A184922 -id:A184922 - cp's OEIS frontend

A003151 Beatty sequence for 1+sqrt(2); a(n) = floor(n*(1+sqrt(2))).

2, 4, 7, 9, 12, 14, 16, 19, 21, 24, 26, 28, 31, 33, 36, 38, 41, 43, 45, 48, 50, 53, 55, 57, 60, 62, 65, 67, 70, 72, 74, 77, 79, 82, 84, 86, 89, 91, 94, 96, 98, 101, 103, 106, 108, 111, 113, 115, 118, 120, 123, 125, 127, 130, 132, 135, 137, 140, 142, 144

Offset: 1

N. J. A. Sloane

Numbers with an odd number of trailing 0's in their minimal representation in terms of the positive Pell numbers (A317204). - Amiram Eldar, Mar 16 2022
From Clark Kimberling, Dec 24 2022: (Start)
This is the first of four sequences that partition the positive integers. Starting with a general overview, suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers. Let u' and v' be their complements, and assume that the following four sequences are infinite:
(1)  u ^ v = intersection of u and v (in increasing order);
(2)  u ^ v';
(3)  u' ^ v;
(4)  u' ^ v'.
Every positive integer is in exactly one of the four sequences.
For A003151, u, v, u', v', are the Beatty sequences given by u(n) = floor(n*sqrt(2)) and v(n) = floor(((1+sqrt(2))/2)*n), so that r = sqrt(2), s = (1+sqrt(2))/2, r' = (2+sqrt(2))/2, s' = 1 + 1/sqrt(2).
Assume that if w is any of the sequences u, v, u', v', then lim_{n->oo} w(n)/n exists and defines the (limiting) density of w. For w = u,v,u',v', denote the densities by r,s,r',s'. Then the densities of sequences (1)-(4) exist, and 1/(r*r') + 1/(r*s') + 1/(s*s') + 1/(s*r') = 1.
(1)  u ^ v = (2, 4, 7, 9, 12, 14, 16, 19, 21, 24, 26, 28, 31, 33, ...) = A003151
(2)  u ^ v' = (1, 5, 8, 11, 15, 18, 22, 25, 29, 32, 35, 39, 42, ...) = A001954
(3)  u' ^ v = (284, 287, 289, 292, 294, 296, 299, 301, 304, 306, ...) = A356135
(4)  u' ^ v' = (3, 6, 10, 13, 17, 20, 23, 27, 30, 34, 37, 40, 44, ...) = A003152
For results of compositions instead of intersections, see A184922. (End)
The indices of the twice squares in the sequence of squares and twice squares: A028982(a(n)) = 2*n^2. - Amiram Eldar, Apr 13 2025

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 1..5000
Shiri Artstein-Avidan, Aviezri S. Fraenkel and Vera T. Sós, A two-parameter family of an extension of Beatty sequences, Discrete Math., Vol. 308, No. 20 (2008), pp. 4578-4588; preprint.
Leonard Carlitz, Richard Scoville, and Verner E. Hoggatt, Jr., Pellian representatives, Fib. Quart., Vol. 10, No. 5 (1972), pp. 449-488.
Benoit Cloitre, N. J. A. Sloane, and Matthew J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seq., Vol. 6 (2003), Article 03.2.2; arXiv preprint, arXiv:math/0305308 [math.NT], 2003.
Joshua N. Cooper and Alexander W. N. Riasanovsky, On the Reciprocal of the Binary Generating Function for the Sum of Divisors, J. Int. Seq., Vol. 16 (2013), Article 13.1.8; preprint, 2012.
R. J. Mathar, Graphical representation among sequences closely related to this one (cf. N. J. A. Sloane, "Families of Essentially Identical Sequences").
Luke Schaeffer, Jeffrey Shallit, and Stefan Zorcic, Beatty Sequences for a Quadratic Irrational: Decidability and Applications, arXiv:2402.08331 [math.NT], 2024. See pp. 17-18.
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence).
Index entries for sequences related to Beatty sequences.

Complement of A003152.
Equals A001951(n) + n.
Cf. A028982, A109250, A317204.
The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A003151 as the parent: A003151, A001951, A001952, A003152, A006337, A080763, A082844 (conjectured), A097509, A159684, A188037, A245219 (conjectured), A276862. - N. J. A. Sloane, Mar 09 2021
Bisections: A197878, A215247.
Cf. A001954, A356135, A184922, A341239.

Table[Floor[n*(1 + Sqrt[2])], {n, 1, 50}] (* G. C. Greubel, Jul 02 2017 *)

for(n=1,50, print1(floor(n*(1 + sqrt(2))), ", ")) \\ G. C. Greubel, Jul 02 2017

from math import isqrt
def A003151(n): return n+isqrt(n*n<<1) # Chai Wah Wu, Aug 03 2022

a(1) = 2; for n>1, a(n+1) = a(n)+3 if n is already in the sequence, a(n+1) = a(n)+2 otherwise.

A341239 a(n) = floor(rfloor(sn)), where r = 1 + sqrt(2) and s = sqrt(2).

Original entry on oeis.org

2, 4, 9, 12, 16, 19, 21, 26, 28, 33, 36, 38, 43, 45, 50, 53, 57, 60, 62, 67, 70, 74, 77, 79, 84, 86, 91, 94, 98, 101, 103, 108, 111, 115, 118, 120, 125, 127, 132, 135, 137, 142, 144, 149, 152, 156, 159, 161, 166, 168, 173, 176, 178, 183, 185, 190, 193, 197

Offset: 1

Clark Kimberling, Feb 07 2021

nonn

Conjecture: 1 < r*s*n - a(n) < 3 for n >= 1.
From Clark Kimberling, Dec 27 2022: (Start)
This is the first of four sequences that partition the positive integers. Starting with a general overview, suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers. Let u' and v' be their (increasing) complements, and consider these four sequences:
(1) v o u, defined by (v o u)(n) = v(u(n));
(2) u o v';
(3) v o u';
(4) v' o u'.
Every positive integer is in exactly one of the four sequences. For the reverse composites, u o v, u o v', u' o v, u' o v', see A184922.
Assume that if w is any of the sequences u, v, u', v', then lim_{n->oo} w(n)/n exists and defines the (limiting) density of w. For w = u,v,u',v', denote the densities by r,s,r',s'. Then the densities of sequences (1)-(4) exist, and
1/(r*r') + 1/(r*s') + 1/(s*s') + 1/(s*r') = 1.
For A341239, u, v, u', v', are the Beatty sequences given by u(n) = floor(n*sqrt(2)) and v(n) = floor((1+sqrt(2))/2)*n), so that r = sqrt(2), s = (1+sqrt(2))/2, r' = (2+sqrt(2))/2, s' = 1 + 1/sqrt(2).
(1) v o u = (2, 4, 9, 12, 16, 19, 21, 26, 28, 33, 36, 38, ...) = A341239
(2) v' o u = (1, 3, 6, 8, 11, 13, 15, 18, 20, 23, 25, 27, ...) = A286666
(3) v o u' = (7, 14, 24, 31, 41, 48, 55, 65, 72, 82, 89, ...) = A188383
(4) v' o u' = (5, 10, 17, 22, 29, 34, 39, 46, 51, 58, 63, ...) = A098021
(End)

Cf. A184922, A341240; A098021, A188383, A341239, A286666.

z = 140; r = 1 + Sqrt[2]; s = Sqrt[2]; f[x_] := Floor[r*Floor[s*x]];
Table[f[n], {n, 1, z}]

a(n) = floor(r*floor(s*n)), where r = 1 + sqrt(2) and s = sqrt(2).

A184920 a(n) = n+[sn/r]+[tn/r]+[u*n/r], where []=floor and r=2^(1/2), s=r+1, t=r+2, u=r+3.

