A028356 -id:A028356 - cp's OEIS frontend

A175921 Period 5: repeat [1, 2, 2, -1, 1].

1, 2, 2, -1, 1, 1, 2, 2, -1, 1, 1, 2, 2, -1, 1, 1, 2, 2, -1, 1, 1, 2, 2, -1, 1, 1, 2, 2, -1, 1, 1, 2, 2, -1, 1, 1, 2, 2, -1, 1, 1, 2, 2, -1, 1, 1, 2, 2, -1, 1, 1, 2, 2, -1, 1, 1, 2, 2, -1, 1, 1, 2, 2, -1, 1, 1, 2, 2, -1, 1, 1, 2, 2, -1, 1

Offset: 1

Jaroslav Krizek, Oct 17 2010

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1).

Cf. A040001, A028356, A142150, A000012, A175922.

PadRight[{},120,{1,2,2,-1,1}] (* Harvey P. Dale, Dec 30 2016 *)

a(n)=[1,1,2,2,-1][n%5+1] \\ Charles R Greathouse IV, Mar 12 2017

Edited by Joerg Arndt, Sep 16 2013

A262734 Period 16: repeat (1, 2, 3, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 4, 3, 2).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Sep 29 2015

Decimal expansion of 111111112/900000009.

For n which lies in the interval [16*(k-1), 8*(2*k-1)], where k>0 -> pattern {1, 2, 3, 4, 5, 6, 7, 8, 9}; for n which lies in the interval [16*k - 7, 16*k - 1], where k>0 -> pattern {8, 7, 6, 5, 4, 3, 2}.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,-1,1).

Cf. A158289, A177274, A010889, A138531, A181975, A199264, A068073, A028356.

&cat[[1,2,3,4,5,6,7,8,9,8,7,6,5,4,3,2]: n in [0..10]]; // Vincenzo Librandi, Sep 29 2015

LinearRecurrence[{1, 0, 0, 0, 0, 0, 0, -1, 1}, {1, 2, 3, 4, 5, 6, 7, 8, 9}, 120] (* Vincenzo Librandi, Sep 29 2015 *)

Vec(-(2*x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1)/((x-1)*(x^8+1)) + O(x^100)) \\ Colin Barker, Sep 29 2015

111111112/900000009. \\ Altug Alkan, Sep 29 2015

vector(200, n, default(realprecision, n+2); floor(111111112/900000009*10^n)%10) \\ Altug Alkan, Nov 12 2015

-1 + a(16*(k - 1)) = -2 + a(8*k + 3*(-1)^k - 4) = -3 + a(2*(4*k + (-1)^k - 2)) = -4 + a(8*k + (-1)^k - 4) = -5 + a(4*(2*k - 1)) = -6 + a(8*k - (-1)^k - 4) = -7 + a(-2*(-4*k + (-1)^k + 2)) = -8 + a(8*k - 3*(-1)^k - 4) = -9 + a(8*(2*k - 11)) = 0, for k>0.
a(0) = 1, a(n) = a(n+1) - 1, for 16*(k - 1) <= n < 8*(2*k - 1), and a(n) = a(n + 1) + 1, for 8*(2*k - 1) <= n < 16*k, where k>0.
From Colin Barker, Sep 29 2015: (Start)
a(n) = a(n-1) - a(n-8) + a(n-9) for n>8.
G.f.: -(2*x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1) / ((x-1)*(x^8+1)). (End)

A321301 Lexicographically last sequence of positive integers whose terms can be grouped and summed to produce the natural numbers as well as the prime numbers.

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 5, 2, 4, 7, 8, 5, 4, 10, 3, 8, 11, 1, 13, 9, 5, 15, 9, 7, 17, 7, 11, 19, 7, 13, 21, 7, 15, 23, 5, 19, 25, 3, 23, 27, 3, 25, 29, 5, 25, 31, 5, 27, 33, 7, 27, 35, 9, 27, 37, 9, 29, 39, 11, 29, 41, 13, 29, 43, 17, 27, 45, 25, 21, 47, 33, 15, 49

Offset: 1

Rémy Sigrist, Nov 03 2018

More formally:
- let S be the set of sequences of positive integers with positive indices,
- for any u and v in S, the terms of u can be grouped and summed to produce v iff there is an element w in S such that for any n > 0:
      v(n) = Sum_{i=1..w(n)} u(i + Sum_{j=1..n-1} w(j)),
      or: Sum_{i=1..Sum_{j=1..n} w(j)} u(i) = Sum_{k=1..n} v(k),
      (the sequence w gives the number of terms in each group)
- the set S with the binary relation R "u can be grouped and summed to produce v" is a partially ordered set,
- in particular, A028356 is R-related to A000027,
- for any u in S, A000012 is R-related to u (A000012 is the least element of S with respect to R),
- for any u and v, let L(u, v) denote the lexicographically last element of S that is R-related both to u and to v,
- for any u, v and w in S, the function L satisfies:
       L(u, u) = u,
       L(u, v) = L(v, u),
       L(u, L(v, w)) = L(L(u, v), w),
       L(A000012, u) = A000012,
- this sequence corresponds to L(A000027, A000040).

			The first terms of this sequence, alongside the groups summing to the first natural numbers and to the first prime numbers, are:
                  +-+---+-----+-------+---------+-----------+-------------+
- Natural numbers |1| 2 |  3  |   4   |    5    |     6     |      7      | ...
                  +-+-+-+---+-+-------+---------+---+-------+-------------+
- This sequence   |1|1|1| 2 |1|   4   |    5    | 2 |   4   |      7      | ...
                  +-+-+-+---+-+-------+---------+---+-------+-------------+
- Prime numbers   | 2 |  3  |    5    |      7      |          11         | ...
                  +---+-----+---------+-------------+---------------------+

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A321301

Cf. A000012, A000027, A000040, A028356.

PARI
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A175921 Period 5: repeat [1, 2, 2, -1, 1].

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A262734 Period 16: repeat (1, 2, 3, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 4, 3, 2).

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A321301 Lexicographically last sequence of positive integers whose terms can be grouped and summed to produce the natural numbers as well as the prime numbers.

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