A352103 -id:A352103 - cp's OEIS frontend

A352109 Starts of runs of 3 consecutive lazy-tribonacci-Niven numbers (A352107).

175, 1183, 2259, 5290, 12969, 21130, 51820, 70629, 78090, 79540, 81818, 129648, 160224, 169234, 180908, 228240, 238574, 249494, 278628, 332891, 376335, 383866, 398650, 399644, 454090, 550380, 565200, 683448, 683604, 694274, 728895, 754390, 782110, 809830, 837550

Offset: 1

Amiram Eldar, Mar 05 2022

			175 is a term since 175, 176 and 177 are all divisible by the number of terms in their maximal tribonacci representation:
    k  A352103(k)  A352104(k)  k/A352104(k)
  ---  ----------  ----------  ------------
  175    11111110           7            25
  176    11111111           8            22
  177   100100100           3            59

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

Cf. A352103, A352104.
Subsequence of A352107 and A352108.
A352110 is a subsequence.
Similar sequences: A154701, A328206, A328210, A328214, A330932, A331087, A333428, A334310, A331822, A342428, A344343, A351716, A351721, A352091.

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]]; lazyTriboNivenQ[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 == {}, False, Divisible[n, Total[v[[i[[1, 1]] ;; -1]]]]]]; seq[count, nConsec_] := Module[{tri = lazyTriboNivenQ /@ Range[nConsec], s = {}, c = 0, k = nConsec + 1}, While[c < count, If[And @@ tri, c++; AppendTo[s, k - nConsec]]; tri = Join[Rest[tri], {lazyTriboNivenQ[k]}]; k++]; s]; seq[30, 3]

A352110 Starts of runs of 4 consecutive lazy-tribonacci-Niven numbers (A352107).

Original entry on oeis.org

1081455, 1976895, 2894175, 5886255, 6906912, 15604110, 16588752, 19291479, 20387232, 25919439, 32394942, 34801557, 35654175, 36813582, 36907899, 39117219, 41407392, 43520832, 46181055, 47954499, 52145952, 54524319, 54815397, 56733639, 57775102, 58942959, 59292177

Offset: 1

Amiram Eldar, Mar 05 2022

Conjecture: There are no runs of 5 consecutive lazy-tribonacci-Niven numbers (checked up to 6*10^9).

			1081455 is a term since 1081455, 1081456, 1081457 and 1081458 are all divisible by the number of terms in their maximal tribonacci representation:
        k               A352103(k)   A352104(k)    k/A352104(k)
  -------  -----------------------   ----------    ------------
  1081455  10101011011110110011110           15           72097
  1081456  10101011011110110011111           16           67591
  1081457  10101011011110110100100           13           83189
  1081458  10101011011110110100101           14           77247

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A352103, A352104.
Subsequence of A352107, A352108 and A352109.
Similar sequences: A141769, A328211, A328207, A328215, A330933, A331824, A334311, A342429, A344344, A352092.

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]

A385436 Tribonacci array of the second kind, read by upward antidiagonals.

Original entry on oeis.org

0, 2, 1, 4, 5, 3, 6, 8, 10, 7, 9, 12, 16, 20, 14, 11, 18, 23, 31, 38, 27, 13, 21, 34, 44, 58, 71, 51, 15, 25, 40, 64, 82, 108, 132, 95, 17, 29, 47, 75, 119, 152, 200, 244, 176, 19, 32, 54, 88, 139, 220, 281, 369, 450, 325, 22, 36, 60, 101, 163, 257, 406, 518, 680

Offset: 1

A.H.M. Smeets, Jun 28 2025

The array is, as a sequence, a permutation of the nonnegative integers; however it does not satisfy the conditions for interspersion and dispersion as given by Eric Weisstein's World of Mathematics. However, when all terms are increased by 1, it does satisfy the conditions for interspersion and dispersion!
Rows satisfy the recurrence: T(m,k) = 2*T(m,k-1) - T(m,k-4) for all k>4.
This array belongs to a family of Wythoff like arrays, based on binary number representations like the greedy and lazy Fibonacci number representations (see A035513 and A372501 for arrays), greedy and lazy Narayana number representations (A136189 for the array related to greedy representation).
The array is related to the lazy tribonacci number representation A352103. The first column lists the even numbers, i.e., for wich 0 suffix A352103(T(m,1)). The odd numbers are represented in the columns k > 1: A352103(T(m,k)) = A352103(T(m,1)) + 1^(k-1). Here + stands for concatenation and ^ stands for repeated concatenation.

			Array including some prepended columns (p = 1..4):
  p=4 p=3 p=2 p=1 | k=1 k=2 k=3  k=4  k=5  k=6  k=7   k=8   k=9  k=10
   -2  -1  -1  -1 |   0   1   3    7   14   27   51    95   176   325
   -2  -1   0   0 |   2   5  10   20   38   71  132   244   450   829
   -2   0   0   1 |   4   8  16   31   58  108  200   369   680
   -2   0   1   2 |   6  12  23   44   82  152  281   518
   -1   0   2   4 |   9  18  34   64  119  220  406   748
   -1   1   2   5 |  11  21  40   75  139  257  474   873
   -1   1   3   6 |  13  25  47   88  163  301  555  1022
   -1   1   4   7 |  15  29  54  101  187  345  636  1171
   -1   2   4   8 |  17  32  60  112  207  382  704  1296
   -1   2   5   9 |  19  36  67  125
    0   2   6  11 |  22  42  78  145
Each row of the array satisfies the recurrence relation T(m,k) = 2*T(m,k-1) - T(m,k-4); from this, the prepended columns are obtained by rowwise backward recursion.

A.H.M. Smeets, Table of n, a(n) for n = 1..20100 (first 200 antidiagonals).
Larry Ericksen and Peter G. Anderson, Patterns in differences between rows in k-Zeckendorf arrays, The Fibonacci Quarterly, Vol. 50, No. 1 (February 2012), pp. 11-18.
Clark Kimberling, The Zeckendorf array equals the Wythoff array, Fibonacci Quarterly 33 (1995) 3-8.
Eric Weisstein's World of Mathematics, Interspersion.
Eric Weisstein's World of Mathematics, Sequence Dispersion.

Cf. A035513, A136189, A372501, A027084 (m=1), A351631 (k=1), A352103.

Prepended columns: A385455 (p=1), A385532 (p=2), A385533 (p=3).

def ToDual_111_Zeck(n):
    if n == 0:
        return "0"
    f0, f1, f2, sf = 1, 0, 0, 0
    while n > sf:
        f0, f1, f2 = f0+f1+f2, f0, f1
        sf += f0
    r, s = sf-n, "1"
    while f0 > 1:
        f0, f1, f2 = f1, f2, f0-f1-f2
        r, s = r%f0, s+str(1-r//f0)
    return s
def From_111_Zeck(s):
    f0, f1, f2, i, n = 1, 1, 0, len(s), 0
    while i > 0:
        i -= 1
        f0, f1, f2, n = f0+f1+f2, f0, f1, n+int(s[i])*f0
    return n
d, a, n, c1 = 0, 0, 0, []
while d < 11:
    s = ToDual_111_Zeck(a)
    if s[len(s)-1] == "0": # == even
        n, d = n+1, d+1
        print(a, end = ", ")
        i, c1, p1 = d-1, c1+[s], ""
        while i > 0:
            n, i, p1 = n+1, i-1, p1+"1"
            print(From_111_Zeck(c1[i]+p1), end = ", ")
    a += 1

A351631 The numbers that are not doubled in column -1 of the extended Trithoff (tribonacci) array.

Original entry on oeis.org

0, 2, 4, 6, 9, 11, 13, 15, 17, 19, 22, 24, 26, 28, 30, 33, 35, 37, 39, 41, 43, 46, 48, 50, 53, 55, 57, 59, 61, 63, 66, 68, 70, 72, 74, 77, 79, 81, 83, 85, 87, 90, 92, 94, 96, 98, 100, 103, 105, 107, 109, 111, 114, 116, 118, 120, 122, 124, 127, 129, 131, 134, 136, 138, 140, 142, 144, 147, 149, 151

Offset: 1

Tanya Khovanova and PRIMES STEP Senior group, May 04 2022

nonn

Excluding zeros, these are the indices of letters b and c in the tribonacci word.
The complement of A003144: the indices of the letter a in the tribonacci word. These numbers are doubled in column -1 in the extended Trithoff array.
Also integers with "even" lazy tribonacci representation A352103, and first column of A385436. - A.H.M. Smeets, Jun 29 2025

A.H.M. Smeets, Table of n, a(n) for n = 1..20000

Cf. A000073, A003144, A003145, A003146, A136175, A353083, A353084, A353086, A353090.

def ToDual_111_Zeck(n):
    if n == 0:
        return "0"
    f0, f1, f2, sf = 1, 0, 0, 0
    while n > sf:
        f0, f1, f2 = f0+f1+f2, f0, f1
        sf += f0
    r, s = sf-n, "1"
    while f0 > 1:
        f0, f1, f2 = f1, f2, f0-f1-f2
        r, s = r%f0, s+str(1-r//f0)
    return s
n, a = 0, 0
while n < 70:
    s = ToDual_111_Zeck(a)
    if s[len(s)-1] == "0": # == even
        n += 1
        print(a, end = ", ")
a += 1 # A.H.M. Smeets, Jun 28 2025

