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A202349 Lexicographically first sequence such that the sequence and its first and second differences share no terms, and the 3rd differences are equal to the original sequence. 1
1, 3, 9, 20, 39, 75, 148, 297, 597, 1196, 2391, 4779, 9556, 19113, 38229, 76460, 152919, 305835, 611668, 1223337, 2446677, 4893356, 9786711, 19573419, 39146836, 78293673, 156587349, 313174700, 626349399, 1252698795, 2505397588, 5010795177, 10021590357 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The sequence is completely determined by its first 3 terms. If the first terms are x, y, z, then the following terms are 2*x-3*y+3*z, 6*x-7*y+6*z, 12*x-12*y+11*z, 22*x-21*y+21*z, 42*x-41*y+42*z, 84*x-84*y+85*z, 170*x-171*y+171*z, 342*x-343*y+342*z. - Giovanni Resta, Jun 21 2016

Is it a theorem that, if x,y,z = 1,3,9, the sequence has the desired properties, or is it just a conjecture? - N. J. A. Sloane, Jun 21 2016

From Charlie Neder, Jan 10 2019: (Start)

No two terms among this sequence and its first and second differences are equal.

Proof: Representing the first and second differences by b(n) and c(n), we have that a-b is [-1, -3, -2, 1, 3, 2] with period 6, a-c is [-3, -2, -1, 3, 2, 1] with period 6, and b-c is [-2, 1, 3, 2, -1, -3] with period 6. Therefore, no two terms at the same index are equal. Since the sequence is forced to grow exponentially, only the first few terms need to be checked to confirm that no two terms at different indices are equal, proving the criterion always holds. (End)

LINKS

David A. Corneth, Table of n, a(n) for n = 1..3318

Sequence and first differences include all numbers, etc.

Index entries for linear recurrences with constant coefficients, signature (3,-3,2).

FORMULA

From Colin Barker, Jan 11 2019: (Start)

G.f.: x*(1 + 3*x^2) / ((1 - 2*x)*(1 - x + x^2)).

a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3) for n>2.

(End)

EXAMPLE

  1 3 9  20  39  75  148   297   597   1196

   2 6 11  19  36  73   149   300   599

    4 5   8  17  37  76   151    299

     1  3   9  20  39   75    148   <-- the starting sequence

MATHEMATICA

d = Differences; i = Intersection; sol = Solve[d@ d@ d@ Array[x, 50] == Array[x, 47], Array[x, 47, 4]][[1]]; a = (Array[x, 50] /. sol) /. {x[1] -> 1, x[2] -> 3, x[3] -> 9}; Print["Check = ", {i[a, d@ a], i[a, d@ d@ a], i[d@ a, d@ d@ a]}]; a (* Giovanni Resta, Jun 21 2016 *)

PROG

(PARI) first(n) = {n = max(n, 4); my(res = vector(n)); for(i = 1, 3, res[i] = 3^(i - 1)); for(i = 4, n, res[i] = 3 * res[i - 1] - 3 * res[i - 2] + 2 * res[i - 3]); res } \\ David A. Corneth, Jan 11 2019

(PARI) Vec(x*(1 + 3*x^2) / ((1 - 2*x)*(1 - x + x^2)) + O(x^40)) \\ Colin Barker, Jan 12 2019

CROSSREFS

Cf. A024493, A130781.

For many similar sequences, see the Index link.

Sequence in context: A037257 A145068 A293357 * A192951 A027114 A145070

Adjacent sequences:  A202346 A202347 A202348 * A202350 A202351 A202352

KEYWORD

nonn,base,easy

AUTHOR

Eric Angelini, Jun 21 2016

EXTENSIONS

a(18)-a(33) from Giovanni Resta, Jun 21 2016

STATUS

approved

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Last modified June 26 00:10 EDT 2019. Contains 324367 sequences. (Running on oeis4.)