A130311 -id:A130311 - cp's OEIS frontend

A351717 Numbers whose maximal (or lazy) Lucas representation (A130311) is palindromic.

0, 2, 3, 5, 6, 10, 12, 14, 17, 20, 28, 30, 34, 36, 42, 46, 56, 61, 75, 77, 85, 92, 94, 101, 107, 115, 122, 128, 150, 166, 176, 198, 200, 211, 219, 233, 244, 246, 260, 271, 277, 288, 296, 310, 321, 345, 360, 396, 405, 441, 469, 484, 520, 522, 544, 562, 570, 588

Offset: 1

Amiram Eldar, Feb 17 2022

A001610(n) = Lucas(n+1) - 1 is a term for all n, since A001610(0) = 0 has the representation 0 and the representation of Lucas(n+1) - 1 is n 1's for n > 0.

			The first 10 terms are:
   n  a(n)  A130311(a(n))
   ----------------------
   1   0               0
   2   2               1
   3   3              11
   4   5             101
   5   6             111
   6  10            1111
   7  12           10101
   8  14           11011
   9  17           11111
  10  20          101101

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for sequences related to palindromes.

Cf. A000032, A001610, A130311.

Similar sequences: A002113, A006995, A014190, A094202, A331191, A351712.

lazy = Select[IntegerDigits[Range[6000], 2], SequenceCount[#, {0, 0}] == 0 &]; t = Total[# * Reverse @ LucasL[Range[0, Length[#] - 1]]] & /@ lazy; s = FromDigits /@ lazy[[TakeWhile[Flatten[FirstPosition[t, #] & /@ Range[Max[t]]], NumberQ]]]; Join[{0}, Position[s, _?PalindromeQ] // Flatten]

A351719 Lazy-Lucas-Niven numbers: numbers divisible by the number of terms in their maximal (or lazy) representation in terms of the Lucas numbers (A130311).

Original entry on oeis.org

1, 2, 4, 6, 9, 12, 16, 20, 25, 40, 42, 54, 60, 66, 78, 84, 91, 96, 104, 112, 120, 126, 144, 154, 161, 168, 175, 176, 180, 182, 184, 192, 203, 210, 216, 217, 224, 232, 234, 240, 243, 264, 270, 280, 288, 304, 306, 310, 315, 320, 322, 328, 336, 344, 350, 360, 378

Offset: 1

Amiram Eldar, Feb 17 2022

Numbers k such that A131343(k) | k.

			6 is a term since its maximal Lucas representation, A130311(6) = 111, has A131343(6) = 3 1's and 6 is divisible by 3.

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

Cf. A000032, A130311, A131343.
Subsequences: A351720, A351721.
Similar sequences: A005349, A049445, A064150, A064438, A064481, A118363, A328208, A328212, A331085, A333426, A342726, A334308, A331728, A342426, A344341, A351714.

lazy = Select[IntegerDigits[Range[3000], 2], SequenceCount[#, {0, 0}] == 0 &]; t = Total[# * Reverse @ LucasL[Range[0, Length[#] - 1]]] & /@ lazy; s = FromDigits /@ lazy[[TakeWhile[Flatten[FirstPosition[t, #] & /@ Range[Max[t]]], NumberQ]]]; Position[Divisible[Range[Length[s]], Plus @@@ IntegerDigits[s]], True] // Flatten

A131343 Number of 1's in maximal Lucas representation (A130311) of n.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 3, 3, 4, 4, 4, 4, 5, 4, 4, 4, 5, 4, 4, 5, 5, 5, 5, 6, 4, 4, 5, 5, 5, 5, 6, 5, 5, 5, 6, 5, 5, 6, 6, 6, 6, 7, 5, 5, 5, 6, 5, 5, 6, 6, 6, 6, 7, 5, 5, 6, 6, 6, 6, 7, 6, 6, 6, 7, 6, 6, 7, 7, 7, 7, 8, 5, 5, 6, 6, 6, 6, 7, 6, 6, 6, 7

Offset: 0

Casey Mongoven, Jun 29 2007

Amiram Eldar, Table of n, a(n) for n = 0..10000
Ron Knott, Using the Fibonacci numbers to represent whole numbers.

Cf. A000032, A007895, A130311, A116543.

lazy = Select[IntegerDigits[Range[400], 2], SequenceCount[#, {0, 0}] == 0 &]; t = Total[# * Reverse@LucasL[Range[0, Length[#] - 1]]] & /@ lazy; Join[{0}, Plus@@@ lazy[[TakeWhile[Flatten[FirstPosition[t, #] & /@ Range[Max[t]]], NumberQ]]]] (* Amiram Eldar, Feb 17 2022 *)

a(n) = A007953(A130311(n)). - Amiram Eldar, Feb 17 2022

a(0) and more terms from Amiram Eldar, Feb 17 2022

A131341 Number of 0's in maximal Lucas representation (A130311) of n.

Original entry on oeis.org

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

Offset: 1

Casey Mongoven, Jun 29 2007, corrected Mar 23 2008

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Ron Knott, Using the Fibonacci numbers to represent whole numbers.

Cf. A131340, A000032, A102364.

lazy = Select[IntegerDigits[Range[400], 2], SequenceCount[#, {0, 0}] == 0 &]; t = Total[#*Reverse@LucasL[Range[0, Length[#] - 1]]] & /@ lazy; Plus @@@ (1 - lazy[[TakeWhile[Flatten[FirstPosition[t, #] & /@ Range[Max[t]]], NumberQ]]]) (* Amiram Eldar, Jul 05 2025 *)

More terms from Amiram Eldar, Jul 05 2025

A133771 Number of runs (of equal bits) in the maximal Lucas binary (A130311) representation of n.

Original entry on oeis.org

2, 1, 1, 2, 3, 1, 3, 2, 3, 1, 4, 5, 3, 3, 2, 3, 1, 5, 4, 5, 3, 4, 5, 3, 3, 2, 3, 1, 6, 7, 5, 5, 4, 5, 3, 5, 4, 5, 3, 4, 5, 3, 3, 2, 3, 1, 7, 6, 7, 5, 6, 7, 5, 5, 4, 5, 3, 6, 7, 5, 5, 4, 5, 3, 5, 4, 5, 3, 4, 5, 3, 3, 2, 3, 1, 8, 9, 7, 7, 6, 7, 5, 7, 6, 7, 5, 6, 7, 5, 5, 4, 5, 3, 7, 6, 7, 5, 6, 7, 5, 5, 4, 5, 3, 6

Offset: 1

Casey Mongoven, Sep 23 2007; corrected Mar 23 2008

nonn

			A130311(19)=101110 because 11+4+3+1=19 (a sum of Lucas numbers); this representation has four runs: 1,0,111,0. So a(19)=4.

Zeckendorf, E., Représentation des nombres naturels par une somme des nombres de Fibonacci ou de nombres de Lucas, Bull. Soc. Roy. Sci. Liège 41, 179-182, 1972.

Ron Knott, Using Powers of Phi to represent Integers.

Cf. A133770, A130311.

The b-file submitted by Casey Mongoven did not match the terms of the sequence, so I have deleted it. Of course it may be that the sequence is wrong and the b-file was correct. Should be rechecked. - N. J. A. Sloane, Nov 10 2010

A352103 a(n) is the maximal (or lazy) tribonacci representation of n using a binary system of vectors not containing three consecutive 0's.