A356899 Nonnegative numbers whose minimal and maximal tribonacci representations are the same.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 15, 16, 17, 18, 19, 21, 22, 23, 28, 29, 30, 32, 33, 34, 35, 36, 39, 40, 41, 42, 43, 52, 53, 54, 55, 56, 59, 60, 61, 62, 63, 65, 66, 67, 72, 73, 74, 76, 77, 78, 79, 80, 96, 97, 98, 99, 100, 102, 103, 104, 109, 110, 111, 113

Offset: 1

Amiram Eldar, Sep 03 2022

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

Cf. A278038, A352103.
A089068 is a subsequence.
Similar sequence: A000071 (numbers whose Zeckendorf and dual Zeckendorf representations are the same).

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]];
tribmin[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]; FromDigits@IntegerDigits[Total[2^(s - 1)], 2]];
tribmax[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]]]]];
Select[Range[0, 150], tribmin[#] == tribmax[#] &]

A357121 Irregular triangle T(n, k), n > 0, k = 1..A352104(n); the n-th row contains, in ascending order, the terms in the lazy tribonacci representation of n.

Original entry on oeis.org

1, 2, 1, 2, 4, 1, 4, 2, 4, 1, 2, 4, 1, 7, 2, 7, 1, 2, 7, 4, 7, 1, 4, 7, 2, 4, 7, 1, 2, 4, 7, 2, 13, 1, 2, 13, 4, 13, 1, 4, 13, 2, 4, 13, 1, 2, 4, 13, 1, 7, 13, 2, 7, 13, 1, 2, 7, 13, 4, 7, 13, 1, 4, 7, 13, 2, 4, 7, 13, 1, 2, 4, 7, 13, 4, 24, 1, 4, 24, 2, 4, 24

Offset: 1

Rémy Sigrist, Sep 12 2022

See A357120 for the sequence corresponding to greedy tribonacci representations.

			Triangle T(n, k) begins:
     1: [1]
     2: [2]
     3: [1, 2]
     4: [4]
     5: [1, 4]
     6: [2, 4]
     7: [1, 2, 4]
     8: [1, 7]
     9: [2, 7]
    10: [1, 2, 7]
    11: [4, 7]
    12: [1, 4, 7]
    13: [2, 4, 7]
    14: [1, 2, 4, 7]
    15: [2, 13]

Rémy Sigrist, PARI program
Index entries for sequences related to Zeckendorf expansion of n

Cf. A000073, A080843, A112309, A352103, A352104, A357120.

PARI
```
See Links section.
```

T(n, 1) = 2^A080843(n-1).

A352106 Numbers whose binary and maximal tribonacci representations are both palindromic.

Original entry on oeis.org

0, 1, 3, 5, 7, 27, 51, 325, 2193, 3735, 23709, 35889, 53835, 589833, 1294265, 17291201, 80719769, 1274288105, 23157444917, 23635236877, 230684552043, 1218891196337, 1722894010643, 2544113575977, 93096801594005, 175482093541881, 256924005422487, 372295593308821

Offset: 1

Amiram Eldar, Mar 05 2022

			The first 5 terms are:
   n  a(n)  A007088(a(n))  A352103(a(n))
   -  ----  -------------  -------------
   1     0              0              0
   2     1              1              1
   3     3             11             11
   4     5            101            101
   5     7            111            111
   6    27          11011          11111
   7    51         110011         111111
   8   325      101000101      111111111
   9  2193   100010010001  1001101011001
  10  3735   111010010111  1111111111111

Intersection of A006995 and A352105.
Cf. A007088, A352103.
Similar sequences: A095309, A331193, A331894, A351713, A351718, A352088.

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]]; lazyTribPalQ[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 == {}, True, PalindromeQ[FromDigits[v[[i[[1, 1]] ;; -1]]]]]]; Join[{0}, Select[Range[1, 10^5, 2], PalindromeQ[IntegerDigits[#, 2]] && lazyTribPalQ[#] &]]

cp's OEIS Frontend

A352109 Starts of runs of 3 consecutive lazy-tribonacci-Niven numbers (A352107).

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A352110 Starts of runs of 4 consecutive lazy-tribonacci-Niven numbers (A352107).

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

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A385436 Tribonacci array of the second kind, read by upward antidiagonals.

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A351631 The numbers that are not doubled in column -1 of the extended Trithoff (tribonacci) array.

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A356899 Nonnegative numbers whose minimal and maximal tribonacci representations are the same.

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A357121 Irregular triangle T(n, k), n > 0, k = 1..A352104(n); the n-th row contains, in ascending order, the terms in the lazy tribonacci representation of n.

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A352106 Numbers whose binary and maximal tribonacci representations are both palindromic.

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