A317204 -id:A317204 - cp's OEIS frontend

A265744 a(n) is the number of Pell numbers (A000129) needed to sum to n using the greedy algorithm (A317204).

0, 1, 1, 2, 2, 1, 2, 2, 3, 3, 2, 3, 1, 2, 2, 3, 3, 2, 3, 3, 4, 4, 3, 4, 2, 3, 3, 4, 4, 1, 2, 2, 3, 3, 2, 3, 3, 4, 4, 3, 4, 2, 3, 3, 4, 4, 3, 4, 4, 5, 5, 4, 5, 3, 4, 4, 5, 5, 2, 3, 3, 4, 4, 3, 4, 4, 5, 5, 4, 5, 1, 2, 2, 3, 3, 2, 3, 3, 4, 4, 3, 4, 2, 3, 3, 4, 4, 3, 4, 4, 5, 5, 4, 5, 3, 4, 4, 5, 5, 2, 3, 3, 4, 4, 3, 4, 4, 5, 5, 4, 5, 3

Offset: 0

Antti Karttunen, Dec 17 2015

a(0) = 0, because no numbers are needed to form an empty sum, which is zero.

It would be nice to know for sure whether this sequence also gives the least number of Pell numbers that add to n, i.e., that there cannot be even better nongreedy solutions.

A. F. Horadam, Zeckendorf representations of positive and negative integers by Pell numbers, Applications of Fibonacci Numbers, Springer, Dordrecht, 1993, pp. 305-316.

Antti Karttunen, Table of n, a(n) for n = 0..13860
L. Carlitz, Richard Scoville, and V. E. Hoggatt, Jr., Pellian Representations, The Fibonacci Quarterly, Vol. 10, No. 5 (1972), pp. 449-488.

Cf. A000129, A007953, A317204.
Cf. also A014420, A053610, A265404, A265743, A265745.
Similar sequences: A007895, A116543, A278043.

pell[1] = 1; pell[2] = 2; pell[n_] := pell[n] = 2*pell[n - 1] + pell[n - 2]; a[n_] := Module[{s = {}, m = n, k}, While[m > 0, k = 1; While[pell[k] <= m, k++]; k--; AppendTo[s, k]; m -= pell[k]; k = 1]; Plus @@ IntegerDigits[Total[3^(s - 1)], 3]]; Array[a, 100, 0] (* Amiram Eldar, Mar 12 2022 *)

a(n) = A007953(A317204(n)). - Amiram Eldar, Mar 12 2022

A352320 Pell-Niven numbers: numbers that are divisible by the sum of the digits in their minimal (or greedy) representation in terms of the Pell numbers (A317204).

Original entry on oeis.org

1, 2, 4, 5, 6, 9, 10, 12, 14, 15, 18, 20, 24, 28, 29, 30, 33, 34, 36, 39, 40, 42, 44, 48, 50, 58, 60, 63, 64, 68, 70, 72, 82, 84, 87, 88, 90, 92, 96, 110, 111, 112, 115, 116, 120, 125, 126, 135, 140, 141, 144, 155, 164, 165, 168, 169, 170, 174, 180, 183, 184, 186

Offset: 1

Amiram Eldar, Mar 12 2022

Numbers k such that A265744(k) | k.

All the positive Pell numbers (A000129) are terms.

			6 is a term since its minimal Pell representation, A317204(6) = 101, has A265744(6) = 2 1's and 6 is divisible by 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000129, A265744, A317204.
Subsequences: A352321, A352322.
Similar sequences: A005349, A049445, A064150, A064438, A064481, A118363, A328208, A328212, A331085, A333426, A342726, A334308, A331728, A342426, A344341, A351714, A351719, A352089, A352107.

pell[1] = 1; pell[2] = 2; pell[n_] := pell[n] = 2*pell[n - 1] + pell[n - 2]; q[n_] := Module[{s = {}, m = n, k}, While[m > 0, k = 1; While[pell[k] <= m, k++]; k--; AppendTo[s, k]; m -= pell[k]; k = 1]; Divisible[n, Plus @@ IntegerDigits[ Total[3^(s - 1)], 3]]]; Select[Range[200], q]

A352319 Numbers whose minimal (or greedy) Pell representation (A317204) is palindromic.

Original entry on oeis.org

0, 1, 3, 6, 8, 13, 20, 30, 35, 40, 44, 49, 71, 88, 102, 119, 170, 182, 194, 204, 216, 238, 242, 254, 266, 276, 288, 409, 450, 484, 525, 559, 580, 621, 655, 696, 986, 1015, 1044, 1068, 1097, 1150, 1160, 1189, 1218, 1242, 1271, 1334, 1363, 1392, 1396, 1425, 1454

Offset: 1

Amiram Eldar, Mar 12 2022

A052937(n) = A000129(n+1)+1 is a term for n>0, since its minimal Pell representation is 10...01 with n-1 0's between two 1's.
A048739 is a subsequence since these are repunit numbers in the minimal Pell representation.
A001109 is a subsequence. The minimal Pell representation of A001109(n), for n>1, is 1010...01, with n-1 0's interleaved with n 1's.

			The first 10 terms are:
   n  a(n)  A317204(a(n))
  --  ----  -------------
   1     0              0
   2     1              1
   3     3             11
   4     6            101
   5     8            111
   6    13           1001
   7    20           1111
   8    30          10001
   9    35          10101
  10    40          10201

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000129, A317204.
Subsequences: A001109, A048739, A052937 \ {2}.
Similar sequences: A002113, A006995, A014190, A094202, A331191, A351712, A351717, A352087.

pell[1] = 1; pell[2] = 2; pell[n_] := pell[n] = 2*pell[n - 1] + pell[n - 2]; q[n_] := Module[{s = {}, m = n, k}, While[m > 0, k = 1; While[pell[k] <= m, k++]; k--; AppendTo[s, k]; m -= pell[k]; k = 1]; PalindromeQ[IntegerDigits[Total[3^(s - 1)], 3]]]; Select[Range[0, 1500], q]

A352427 a(n) is the number of trailing 0's in the minimal representation of n in terms of the positive Pell numbers (A317204).

Original entry on oeis.org

0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 0, 3, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 0, 3, 0, 1, 0, 1, 4, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 0, 3, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 0, 3, 0, 1, 0, 1, 4, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 0, 5, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 0, 3, 0, 1, 0, 1

Offset: 1

Amiram Eldar, Mar 16 2022

The asymptotic density of the occurrences of 0 is sqrt(2)-1 and of the occurrences of k = 1, 2, ... is 2*(sqrt(2)-1)^(k+1).

The asymptotic mean of this sequence is 1 and its asymptotic variance is sqrt(2).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000129, A003151, A003152, A286666, A286667, A317204.

Similar sequences: A003849 (dual Zeckendorf), A035614 (Zeckendorf), A230403 (factorial), A276084 (primorial), A278045 (tribonacci).

pell[1] = 1; pell[2] = 2; pell[n_] := pell[n] = 2*pell[n - 1] + pell[n - 2]; a[n_] := Module[{s = {}, m = n, k}, While[m > 0, k = 1; While[pell[k] <= m, k++]; k--; AppendTo[s, k]; m -= pell[k]; k = 1]; IntegerExponent[Total[3^(s - 1)], 3]]; Array[a, 100]

a(A000129(n)) = n-1 for n>=1.
a(n) = 0 if and only if n is in A286666.
a(n) > 0 if and only if n is in A286667.
a(n) == 0 (mod 2) if and only if n is in A003152.
a(n) == 1 (mod 2) if and only if n is in A003151.

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

Original entry on oeis.org

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.

A003152 A Beatty sequence: a(n) = floor(n*(1+1/sqrt(2))).

Original entry on oeis.org

1, 3, 5, 6, 8, 10, 11, 13, 15, 17, 18, 20, 22, 23, 25, 27, 29, 30, 32, 34, 35, 37, 39, 40, 42, 44, 46, 47, 49, 51, 52, 54, 56, 58, 59, 61, 63, 64, 66, 68, 69, 71, 73, 75, 76, 78, 80, 81, 83, 85, 87, 88, 90, 92, 93, 95, 97, 99, 100, 102, 104, 105, 107, 109, 110, 112, 114, 116

Offset: 1

N. J. A. Sloane

Numbers with an even number of trailing 0's in their minimal representation in terms of the positive Pell numbers (A317204). - Amiram Eldar, Mar 16 2022

The indices of the squares in the sequence of squares and twice squares: A028982(a(n)) = 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..10000
Leonard Carlitz, Richard Scoville, and Verner E. Hoggatt, Jr., Pellian representations, Fibonacci Quarterly, Vol. 10, No. 5 (1972), pp. 449-488.
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.
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 A003151.
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: A001952, A001954.

[Floor(n*(1+1/Sqrt(2))): n in [1..70]]; // Vincenzo Librandi, Dec 26 2015

Digits := 100: t := evalf(1+sin(Pi/4)): A:= n->floor(t*n): seq(floor((t*n)),n=1..68); # Zerinvary Lajos, Mar 27 2009

Table[Floor[n (1 + 1/Sqrt[2])], {n, 70}] (* Vincenzo Librandi, Dec 26 2015 *)

a(n)=n+sqrtint(2*n^2)\2 \\ Charles R Greathouse IV, Jan 25 2022

from math import isqrt
def A003152(n): return n+isqrt(n**2>>1) # Chai Wah Wu, May 24 2025

A352321 Numbers k such that k and k+1 are both Pell-Niven numbers (A352320).

Original entry on oeis.org

1, 4, 5, 9, 14, 28, 29, 33, 39, 63, 87, 110, 111, 115, 125, 140, 164, 168, 169, 183, 255, 275, 308, 338, 410, 444, 483, 507, 564, 579, 584, 704, 791, 984, 985, 999, 1004, 1024, 1025, 1115, 1134, 1154, 1164, 1211, 1265, 1308, 1323, 1351, 1395, 1415, 1424, 1491

Offset: 1

Amiram Eldar, Mar 12 2022

All the odd-indexed Pell numbers (A001653) are terms.

			4 is a term since 4 and 5 are both Pell-Niven numbers: the minimal Pell representation of 4, A317204(20) = 20, has the sum of digits 2+0 = 2 and 4 is divisible by 2, and the minimal Pell representation of 5, A317204(5) = 100, has the sum of digits 1+0+0 = 1 and 5 is divisible by 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000129, A265744, A317204.
Subsequence of A352320.
Subsequences: A001653, A352322.
Similar sequences: A330927, A328205, A328209, A328213, A330931, A331086, A333427, A334309, A331820, A342427, A344342, A351715, A351720, A352090, A352108.

pell[1] = 1; pell[2] = 2; pell[n_] := pell[n] = 2*pell[n - 1] + pell[n - 2]; q[n_] := Module[{s = {}, m = n, k}, While[m > 0, k = 1; While[pell[k] <= m, k++]; k--; AppendTo[s, k]; m -= pell[k]; k = 1]; Divisible[n, Plus @@ IntegerDigits[ Total[3^(s - 1)], 3]]]; Select[Range[1500], q[#] && q[#+1] &]

A352322 Starts of runs of 3 consecutive Pell-Niven numbers (A352320).

Original entry on oeis.org

4, 28, 110, 168, 984, 1024, 3123, 3514, 5740, 6783, 6923, 8584, 12664, 16744, 18160, 19670, 23190, 23470, 24030, 34503, 34643, 36304, 40384, 45880, 47390, 50910, 51190, 51750, 57607, 61640, 68104, 73600, 78403, 78630, 78910, 79470, 86674, 89360, 95824, 101320

Offset: 1

Amiram Eldar, Mar 12 2022

Conjecture: There are no runs of 4 consecutive Pell-Niven numbers (checked up to 2*10^8).

			4 is a term since 4, 5 and 6 are all Pell-Niven numbers: the minimal Pell representation of 4, A317204(20) = 20, has the sum of digits 2+0 = 2 and 4 is divisible by 2, the minimal Pell representation of 5, A317204(5) = 100, has the sum of digits 1+0+0 = 1 and 5 is divisible by 1, and the minimal Pell representation of 6, A317204(6) = 101, has the sum of digits 1+0+1 = 2 and 6 is divisible by 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000129, A265744, A317204.
A182190 \ {0} is a subsequence.
Subsequence of A352320 and A352321.
Similar sequences: A154701, A328206, A328210, A328214, A330932, A331087, A333428, A334310, A331822, A342428, A344343, A351716, A351721, A352091, A352109.

pell[1] = 1; pell[2] = 2; pell[n_] := pell[n] = 2*pell[n - 1] + pell[n - 2]; pellNivenQ[n_] := Module[{s = {}, m = n, k}, While[m > 0, k = 1; While[pell[k] <= m, k++]; k--; AppendTo[s, k]; m -= pell[k]; k = 1]; Divisible[n, Plus @@ IntegerDigits[Total[3^(s - 1)], 3]]]; seq[count_, nConsec_] := Module[{pn = pellNivenQ /@ Range[nConsec], s = {}, c = 0, k = nConsec + 1}, While[c < count, If[And @@ pn, c++; AppendTo[s, k - nConsec]]; pn = Join[Rest[pn], {pellNivenQ[k]}]; k++]; s]; seq[30, 3]

A352339 a(n) is the maximal (or lazy) Pell representation of n using a ternary system of vectors.

Original entry on oeis.org

0, 1, 10, 11, 20, 21, 22, 110, 111, 120, 121, 122, 210, 211, 220, 221, 1020, 1021, 1022, 1110, 1111, 1120, 1121, 1122, 1210, 1211, 1220, 1221, 2020, 2021, 2022, 2110, 2111, 2120, 2121, 2122, 2210, 2211, 2220, 2221, 2222, 10210, 10211, 10220, 10221, 11020, 11021

Offset: 0

Amiram Eldar, Mar 12 2022

There are 2 well-established systems of giving every nonnegative integer a unique representation as a sum of positive Pell numbers (A000129), 1, 2, 5, 12, 29, 70, ...: the minimal (or greedy) representation (A317204) in which any occurrence of the digit 2 is succeeded by a 0 (i.e., if a Pell number appears twice in the sum, its preceding term in the Pell sequence does not appear), and the maximal (or lazy) representation of n in which any occurrence of the digit 0 is succeeded by a 2 (i.e., if a Pell number does not appear in the sum, its preceding term in the Pell sequence appears twice). [Edited by Amiram Eldar and Peter Munn, Oct 04 2022]

			a(5) = 21 since 5 = 2*2 + 1.
a(6) = 22 since 6 = 2*2 + 2.
a(7) = 110 since 7 = 5 + 2.
We read the first term, 0, like the others, as a list of ternary digits. It has no 1s or 2s in it, so 0 here indicates a sum of 0 Pell numbers. This is called an "empty sum" (see Wiki link) and its total is 0. So 0 represents 0. - _Peter Munn_, Oct 04 2022

Amiram Eldar, Table of n, a(n) for n = 0..10000
A. F. Horadam, Maximal representations of positive integers by Pell numbers, The Fibonacci Quarterly, Vol. 32, No. 3 (1994), pp. 240-244.
OEIS Wiki, Empty sum

Cf. A000129, A317204.

Similar sequences: A104326 (Fibonacci), A130311 (Lucas), A352103 (tribonacci).

pell[1] = 1; pell[2] = 2; pell[n_] := pell[n] = 2*pell[n - 1] + pell[n - 2]; pellp[n_] := Module[{s = {}, m = n, k}, While[m > 0, k = 1; While[pell[k] <= m, k++]; k--; AppendTo[s, k]; m -= pell[k]; k = 1]; IntegerDigits[Total[3^(s - 1)], 3]]; a[n_] := Module[{v = pellp[n]}, nv = Length[v]; i = 1; While[i <= nv - 2, If[v[[i]] > 0 && v[[i + 1]] == 0 && v[[i + 2]] < 2, v[[i ;; i + 2]] += {-1, 2, 1}; If[i > 2, i -= 3]]; i++]; i = Position[v, _?(# > 0 &)]; If[i == {}, 0, FromDigits[v[[i[[1, 1]] ;; -1]]]]]; Array[a, 100, 0]

A286666 Positions of 0 in A286665; complement of A286667.

Original entry on oeis.org

1, 3, 6, 8, 11, 13, 15, 18, 20, 23, 25, 27, 30, 32, 35, 37, 40, 42, 44, 47, 49, 52, 54, 56, 59, 61, 64, 66, 69, 71, 73, 76, 78, 81, 83, 85, 88, 90, 93, 95, 97, 100, 102, 105, 107, 110, 112, 114, 117, 119, 122, 124, 126, 129, 131, 134, 136, 139, 141, 143, 146

Offset: 1

Clark Kimberling, May 13 2017

a(n) - a(n-1) is in {2,3} for n>=2. Conjecture: a(n)/n -> 1 + sqrt(2).
This conjecture follows easily from the fact that (a(n)) is a Beatty sequence, see my comments in A286665. - Michel Dekking, Mar 11 2018
Numbers with no trailing 0's in their minimal representation in terms of the positive Pell numbers (A317204). - Amiram Eldar, Mar 16 2022

			As a word, A286665 = 010110101101010110101101..., in which 0 is in positions 1,3,6,8,11,...

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A171588, A286665, A286667, A317204.

s = Nest[Flatten[# /. {0 -> {0, 0, 1}, 1 -> {0}}] &, {0}, 6] (* A171588 *)
w = StringJoin[Map[ToString, s]]
w1 = StringReplace[w, {"0" -> "01"}]
st = ToCharacterCode[w1] - 48 ; (* A286665 *)
p0 = Flatten[Position[st, 0]];  (* A286666 *)
p1 = Flatten[Position[st, 1]];  (* A286667 *)

cp's OEIS Frontend

A265744 a(n) is the number of Pell numbers (A000129) needed to sum to n using the greedy algorithm (A317204).

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A352320 Pell-Niven numbers: numbers that are divisible by the sum of the digits in their minimal (or greedy) representation in terms of the Pell numbers (A317204).

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A352319 Numbers whose minimal (or greedy) Pell representation (A317204) is palindromic.

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A352427 a(n) is the number of trailing 0's in the minimal representation of n in terms of the positive Pell numbers (A317204).

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

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A003152 A Beatty sequence: a(n) = floor(n*(1+1/sqrt(2))).

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A352321 Numbers k such that k and k+1 are both Pell-Niven numbers (A352320).

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A352322 Starts of runs of 3 consecutive Pell-Niven numbers (A352320).