A214992 -id:A214992 - cp's OEIS frontend

A214999 Power floor sequence of sqrt(5).

2, 4, 8, 17, 38, 84, 187, 418, 934, 2088, 4668, 10437, 23337, 52183, 116684, 260913, 583419, 1304564, 2917093, 6522818, 14585464, 32614088, 72927317, 163070438, 364636584, 815352188, 1823182917, 4076760937, 9115914583

Offset: 0

Clark Kimberling, Nov 10 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 = sqrt(5), and the limit p1(r) = 1.4935514451954997630823098687087959696356...

			a(0) = [r] = 2, where r = sqrt(5); a(1) = [2*r] = 4; a(2) = [4*r] = 8.

Clark Kimberling, Table of n, a(n) for n = 0..250

Cf. A214992, A215091, A218982, A218983.

x = Sqrt[5]; 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]]
Table[p1[n], {n, 0, z}]  (* A214999 *)
Table[p2[n], {n, 0, z}]  (* A215091 *)
Table[p3[n], {n, 0, z}]  (* A218982 *)
Table[p4[n], {n, 0, z}]  (* A218983 *)

a(n) = [x*a(n-1)], where x=sqrt(5), a(0) = [x].

A215091 Power floor-ceiling sequence of sqrt(5).

Original entry on oeis.org

2, 5, 11, 25, 55, 123, 275, 615, 1375, 3075, 6875, 15373, 34375, 76865, 171875, 384325, 859376, 1921624, 4296881, 9608119, 21484407, 48040595, 107422036, 240202975, 537110180, 1201014874, 2685550900, 6005074370, 13427754501

Offset: 0

Clark Kimberling, Nov 10 2012

See A214992 for a discussion of power floor-ceiling sequence and the power floor-ceiling function, p2(x) = limit of a(n,x)/x^n. The present sequence is a(n,r), where r = sqrt(5), and the limit p2(r) = 2.20000329748317471983660768168522753590...

			a(0) = floor(r) = 2, where r = sqrt(5);
a(1) = ceiling(2*r) = 5; a(2) = floor(5*r) = 11.

Clark Kimberling, Table of n, a(n) for n = 0..250

Cf. A214992, A214999, A218982, A218983.

(See A214999.)
nxt[{n_,a_}]:={n+1,If[OddQ[n],Floor[Sqrt[5]*a],Ceiling[Sqrt[5]*a]]}; Transpose[ NestList[nxt,{0,2},30]][[2]] (* Harvey P. Dale, Oct 27 2015 *)

a(n) = ceiling(x*a(n-1)) if n is odd, a(n) = floor(x*a(n-1)) if n is even, where x = sqrt(5) and a(0) = floor(x).

A218982 Power ceiling-floor sequence of sqrt(5).

Original entry on oeis.org

3, 6, 14, 31, 70, 156, 349, 780, 1745, 3901, 8723, 19505, 43615, 97526, 218075, 487630, 1090374, 2438150, 5451870, 12190751, 27259348, 60953755, 136296740, 304768775, 681483699, 1523843876, 3407418494, 7619219380, 17037092470

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 = sqrt(5), and the limit p3(r) = 2.79135723025040661923369247589566824549062...

			a(0) = ceiling(r) = 3, where r = sqrt(5);
a(1) = floor(3*r) = 6; a(2) = ceiling(6*r) = 14.

Clark Kimberling, Table of n, a(n) for n = 0..250

Cf. A214992, A214999, A215091, A218983.

Mathematica
```
(See A214999.)
```

a(n) = floor(x*a(n-1)) if n is odd, a(n) = ceiling(x*a(n-1)) if n is even, where x=sqrt(5) and a(0) = ceiling(x).

A218983 Power ceiling sequence of sqrt(5).

Original entry on oeis.org

3, 7, 16, 36, 81, 182, 407, 911, 2038, 4558, 10192, 22791, 50963, 113957, 254816, 569786, 1274081, 2848932, 6370406, 14244661, 31852031, 71223307, 159260157, 356116538, 796300787, 1780582691, 3981503937, 8902913456

Offset: 0

Clark Kimberling, Nov 10 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 = sqrt(5), and the limit p4(r) = 3.2616480254413398807499379112702935254866963...

See A214999 for the power floor function, p1(x). For comparison of p4 and p1, limit(p4(r)/p1(r)) = 2.183820340393031136325385184014007307594650...

			a(0) = ceiling(r) = 3, where r = sqrt(5);
a(1) = ceiling(3*r) = 7; a(2) = ceiling(7*r ) = 16.

Clark Kimberling, Table of n, a(n) for n = 0..250

Cf. A214992, A214999, A215091, A218982.

(See A214999.)
With[{c=Sqrt[5]},NestList[Ceiling[c #]&,Ceiling[c],30]] (* Harvey P. Dale, Mar 06 2013 *)

a(n) = ceiling(x*a(n-1)), where x=sqrt(5), a(0) = ceiling(x).

A214993 Power floor sequence of (golden ratio)^5.

Original entry on oeis.org

11, 121, 1341, 14871, 164921, 1829001, 20283931, 224952241, 2494758581, 27667296631, 306835021521, 3402852533361, 37738212888491, 418523194306761, 4641493350262861, 51474950047198231, 570865943869443401, 6331000332611075641, 70211869602591275451

Offset: 0

Clark Kimberling, Nov 09 2012

See A214992 for a discussion of power floor sequence and also the power floor function, p1(x) = limit of a(n,x)/x^n. The present sequence is a(n,r), where r = (golden ratio)^5, and the limit p1(r) = (3/22)*(3+2*sqrt(5)).

			a(0) = [r] = [11.0902] = 11, where r = (1+sqrt(5))^5.
a(1) = [11*r] = 121; a(2) = [121*r] = 1341.

Clark Kimberling, Table of n, a(n) for n = 0..250
Yaohui Zhu, Kaiming Sun, Zhengdong Luo, and Lingfeng Wang, Progressive Self-Learning for Domain Adaptation on Symbolic Regression of Integer Sequences, Proc. 39th AAAI Conf. Artif. Intel. (2025) Vol. 39, No. 1, 1692-1699. See p. 1698.
Index entries for linear recurrences with constant coefficients, signature (12,-10,-1).

Cf. A214992, A214994, A049666, A015457.

I:=[11,121,1341]; [n le 3 select I[n] else 12*Self(n-1)-10*Self(n-2)-Self(n-3): n in [1..30]]; // G. C. Greubel, Feb 01 2018

x = GoldenRatio^5; 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}]  (* A214993 *)
Table[p2[n], {n, 0, z}]  (* A049666 *)
Table[p3[n], {n, 0, z}]  (* A015457 *)
Table[p4[n], {n, 0, z}]  (* A214994 *)
LinearRecurrence[{12,-10,-1}, {11,121,1341}, 30] (* G. C. Greubel, Feb 01 2018 *)

Vec((11 - 11*x - x^2) / ((1 - x)*(1 - 11*x - x^2)) + O(x^20)) \\ Colin Barker, Nov 13 2017

a(n) = [x*a(n-1)], where x=((1+sqrt(5))/2)^5, a(0) = [x].
a(n) = 1 (mod 10).
a(n) = 12*a(n-1) - 10*a(n-2) - a(n-3).
G.f.: (11 - 11*x - x^2)/(1 - 12*x + 10*x^2 + x^3).
a(n) = (1/55)*(5 + (300-134*sqrt(5))*((11-5*sqrt(5))/2)^n + 2*(11/2+(5*sqrt(5))/2)^n*(150+67*sqrt(5))). - Colin Barker, Nov 13 2017

A214996 Power floor-ceiling sequence of 2+sqrt(2).