Original entry on oeis.org

0, 1, 10, 11, 100, 101, 110, 111, 1001, 1010, 1011, 1100, 1101, 1110, 1111, 10010, 10011, 10100, 10101, 10110, 10111, 11001, 11010, 11011, 11100, 11101, 11110, 11111, 100100, 100101, 100110, 100111, 101001, 101010, 101011, 101100, 101101, 101110, 101111, 110010

Offset: 0

Amiram Eldar, Mar 05 2022

Each nonnegative integer has 2 unique representations as sums of distinct positive tribonacci numbers (A000073): 1, 2, 4, 7, 13, 24, ...: the minimal (or greedy, A278038) representation in which there are no 3 consecutive 1's (i.e., no 3 consecutive tribonacci numbers appear in the sum), and the maximal (or lazy) representation of n in which no 3 consecutive 0's appear.

			a(5) = 101 = 4 + 1.
a(6) = 110 = 4 + 2.
a(7) = 111 = 4 + 2 + 1.

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A000073, A007088, A003796, A278038.

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

t[1] = 1; t[2] = 2; t[3] = 4; t[n_] := t[n] = t[n - 1] + t[n - 2] + t[n - 3]; trib[n_] := Module[{s = {}, m = n, k}, While[m > 0, k = 1; While[t[k] <= m, k++]; k--; AppendTo[s, k]; m -= t[k]; k = 1]; IntegerDigits[Total[2^(s - 1)], 2]]; a[n_] := Module[{v = trib[n]}, nv = Length[v]; i = 1; While[i <= nv - 3, If[v[[i ;; i + 3]] == {1, 0, 0, 0}, v[[i ;; i + 3]] = {0, 1, 1, 1}; If[i > 3, i -= 4]]; i++]; i = Position[v, _?(# > 0 &)]; If[i == {}, 0, FromDigits[v[[i[[1, 1]] ;; -1]]]]]; Array[a, 100, 0]

a(n) = A007088(A003796(n+1)).

A130310 Minimal (or "greedy") Lucas representation of n, in which L(0) = 2 and L(2) = 3 are not allowed in the same representation (hence the correct representation of the integer 5 is 1010 rather than 101). A binary system of integers with Lucas numbers (A000032) as a base.

Original entry on oeis.org

0, 10, 1, 100, 1000, 1010, 1001, 10000, 10010, 10001, 10100, 100000, 100010, 100001, 100100, 101000, 101010, 101001, 1000000, 1000010, 1000001, 1000100, 1001000, 1001010, 1001001, 1010000, 1010010, 1010001, 1010100, 10000000, 10000010, 10000001, 10000100, 10001000

Offset: 0

Casey Mongoven, May 21 2007

			a(9) = 10001 because 7 + 2 = 9.
a(10) is 10100 because 7 + 3 = 10.

Richard A. Dunlap, The Golden Ratio and Fibonacci Numbers, Singapore, World Scientific, 1997, pp. 73-77.
Edouard Zeckendorf, Représentation des nombres naturels par une somme des nombres de Fibonacci ou de nombres de Lucas, Bull. Soc. Roy. Sci. Liège, Vol. 41 (1972), pp. 179-182.

Amiram Eldar, Table of n, a(n) for n = 0..10000
Ron Knott, Using Powers of Phi to represent Integers.

Cf. A000032, A014417, A116543, A130311.

a[n_] := Module[{s = {}, m = n, k = 1}, While[m > 0, If[m == 1, k = 1; AppendTo[s, k]; m = 0, If[m == 2, k = 0; AppendTo[s, k]; m = 0, While[LucasL[k] <= m, k++]; k--; AppendTo[s, k]; m -= LucasL[k]; k = 1]]]; FromDigits @ IntegerDigits[Total[2^s], 2]]; Array[a, 30, 0] (* Amiram Eldar, Feb 17 2022 *)

Definition corrected by Casey Mongoven, May 29 2010

a(0) and more terms from Amiram Eldar, Feb 17 2022

A351720 Numbers k such that k and k + 1 are both lazy-Lucas-Niven numbers (A351719).

