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A309523 Start with a(1) = 1 and apply certain patterns of operations on a(n-1) to obtain a(n) as described in comments. 1
1, 7, 8, 2, 16, 4, 5, 17, 10, 34, 35, 11, 70, 22, 23, 71, 46, 142, 143, 47, 286, 94, 95, 287, 190, 574, 575, 191, 1150, 382, 383, 1151, 766, 2302, 2303, 767, 4606, 1534, 1535, 4607, 3070, 9214, 9215, 3071, 18430, 6142, 6143, 18431, 12286, 36862 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

a(2) = 7 is obtained from a(1) = 1 by (((1) +1) *3) +1. We abbreviate this to the operation pattern "+1 *3 +1". The 8 patterns for a(3..10), a(11..18) etc. are:

  +1

  +1 /3 -1

  +1 *3 *2 -2

  -1 /3 -1

  +1

  +1 *3 -1

  +1 /3 *2 -2

  +1 *3 +1

A308709 uses similiar, but simpler patterns in blocks of 4 (cf. the example, below). A308709 contains the set {2^k | k>=0} union {3*2^k | k>=0}, so all terms are different. This sequence contains the terms {6*A308709 - 2} union {6*A308709 - 1}, therefore all terms are also different.

LINKS

Colin Barker, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (1,-1,1,0,0,0,0,4,-4,4,-4).

FORMULA

From Colin Barker, Aug 06 2019: (Start)

G.f.: x*(1 + 6*x + 2*x^2 + 15*x^4 - 18*x^5 + 15*x^6 - 10*x^8 + 12*x^9 - 14*x^10) / ((1 - x)*(1 + x^2)*(1 - 2*x^4)*(1 + 2*x^4)).

a(n) = a(n-1) - a(n-2) + a(n-3) + 4*a(n-8) - 4*a(n-9) + 4*a(n-10) - 4*a(n-11) for n>11.

(End)

EXAMPLE

  A308709 | this sequence

          |   1

          |   7    +1 *3 +1

          |   8    +1

          |   2    +1 /3 -1

   3      |  16    +1 *3 *2 -2

   1   /3 |   4    -1 /3 -1

          |   5    +1

          |  17    +1 *3 -1

   2   *2 |  10    +1 /3 *2 -2

   6   *3 |  34    +1 *3 +1

          |  35    +1

          |  11    +1 /3 -1

  12   *2 |  70    +1 *3 *2 -2

   4   /3 |  22    -1 /3 -1

          |  23    +1

          |  71    +1 *3 -1

   8   *2 |  46    +1 /3 *2 -2

  24   *3 | 142    +1 *3 +1

          | 143    +1

MATHEMATICA

LinearRecurrence[{1, -1, 1, 0, 0, 0, 0, 4, -4, 4, -4}, {1, 7, 8, 2, 16, 4, 5, 17, 10, 34, 35}, 50]

PROG

(PARI) Vec(x*(1 + 6*x + 2*x^2 + 15*x^4 - 18*x^5 + 15*x^6 - 10*x^8 + 12*x^9 - 14*x^10) / ((1 - x)*(1 + x^2)*(1 - 2*x^4)*(1 + 2*x^4)) + O(x^40)) \\ Colin Barker, Aug 06 2019

(Perl) use integer;

  my @a; my $n = 1; $a[$n ++] = 1;

  $a[$n ++] =   (($a[$n-1] +1) *3) +1;    #  7

  while ($n < 50) {

    $a[$n ++] = (($a[$n-1] +1)   );       #  8

    $a[$n ++] = (($a[$n-1] +1) /3) -1;    #  2

    $a[$n ++] = (($a[$n-1] +1) *3) *2 -2; # 16

    $a[$n ++] = (($a[$n-1] -1) /3) -1;    #  4

    $a[$n ++] = (($a[$n-1] +1)   );       #  5

    $a[$n ++] = (($a[$n-1] +1) *3) -1;    # 17

    $a[$n ++] = (($a[$n-1] +1) /3) *2 -2; # 10

    $a[$n ++] = (($a[$n-1] +1) *3) +1;    # 34

  } # while

  shift(@a); # remove $a[0]

  print join(", ", @a) . "\n"; # Georg Fischer, Aug 07 2019

(Python 3)

def A309523():

    k, j, a = 0, 0, 1

    def b(a): return a + 1

    def c(a): return a + 2

    def d(a): return a - 1

    def e(a): return a - 2

    def f(a): return a << 1

    def g(a): return a * 3

    def h(a): return a // 3

    O = [c, g, e, b, b, h, d, b, g, f, e, c, h, e, b, b, g, d, b, h, f, e]

    L = [3, 1, 3, 4]

    while True:

        yield(a)

        for _ in range(L[j]):

            a = O[k](a)

            k += 1; k %= 22

        j += 1; j %= 4

a = A309523()

print([next(a) for _ in range(50)]) # Peter Luschny, Aug 06 2019

CROSSREFS

Cf. A308709.

Sequence in context: A011103 A245758 A266566 * A153622 A257576 A064207

Adjacent sequences:  A309520 A309521 A309522 * A309524 A309525 A309526

KEYWORD

nonn,easy

AUTHOR

Georg Fischer, Aug 06 2019

STATUS

approved

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Last modified October 21 18:54 EDT 2019. Contains 328308 sequences. (Running on oeis4.)