Original entry on oeis.org

3, 11, 37, 127, 433, 1479, 5049, 17239, 58857, 200951, 686089, 2342455, 7997641, 27305655, 93227337, 318298039, 1086737481, 3710353847, 12667940425, 43251054007, 147668335177, 504171232695, 1721348260425, 5877050576311, 20065505784393, 68507921984951

Offset: 0

Clark Kimberling, Nov 10 2012

See A214992 for a discussion of power floor-ceiling sequence and power floor-ceiling function, p2(x) = limit of a(n,x)/x^n. The present sequence is a(n,r), where r = 2+sqrt(2), and the limit p2(r) = (11 + 8*sqrt(2))/7.
From Greg Dresden, Jun 02 2020: (Start)
a(n) is the number of ways to tile a 2 X (n+1) strip, with one extra square at the top left corner, using 1 X 1 squares, 2 X 2 squares, and 1 X 2 dominoes (either horizontal or vertical). This picture shows a(1) = 11.
                                                     _
  ||  ||  | |  ||_  ||  ||  ||  | |  | |  ||  ||
  ||| |   | ||| | || || | |_| ||| ||| || | |__| | | |
  ||| |_| ||| ||| ||| ||| |_| |_| ||| |_| |||
(End)

			a(0) = floor(r) = 3, where r = 2+sqrt(2).
a(1) = ceiling(3*r) = 11; a(2) = floor(11*r) = 37.

Clark Kimberling, Table of n, a(n) for n = 0..250
Index entries for linear recurrences with constant coefficients, signature (3,2,-2).

Cf. A214992, A007052, A214997, A007070, A052543.

Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!((3+2*x-2*x^2)/(1-3*x-2*x^2+2*x^3))) // G. C. Greubel, Feb 02 2018

x = 2 + Sqrt[2]; 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}]  (* A007052 *)
Table[p2[n], {n, 0, z}]  (* A214996 *)
Table[p3[n], {n, 0, z}]  (* A214997 *)
Table[p4[n], {n, 0, z}]  (* A007070 *)

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

a(n) = ceiling(x*a(n-1)) if n is odd, a(n) = floor(x*a(n-1)) if n is even, where x = 2+sqrt(2) and a(0) = floor(x).
a(n) = 3*a(n-1) + 2*a(n-2) - 2*a(n-3).
G.f.: (3 + 2*x - 2*x^2)/(1 - 3*x - 2*x^2 + 2*x^3).
a(n) = (1/7)*((-1)^(1+n) + (11-8*sqrt(2))*(2-sqrt(2))^n + (2+sqrt(2))^n*(11+8*sqrt(2))). - Colin Barker, Nov 13 2017

A214997 Power ceiling-floor sequence of 2+sqrt(2).

Original entry on oeis.org

4, 13, 45, 153, 523, 1785, 6095, 20809, 71047, 242569, 828183, 2827593, 9654007, 32960841, 112535351, 384219721, 1311808183, 4478793289, 15291556791, 52208640585, 178251448759, 608588513865, 2077851157943, 7094227604041, 24221208100279, 82696377193033

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(2), and the limit p3(r) = 3.8478612632206289...
a(n) is the number of words over {0,1,2,3} of length n+1 that avoid 23, 32, and 33. As an example, a(2)=45 corresponds to the 45 such words of length 3; these are all 64 words except for the 19 prohibited cases which are 320, 321, 322, 323, 230, 231, 232, 233, 330, 331, 332, 333, 023, 123, 223, 032, 132, 033, 133. - Greg Dresden and Mina BH Arsanious, Aug 09 2023
Let M denote the 4 X 4 matrix = [[1,1,1,1], [1,1,1,1], [1,1,1,0], [1,1,0,0]] and A(n) = the column vector (p(n),q(n),r(n),s(n)) = M^n * A(0), where A(0) = (1,1,1,1), then a(n) = p(n)+q(n)+r(n)+s(n) = p(n+1). - Mina BH Arsanious, Jan 18 2025
Sum_{k=0..n} a(k) = (r(n-2)-3)/2 where r(n) is defined in previous comment. - Mina BH Arsanious, May 21 2025

			a(0) = ceiling(r) = 4, where r = 2+sqrt(2);
a(1) = floor(4*r) = 13; a(2) = ceiling(13*r) = 45.

Clark Kimberling, Table of n, a(n) for n = 0..250
Index entries for linear recurrences with constant coefficients, signature (3,2,-2).

Cf. A214992, A007052, A214996, A007070.

Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!((4 +x-2*x^2)/(1-3*x-2*x^2+2*x^3))); // G. C. Greubel, Feb 01 2018

(See A214996.)
CoefficientList[Series[(4+x-2*x^2)/(1-3*x-2*x^2+2*x^3), {x,0,50}], x] (* G. C. Greubel, Feb 01 2018 *)

Vec((4 + x - 2*x^2) / ((1 + x)*(1 - 4*x + 2*x^2)) + O(x^40)) \\ 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(2) and a(0) = ceiling(x).
a(n) = 3*a(n-1) + 2*a(n-2) - 2*a(n-3).
G.f.: (4 + x - 2*x^2)/(1 - 3*x - 2*x^2 + 2*x^3).
a(n) = (1/14)*(2*(-1)^n + (27-19*sqrt(2))*(2-sqrt(2))^n + (2+sqrt(2))^n*(27+19*sqrt(2))). - Colin Barker, Nov 13 2017

A218988 Power floor sequence of 2+sqrt(8).

Original entry on oeis.org

4, 19, 91, 439, 2119, 10231, 49399, 238519, 1151671, 5560759, 26849719, 129641911, 625966519, 3022433719, 14593600951, 70464138679, 340230958519, 1642780388791, 7932045389239, 38299303112119, 184925394005431, 892898788470199, 4311296729902519

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(8), and the limit p1(r) = 3.8983688904482395322594950087206...

See A218989 for the power floor function, p4. For comparison with p1, limit(p4(r)/p1(r) = 4/3.

			a(0) = [r] = 4, where r = 2+sqrt(8).
a(1) = [4*r] = 19; a(2) = [19*r] = 91.

Clark Kimberling, Table of n, a(n) for n = 0..250
Index entries for linear recurrences with constant coefficients, signature (5,0,-4).

Cf. A214992, A057087, A086347, A218989.

x = 2 + 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}]  (* this sequence *)
t2 = Table[p2[n], {n, 0, z}]  (* A057087 *)
t3 = Table[p3[n], {n, 0, z}]  (* A086347 *)
t4 = Table[p4[n], {n, 0, z}]  (* A218989 *)

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

a(n) = floor(x*a(n-1)), where x=2+sqrt(8), a(0) = floor(x).
a(n) = 5*a(n-1) - 4*a(n-3).
G.f.: (4 - x - 4*x^2) / ((1 - x)*(1 - 4*x - 4*x^2)). [Corrected by Colin Barker, Nov 13 2017]
a(n) = (1/28)*(4 + (54-39*sqrt(2))*(2-2*sqrt(2))^n + (2*(1+sqrt(2)))^n*(54+39*sqrt(2))). - Colin Barker, Nov 13 2017
From Philippe Deléham, Mar 18 2024: (Start)
a(n) = 4*a(n-1) + 4*a(n-2) - 1.
a(n-1) = Sum_{k = 0..n} A370174(n,k)*3^k. (End)

A218989 Power ceiling sequence of 2+sqrt(8).

Original entry on oeis.org

5, 25, 121, 585, 2825, 13641, 65865, 318025, 1535561, 7414345, 35799625, 172855881, 834622025, 4029911625, 19458134601, 93952184905, 453641278025, 2190373851721, 10576060518985, 51065737482825, 246567192007241, 1190531717960265, 5748395639870025

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(8), and the limit p4(r) = (18 + 13*sqrt(2))/2 = 5.1978251872643193763459933449608678602008191971286...

See A218988 for the power floor function, p1(x); for comparison of p1 and p4, we have limit(p4(r)/p1(r) = 4 - sqrt(7).

			a(0) = ceiling(r) = 5, where r = 2+sqrt(8);
a(1) = ceiling(5*r) = 25; a(2) = ceiling(25*r) = 121.

Clark Kimberling, Table of n, a(n) for n = 0..250
Index entries for linear recurrences with constant coefficients, signature (5,0,-4).

Cf. A214992, A057087, A086347, A218988.

Mathematica
```
(See A218988.)
```

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

a(n) = ceiling(x*a(n-1)), where x=2+sqrt(8), a(0) = ceiling(x).
a(n) = 5*a(n-1) - 4*a(n-3).
G.f.: (5 - 4*x^2) / ((1 - x)*(1 - 4*x - 4*x^2)). Corrected by Colin Barker, Nov 13 2017
a(n) = (1/7)*(-1 + (18-13*sqrt(2))*(2-2*sqrt(2))^n + (2*(1+sqrt(2)))^n*(18+13*sqrt(2))). - Colin Barker, Nov 13 2017

A214994 Power ceiling sequence of (golden ratio)^5.

Original entry on oeis.org

12, 134, 1487, 16492, 182900, 2028393, 22495224, 249475858, 2766729663, 30683502152, 340285253336, 3773821288849, 41852319430676, 464149335026286, 5147495004719823, 57086594386944340, 633100033261107564, 7021186960259127545, 77866156596111510560

Offset: 0

Clark Kimberling, Nov 09 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 = (golden ratio)^5, and the limit p4(r) = (1/30)*(105+47*sqrt(5)).

See A214993 for the power floor sequence and power floor function, p1. For comparison with p4, we have p4(r)/p1(r) = (5 + 3*sqrt(5))/10.

			a(0) = ceiling(r) = [11.0902]=12, where r=(1+sqrt(5))^5.
a(1) = ceiling(12) = 134; a(2) = ceiling(134 ) = 1487.

Clark Kimberling, Table of n, a(n) for n = 0..250
Index entries for linear recurrences with constant coefficients, signature (12,-10,-1).

Cf. A214992, A214993, A049666, A015457.

I:=[12,134,1487]; [n le 3 select I[n] else 12*Self(n-1) - 10*Self(n-2) - Self(n-3): n in [1..30]]; // G. C. Greubel, Feb 01 2018

(See A214993.)
LinearRecurrence[{12,-10,-1}, {12,134,1487}, 30] (* G. C. Greubel, Feb 01 2018 *)

Vec((12 - 10*x - x^2) / ((1 - x)*(1 - 11*x - x^2)) + O(x^40)) \\ Colin Barker, Nov 13 2017

a(n) = ceiling(x*a(n-1)), x=((1+sqrt(5))/2)^5, a(0) = ceiling(x).
a(n) = 12*a(n-1) - 10*a(n-2) - a(n-3).
G.f.: (12 - 10*x - x^2)/(1 - 12*x + 10*x^2 + x^3).
a(n) = (1/550)*(-50 + (3325-1487*sqrt(5))*((11-5*sqrt(5))/2)^n + ((11+5*sqrt(5))/2)^n*(3325+1487*sqrt(5))). - Colin Barker, Nov 13 2017

cp's OEIS Frontend

A214999 Power floor sequence of sqrt(5).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A215091 Power floor-ceiling sequence of sqrt(5).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A218982 Power ceiling-floor sequence of sqrt(5).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A218983 Power ceiling sequence of sqrt(5).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A214993 Power floor sequence of (golden ratio)^5.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A214996 Power floor-ceiling sequence of 2+sqrt(2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A214997 Power ceiling-floor sequence of 2+sqrt(2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs