend

A218984 Power floor sequence of 2+sqrt(6).

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A218985 Power ceiling sequence of 2+sqrt(6).

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A218986 Power floor sequence of 2+sqrt(7).

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A218987 Power ceiling sequence of 2+sqrt(7).

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A218991 Power floor sequence of 3+sqrt(10).

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A218992 Power ceiling sequence of 3+sqrt(10).

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Mathematica

Formula

A214995 Power ceiling-floor sequence of (golden ratio)^6.

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