Original entry on oeis.org

7, 15, 24, 31, 40, 48, 55, 64, 73, 82, 89, 97, 106, 113, 122, 130, 140, 147, 155, 164, 171, 180, 188, 195, 205, 213, 222, 229, 238, 246, 253, 262, 271, 280, 287, 295, 304, 311, 320, 328, 335, 345, 353, 362, 369, 378, 386, 393, 402, 411, 420, 427, 435, 444, 451, 460, 468, 478, 485, 493, 502, 509, 518, 526, 533, 543, 551, 560, 567, 575, 584, 591, 600, 608, 618, 625, 633, 642, 649, 658, 666, 673, 683, 691, 700, 707, 716, 724, 731, 740, 749, 758, 765, 773, 782, 789, 798, 806, 816, 823, 831, 840, 847, 856, 864, 871, 880, 889, 898, 905, 913, 922, 929, 938, 946, 956, 963, 971, 980, 987

Offset: 1

Clark Kimberling, Jan 26 2011

nonn

The sequences A184920-A184923 partition the positive integers:
  A184920: 7,15,24,31,40,48,55,64,...
  A184921: 3,8,13,18,23,27,32,37,...
  A184922: 2,5,9,12,16,19,22,26,29,...
  A184923: 1,4,6,10,11,14,17,20,21,...
Jointly rank the sets {h*r}, {i*s}, {j*t}, {k*u}, where h>=1, i>=1, j>=1, k>=1.
The position of n*r in the joint ranking is n+[s*n/r]+[t*n/r]+[u*n/r], and likewise for the positions of n*s, n*t, and n*u.

Cf. A184912, A184921, A184922, A184923.

r=2^(1/2); s=r+1; t=r+2; u=r+3;
a[n_]:=n+Floor[n*s/r]+Floor[n*t/r]+Floor[n*u/r];
b[n_]:=n+Floor[n*r/s]+Floor[n*t/s]+Floor[n*u/s];
c[n_]:=n+Floor[n*r/t]+Floor[n*s/t]+Floor[n*u/t];
d[n_]:=n+Floor[n*r/u]+Floor[n*s/u]+Floor[n*t/u];
Table[a[n],{n,1,120}]  (* A184920 *)
Table[b[n],{n,1,120}]  (* A184921 *)
Table[c[n],{n,1,120}]  (* A184922 *)
Table[d[n],{n,1,120}]  (* A184923 *)

A341240 a(n) = 4a(n-1) - 2a(n-2) + a(n-3) - 4a(n-4) + 2a(n-5) for n >= 7, where a(1) = 1, a(2) = 2, a(3) = 4, a(4) = 12, a(5) = 38, a(6) = 127.

Original entry on oeis.org

1, 2, 4, 12, 38, 127, 432, 1472, 5023, 17148, 58544, 199879, 682428, 2329952, 7954951, 27159900, 92729696, 316598983, 1080936540, 3690548192, 12600319687, 43020182364, 146880090080, 501479995591, 1712159802204, 5845679217632, 19958397266119, 68142230629212

Offset: 1

Clark Kimberling, Feb 07 2021

Index entries for linear recurrences with constant coefficients, signature (4,-2,1,-4,2).

Cf. A184922, A339828, A341239.

z = 50; r = 1 + Sqrt[2]; s = Sqrt[2]; f[x_] := Floor[r*Floor[s*x]];
Table[f[n], {n, 1, z}] (* A341239 *)
a[1] = 1; a[n_] := f[a[n - 1]];
Table[a[n], {n, 1, z}] (* A341240 *)
LinearRecurrence[{4,-2,1,-4,2},{1,2,4,12,38,127},30] (* Harvey P. Dale, Jun 14 2022 *)

Let f(n) = floor(r*floor(s*n)) = A184922(n), where r = 1 + sqrt(2) and s = sqrt(2). Let a(1) = 1. Then a(n) = f(a(n-1)) for n >= 2.

G.f.: x*(1 - 2*x - 2*x^2 - x^3 + x^5)/(1 - 4*x + 2*x^2 - x^3 + 4*x^4 - 2*x^5). - Stefano Spezia, Feb 11 2021

A339828 a(n) = 4a(n-1) - 2a(n-2) + a(n-3) - 4*a(n-4) + a(n-5) for n >= 6, where a(1) = 1, a(2) = 3, a(3) = 5, a(4) = 16, a(5) = 53.

Original entry on oeis.org

1, 2, 5, 16, 53, 179, 610, 2081, 7103, 24250, 82793, 282671, 965098, 3295049, 11249999, 38409898, 131139593, 447738575, 1528675114, 5219223305, 17819542991, 60839725354, 207719815433, 709199811023, 2421359613226, 8267038830857, 28225436096975, 96367666726186

Offset: 1

Clark Kimberling, Feb 07 2021

nonn

Index entries for linear recurrences with constant coefficients, signature (4,-2,1,-4,2).

Cf. A184922, A341239, A341240.

z = 40; r = Sqrt[2]; s = 1 + Sqrt[2]; f[x_] := Floor[r*Floor[s*x]];
Table[f[n], {n, 1, z}] m (* A184922 *)
a[1] = 1; a[n_] := f[a[n - 1]]; Table[a[n], {n, 1, z}] (* A339828 *)

Let f(n) = floor(r*floor(s*n)) = A184922(n), where r = sqrt(2) and s = r + 1. Let a(1) = 1. Then a(n) = f(a(n-1)) for n >= 2.
Also, a(n) = 4*a(n-1) - 2*a(n-2) + a(n-3) - 4*a(n-4) + a(n-5) for n >= 6, where a(1) = 1, a(2) = 3, a(3) = 5, a(4) = 16, a(5) = 53.
G.f.: x*(-x^4 + x^3 + x^2 + 2*x - 1)/((x - 1)*(x^2 + x + 1)*(2*x^2 - 4*x + 1)). - Chai Wah Wu, Feb 15 2021

A184921 a(n) = n+[rn/s]+[tn/s]+[u*n/s], where []=floor and r=2^(1/2), s=r+1, t=r+2, u=r+3.

Original entry on oeis.org

3, 8, 13, 18, 23, 27, 32, 37, 42, 47, 52, 56, 61, 66, 71, 76, 81, 85, 90, 95, 100, 105, 110, 114, 119, 124, 129, 134, 139, 143, 148, 153, 158, 163, 167, 172, 177, 182, 187, 192, 196, 201, 206, 211, 216, 221, 225, 230, 235, 240, 245, 250, 254, 259, 264, 269, 274, 279, 283, 288, 293, 298, 303, 308, 312, 317, 322, 327, 332, 336, 341, 346, 351, 356, 361, 365, 370, 375, 380, 385, 390, 394, 399, 404, 409, 414, 419, 423, 428, 433, 438, 443, 448, 452, 457, 462, 467, 472, 477, 481, 486, 491, 496, 501, 505, 510, 515, 520, 525, 530, 534, 539, 544, 549, 554, 559, 563, 568, 573, 578

Offset: 1

Clark Kimberling, Jan 26 2011

nonn

The sequences A184920-A184923 partition the positive integers:
  A184920: 7,15,24,31,40,48,55,64,...
  A184921: 3,8,13,18,23,27,32,37,...
  A184922: 2,5,9,12,16,19,22,26,29,...
  A184923: 1,4,6,10,11,14,17,20,21,...
Jointly rank the sets {h*r}, {i*s}, {j*t}, {k*u}, where h>=1, i>=1, j>=1, k>=1.
The position of n*s in the joint ranking is n+[r*n/s]+[t*n/s]+[u*n/s], and likewise for the positions of n*r, n*t, and n*u.

Cf. A184912, A184920, A184922, A184923.

Mathematica
```
(* See A184920 *)
```

A184923 a(n) = n+[rn/u]+[sn/u]+[t*n/u], where []=floor and r=2^(1/2), s=r+1, t=r+2, u=r+3.

Original entry on oeis.org

1, 4, 6, 10, 11, 14, 17, 20, 21, 25, 28, 30, 34, 35, 38, 41, 44, 45, 49, 51, 54, 58, 59, 62, 65, 68, 69, 72, 75, 78, 79, 83, 86, 88, 92, 93, 96, 99, 102, 103, 107, 109, 112, 116, 117, 120, 123, 126, 127, 131, 133, 136, 137, 141, 144, 146, 150, 151, 154, 157, 160, 161, 165, 168, 170, 174, 175, 178, 181, 184, 185, 189, 191, 194, 198, 199, 202, 204, 208, 209, 212, 215, 218, 219, 223, 226, 228, 232, 233, 236, 239, 242, 243, 247, 249, 252, 256, 257, 260, 263, 266, 267, 270, 273, 276, 277, 281, 284, 286, 290, 291, 294, 297, 300, 301, 305, 307, 310, 314, 315

Offset: 1

Clark Kimberling, Jan 26 2011

nonn

The sequences A184920-A184923 partition the positive integers:
  A184920: 7,15,24,31,40,48,55,64,...
  A184921: 3,8,13,18,23,27,32,37,...
  A184922: 2,5,9,12,16,19,22,26,29,...
  A184923: 1,4,6,10,11,14,17,20,21,...
Jointly rank the sets {h*r}, {i*s}, {j*t}, {k*u}, where h>=1, i>=1, j>=1, k>=1.
The position of n*u in the joint ranking is n+[r*n/u]+[s*n/u]+[t*n/u], and likewise for the positions of n*s, n*t, and n*u.

Cf. A184912, A184920, A184921, A184922.

Mathematica
```
(* See A184920 *)
```

A356219 Intersection of A001952 and A003151.

Original entry on oeis.org

284, 287, 289, 292, 294, 296, 299, 301, 304, 306, 309, 311, 313, 316, 318, 321, 323, 325, 328, 330, 333, 335, 337, 340, 342, 345, 347, 350, 352, 354, 357, 359, 362, 364, 366, 369, 371, 374, 376, 379, 381, 383, 386, 388, 391, 393, 395, 398, 400

Offset: 1

Clark Kimberling, Nov 13 2022

This is the third of four sequences that partition the positive integers. Starting with a general overview, suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers. Let u' and v' be their complements, and assume that the following four sequences are infinite:
(1)  u ^ v = intersection of u and v (in increasing order);
(2)  u ^ v';
(3)  u' ^ v;
(4)  u' ^ v'.
Every positive integer is in exactly one of the four sequences.
Assume that if w is any of the sequences u, v, u', v', then lim_{n->oo} w(n)/n exists and defines the (limiting) density of w. For w = u,v,u',v', denote the densities by r,s,r',s'. Then the densities of sequences (1)-(4) exist, and 1/(r*r') + 1/(r*s') + 1/(s*s') + 1/(s*r') = 1.
For A356219, u, v, u', v', are the Beatty sequences given by u(n) = floor(n*sqrt(2)) and v(n) = floor(((1+sqrt(2))/2)*n), so that r = sqrt(2), s = (1+sqrt(2))/2, r' = (2+sqrt(2))/2, s' = 1 + 1/sqrt(2).

			(1)  u ^ v = (2, 4, 7, 9, 12, 14, 16, 19, 21, 24, 26, 28, 31, 33, ...) = A003151.
(2)  u ^ v' = (1, 5, 8, 11, 15, 18, 22, 25, 29, 32, 35, 39, 42, ...) = A001954.
(3)  u' ^ v = (284, 287, 289, 292, 294, 296, 299, 301, 304, 306, ...) = A356219.
(4)  u' ^ v' = (3, 6, 10, 13, 17, 20, 23, 27, 30, 34, 37, 40, 44, ...) = A003152.

Cf. A001951, A001952, A003151, A003152, A001954, A184922 (results of compositions instead of intersections), A341239 (reversed compositions).

z = 200;
r = Sqrt[2]; u = Table[Floor[n*r], {n, 1, z}]  (* A001951 *)
u1 = Take[Complement[Range[1000], u], z]  (* A001952 *)
r1 = 1 + Sqrt[2]; v = Table[Floor[n*r1], {n, 1, z}]  (* A003151 *)
v1 = Take[Complement[Range[1000], v], z]  (* A003152 *)
t1 = Intersection[u, v]    (* A003151 *)
t2 = Intersection[u, v1]   (* A001954 *)
t3 = Intersection[u1, v]   (* A356219 *)
t4 = Intersection[u1, v1]  (* A001952 *)

A359351 a(n) = A001952(A003151(n)).