Original entry on oeis.org

1, 175, 216, 399, 656, 729, 737, 759, 1000, 1991, 2716, 2820, 2925, 3970, 4068, 4224, 4499, 4641, 5316, 5819, 6565, 6720, 6902, 7890, 9840, 10751, 11843, 12194, 12614, 13034, 13272, 14909, 15483, 15495, 16029, 17234, 17444, 17731, 18074, 18885, 19305, 19669, 20188

Offset: 1

Amiram Eldar, Feb 17 2022

			175 is a term since 175 and 176 are both lazy-Lucas-Niven numbers: the maximal Lucas representation of 175, A130311(175) = 1110110101, has 7 1's and 175 is divisible by 5, and the maximal Lucas representation of 176, A130311(7) = 1110110111, has 8 1's and 176 is divisible by 8.

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

Cf. A000032, A130311, A131343.
Subsequence of A351719.
A351721 is a subsequence.
Similar sequences: A330927, A328205, A328209, A328213, A330931, A331086, A333427, A334309, A331820, A342427, A344342, A351715.

lazy = Select[IntegerDigits[Range[10^6], 2], SequenceCount[#, {0, 0}] == 0 &]; t = Total[#*Reverse@LucasL[Range[0, Length[#] - 1]]] & /@ lazy; s = FromDigits /@ lazy[[TakeWhile[Flatten[FirstPosition[t, #] & /@ Range[Max[t]]], NumberQ]]]; SequencePosition[Divisible[Range[Length[s]], Plus @@@ IntegerDigits[s]], {True, True}][[;; , 1]]

A351721 Starts of runs of 3 consecutive lazy-Lucas-Niven numbers (A351719).

Original entry on oeis.org

607068, 640618, 665720, 900921, 1000880, 1375940, 1505878, 1537250, 1924224, 1938508, 1966338, 2527998, 3394224, 3935424, 4242624, 4476624, 4747224, 4794624, 5351367, 5401824, 5526024, 5636356, 5992298, 6103900, 6343298, 7028362, 7113024, 8879424, 8998262, 9431424

Offset: 1

Amiram Eldar, Feb 17 2022

Conjecture: There are no runs of 4 consecutive lazy-Lucas-Niven numbers (checked up to 3*10^9).

			607068 is a term since 607068, 607069 and 607070 are all divisible by the number of terms in their maximal representation:
     k                   A130311(k)  A131343(k)  k/A131343(k)
-------------------------------------------------------------
607068  111010101010101011110111101         18          33726
607069  111010101010101011110111111         19          31951
607070  111010101010101011111010110         17          35710

Amiram Eldar, Table of n, a(n) for n = 1..5000

Cf. A000032, A130311, A131343.
Subsequence of A351719 and A351720.
Similar sequences: A154701, A328206, A328210, A328214, A330932, A331087, A333428, A334310, A331822, A342428, A344343, A351716.

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]

cp's OEIS Frontend

A351717 Numbers whose maximal (or lazy) Lucas representation (A130311) is palindromic.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A351719 Lazy-Lucas-Niven numbers: numbers divisible by the number of terms in their maximal (or lazy) representation in terms of the Lucas numbers (A130311).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A131343 Number of 1's in maximal Lucas representation (A130311) of n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A131341 Number of 0's in maximal Lucas representation (A130311) of n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Extensions

A133771 Number of runs (of equal bits) in the maximal Lucas binary (A130311) representation of n.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Extensions

A352103 a(n) is the maximal (or lazy) tribonacci representation of n using a binary system of vectors not containing three consecutive 0's.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A130310 Minimal (or "greedy") Lucas representation of n, in which L(0) = 2 and L(2) = 3 are not allowed in the same representation (hence the correct representation of the integer 5 is 1010 rather than 101). A binary system of integers with Lucas numbers (A000032) as a base.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

Extensions

A351720 Numbers k such that k and k + 1 are both lazy-Lucas-Niven numbers (A351719).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica