A213816 -id:A213816 - cp's OEIS frontend

A341456 Let T be the set of sequences {t(k), k >= 0} such that for any k >= 3, t(k) = t(k-1) + t(k-2) + t(k-3); a(n) is the least possible value of t(0) + t(1) + t(2) for an element t of T containing n.

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

Offset: 0

Rémy Sigrist, Feb 12 2021

nonn

This sequence is a variant of A249783; here we consider tribonacci-like sequences, there Fibonacci like sequences. The scatterplots of these sequences both present polygonal shapes emerging from the origin.

			The first terms of the elements t of T such that t(0) + t(1) + t(2) <= 2 are:
  t(0)+t(1)+t(2)  t(0)  t(1)  t(2)  t(3)  t(4)  t(5)  t(6)  t(7)  t(8)  t(9)
  --------------  ----  ----  ----  ----  ----  ----  ----  ----  ----  ----
               0     0     0     0     0     0     0     0     0     0     0
               1     0     0     1     1     2     4     7    13    24    44
               1     0     1     0     1     2     3     6    11    20    37
               1     1     0     0     1     1     2     4     7    13    24
               2     0     0     2     2     4     8    14    26    48    88
               2     0     1     1     2     4     7    13    24    44    81
               2     0     2     0     2     4     6    12    22    40    74
               2     1     0     1     2     3     6    11    20    37    68
               2     1     1     0     2     3     5    10    18    33    61
               2     2     0     0     2     2     4     8    14    26    48
- so a(0) = 0,
     a(1) = a(2) = a(3) = a(4) = a(6) = a(7) = a(11) = 1,
     a(5) = = a(8) = a(10) = 2.

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, Scatterplot of the first 10000000 terms
Rémy Sigrist, PARI program for A341456

Cf. A000073, A001590, A213816, A249783.

PARI
```
See Links section.
```

a(n) = 0 iff n = 0.
a(n) = 1 iff n belongs to A213816.
a(n) <= n.

A274759 Modified quadranacci series.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 2, 2, 3, 4, 4, 6, 7, 8, 12, 14, 15, 23, 27, 29, 44, 52, 56, 85, 100, 108, 164, 193, 208, 316, 372, 401, 609, 717, 773, 1174, 1382, 1490, 2263, 2664, 2872, 4362, 5135, 5536, 8408, 9898, 10671, 16207, 19079, 20569, 31240, 36776, 39648, 60217, 70888, 76424, 116072, 136641

Offset: 0

G. C. Greubel, Jul 04 2016

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Ian Bruce, A Modified Tribonacci Sequence, The Fibonacci Quarterly 22, no.3 (1984):244-246
Index entries for linear recurrences with constant coefficients, signature (0,0,1,0,0,1,0,0,1,0,0,1).

Cf. A213816.

CoefficientList[Series[x*(1 + x + x^2 + x^4 + x^5 + x^8)/(1 - x^3 - x^6 - x^9 - x^12), {x, 0, 25}], x] (* or *) LinearRecurrence[{0,0,1,0,0,1,0,0, 1,0,0,1},{0,1,1,1,1,2,2,2,3,4,4,6}, 50]

concat(0, Vec(x*(1+x+x^2+x^4+x^5+x^8)/(1-x^3-x^6-x^9-x^12) + O(x^99))) \\ Altug Alkan, Jul 04 2016

a(3n) = a(3n-3) + a(3n-6) + a(3n-9) + a(3n-12).
a(3n + 2) = a(3n + 1) + a(3n - 2).
a(3n + 3) = a(3n + 1) + a(3n - 1).
a(3n + 4) = a(3n + 1) + a(3n).
G.f.: x*(1 + x + x^2 + x^4 + x^5 + x^8)/(1 - x^3 - x^6 - x^9 - x^12).

A308189 Numbers of the form t_n or t_n + t_{n+1} where {t_n} are the tribonacci numbers A000073.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 11, 13, 20, 24, 37, 44, 68, 81, 125, 149, 230, 274, 423, 504, 778, 927, 1431, 1705, 2632, 3136, 4841, 5768, 8904, 10609, 16377, 19513, 30122, 35890, 55403, 66012, 101902, 121415, 187427, 223317, 344732, 410744, 634061, 755476, 1166220, 1389537, 2145013, 2555757, 3945294, 4700770, 7256527

Offset: 1

N. J. A. Sloane, Jun 09 2019

Orders of squares in the ternary tribonacci word A080843.

This is A213816 with duplicates removed.