Original entry on oeis.org

6, 13, 23, 30, 40, 47, 54, 64, 71, 81, 88, 95, 105, 112, 122, 129, 139, 146, 153, 163, 170, 180, 187, 194, 204, 211, 221, 228, 238, 245, 252, 262, 269, 279, 286, 293, 303, 310, 320, 327, 334, 344, 351, 361, 368, 378, 385, 392, 402, 409, 419, 426, 433, 443

Offset: 1

Clark Kimberling, Dec 27 2022

This is the third of four sequences that partition the positive integers. Starting with a general overview, suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers.   For details, see A184922.
(1)  u o v = (2, 5, 9, 12, 16, 19, 22, 26, 29, 33, 36, 39, 43, ...) = A184922
(2)  u o v' = (1, 4, 7, 8, 11, 14, 15, 18, 21, 24, 25, 28, 31, ...) = A188376
(3)  u' o v = (6, 13, 23, 30, 40, 47, 54, 64, 71, 81, 88, 95, ...) = A359351
(4)  u' o v' = (3, 10, 17, 20, 27, 34, 37, 44, 51, 58, 61, 68, ...) = A188396

Cf. A001951, A001952, A003151 (intersections instead of the rersults of composition), A003152, A184922, A188376, A356136, A188396, A341239 (results of reversed composition).

z = 1200; zz = 150;
u = Table[Floor[n Sqrt[2]], {n, 1, z}];
u1 = Complement[Range[Max[u]], u];
v = Table[Floor[n (1 + Sqrt[2])], {n, 1, z}];
v1 = Complement[Range[Max[v]], v];
Table[u[[v[[n]]]], {n, 1, zz}];   (* A184922 *)
Table[u[[v1[[n]]]], {n, 1, zz}];  (* A188376 *)
Table[u1[[v[[n]]]], {n, 1, zz}];  (* A359351 *)
Table[u1[[v1[[n]]]], {n, 1, zz}]; (* A188396 *)

cp's OEIS Frontend

A003151 Beatty sequence for 1+sqrt(2); a(n) = floor(n*(1+sqrt(2))).

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A341239 a(n) = floor(r*floor(s*n)), where r = 1 + sqrt(2) and s = sqrt(2).

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A184920 a(n) = n+[s*n/r]+[t*n/r]+[u*n/r], where []=floor and r=2^(1/2), s=r+1, t=r+2, u=r+3.

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A341240 a(n) = 4*a(n-1) - 2*a(n-2) + a(n-3) - 4*a(n-4) + 2*a(n-5) for n >= 7, where a(1) = 1, a(2) = 2, a(3) = 4, a(4) = 12, a(5) = 38, a(6) = 127.

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A339828 a(n) = 4*a(n-1) - 2*a(n-2) + a(n-3) - 4*a(n-4) + a(n-5) for n >= 6, where a(1) = 1, a(2) = 3, a(3) = 5, a(4) = 16, a(5) = 53.

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A184921 a(n) = n+[r*n/s]+[t*n/s]+[u*n/s], where []=floor and r=2^(1/2), s=r+1, t=r+2, u=r+3.

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A184923 a(n) = n+[r*n/u]+[s*n/u]+[t*n/u], where []=floor and r=2^(1/2), s=r+1, t=r+2, u=r+3.

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A356219 Intersection of A001952 and A003151.

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A359351 a(n) = A001952(A003151(n)).

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A341239 a(n) = floor(rfloor(sn)), where r = 1 + sqrt(2) and s = sqrt(2).

A184920 a(n) = n+[sn/r]+[tn/r]+[u*n/r], where []=floor and r=2^(1/2), s=r+1, t=r+2, u=r+3.

A341240 a(n) = 4a(n-1) - 2a(n-2) + a(n-3) - 4a(n-4) + 2a(n-5) for n >= 7, where a(1) = 1, a(2) = 2, a(3) = 4, a(4) = 12, a(5) = 38, a(6) = 127.

A339828 a(n) = 4a(n-1) - 2a(n-2) + a(n-3) - 4*a(n-4) + a(n-5) for n >= 6, where a(1) = 1, a(2) = 3, a(3) = 5, a(4) = 16, a(5) = 53.

A184921 a(n) = n+[rn/s]+[tn/s]+[u*n/s], where []=floor and r=2^(1/2), s=r+1, t=r+2, u=r+3.

A184923 a(n) = n+[rn/u]+[sn/u]+[t*n/u], where []=floor and r=2^(1/2), s=r+1, t=r+2, u=r+3.