A341456 -id:A341456 - cp's OEIS frontend

A341285 Let B be the set of sequences of positive integers {b(k), k >= 0} such that for some k > 0 (necessarily unique) and any m >= 0, b(m+k) = b(m) + b(m+1) + ... + b(m+k-1); let g(b) = b(0); a(n) is the least value of g(b) for an element b of B containing n.

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

Offset: 1

Rémy Sigrist, Feb 16 2021

nonn

This sequence is a generalization of A249783 and A341456 to the set of "k-bonacci sequences of positive integers".

			The first terms of the elements b of B such that g(b) <= 3 are:
  g(b)  b(0)  b(1)  b(2)  b(3)  b(4)  b(5)  b(6)  b(7)  b(8)  b(9)
  ----  ----  ----  ----  ----  ----  ----  ----  ----  ----  ----
     1     1     1     1     1     1     1     1     1     1     1
     2     2     2     2     2     2     2     2     2     2     2
     2     1     1     2     3     5     8    13    21    34    55
     3     3     3     3     3     3     3     3     3     3     3
     3     1     2     3     5     8    13    21    34    55    89
     3     2     1     3     4     7    11    18    29    47    76
     3     1     1     1     3     5     9    17    31    57   105
- so a(1) = 1,
     a(2) = a(3) = a(5) = a(8) = 2,
     a(4) = a(7) = a(9) = a(11) = a(17) = a(18) = 3.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, C program for A341285

Cf. A020695, A070939, A249783, A341456, A341699.

C
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See Links section.
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a(n) <= n.
a(m*n) <= m*a(n).
a(n) = 2 iff n belongs to A020695.
a(n) = A070939(A341699(n)).

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

cp's OEIS Frontend

A341285 Let B be the set of sequences of positive integers {b(k), k >= 0} such that for some k > 0 (necessarily unique) and any m >= 0, b(m+k) = b(m) + b(m+1) + ... + b(m+k-1); let g(b) = b(0); a(n) is the least value of g(b) for an element b of B containing n.

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