Colin Barker, Table of n, a(n) for n = 1..1000
Hamoon Mousavi and Jeffrey Shallit, Mechanical Proofs of Properties of the Tribonacci Word, arXiv:1407.5841 [cs.FL], 2014.
H. Mousavi and J. Shallit, Mechanical Proofs of Properties of the Tribonacci Word, In: Manea F., Nowotka D. (eds) Combinatorics on Words. WORDS 2015. Lecture Notes in Computer Science, vol 9304. Springer, 2015, pp. 170-190.
Index entries for linear recurrences with constant coefficients, signature (0,1,0,1,0,1).

Cf. A000073, A001590, A080843, A092782, A213816.

LinearRecurrence[{0,1,0,1,0,1},{0,1,2,3,4,6,7,11},100] (* Paolo Xausa, Nov 14 2023 *)

concat(0, Vec(x^2*(1 + 2*x + 2*x^2 + 2*x^3 + 2*x^4 + x^5 + x^6) / (1 - x^2 - x^4 - x^6) + O(x^50))) \\ Colin Barker, Jun 11 2019

From Colin Barker, Jun 11 2019: (Start)
G.f.: x^2*(1 + 2*x + 2*x^2 + 2*x^3 + 2*x^4 + x^5 + x^6) / (1 - x^2 - x^4 - x^6).
a(n) = a(n-2) + a(n-4) + a(n-6) for n>8.
(End)

A341474 Let T be the set of sequences {t(k), k >= 0} such that for any k >= 3, t(k) = t(k-1) + t(k-2) + t(k-3); a(n) is the least possible value of t(0)^2 + t(1)^2 + t(2)^2 for an element t of T containing n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 4, 3, 2, 1, 4, 1, 4, 5, 5, 3, 2, 5, 1, 6, 4, 6, 1, 5, 4, 5, 9, 5, 9, 3, 10, 2, 10, 5, 8, 1, 6, 9, 4, 17, 6, 13, 1, 11, 5, 13, 4, 9, 5, 9, 16, 5, 18, 9, 14, 3, 14, 10, 9, 2, 12, 10, 5, 21, 8, 19, 1, 17, 6, 19, 9, 10, 4, 17, 17, 6, 26, 13

Offset: 0

Rémy Sigrist, Feb 13 2021

This sequence is a variant of A286327; here we consider tribonacci-like sequences, there Fibonacci like sequences. The scatterplots of these sequences are similar.

			The first terms of the elements t of T such that t(0)^2 + t(1)^2 + t(2)^2 <= 4 are:
  t(0)^2+t(1)^2+t(3)^2  t(0)  t(1)  t(2)  t(3)  t(4)  t(5)  t(6)  t(7)  t(8)  t(9)
  --------------------  ----  ----  ----  ----  ----  ----  ----  ----  ----  ----
                     0     0     0     0     0     0     0     0     0     0     0
                     1     0     0     1     1     2     4     7    13    24    44
                     1     0     1     0     1     2     3     6    11    20    37
                     1     1     0     0     1     1     2     4     7    13    24
                     2     0     1     1     2     4     7    13    24    44    81
                     2     1     0     1     2     3     6    11    20    37    68
                     2     1     1     0     2     3     5    10    18    33    61
                     3     1     1     1     3     5     9    17    31    57   105
                     4     0     0     2     2     4     8    14    26    48    88
                     4     0     2     0     2     4     6    12    22    40    74
                     4     2     0     0     2     2     4     8    14    26    48
- so a(0) = 0,
     a(1) = a(2) = a(3) = a(4) = a(6) = a(7) = a(11) = 1,
     a(5) = a(10) = a(18) = 2,
     a(9) = a(17) = 3,
     a(8) = a(12) = a(14) = 4.

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, Scatterplot of the first 100000 terms
Rémy Sigrist, Scatterplot of the first 1000000 terms
Rémy Sigrist, PARI program for A341474

Cf. A000073, A001590, A213816, A286327, A341456.

PARI
```
See Links section.
```

a(n) = 0 iff n = 0.
a(n) = 1 iff n belongs to A213816.
a(n) <= n^2.

cp's OEIS Frontend

A341456 Let T be the set of sequences {t(k), k >= 0} such that for any k >= 3, t(k) = t(k-1) + t(k-2) + t(k-3); a(n) is the least possible value of t(0) + t(1) + t(2) for an element t of T containing n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A274759 Modified quadranacci series.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A308189 Numbers of the form t_n or t_n + t_{n+1} where {t_n} are the tribonacci numbers A000073.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A341474 Let T be the set of sequences {t(k), k >= 0} such that for any k >= 3, t(k) = t(k-1) + t(k-2) + t(k-3); a(n) is the least possible value of t(0)^2 + t(1)^2 + t(2)^2 for an element t of T containing n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula