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From P-positions in a certain game.
+10
2
0, 1, 3, 4, 5, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77
FORMULA
Let a(n) = this sequence, b(n) = A081691. Then a(n) = mex{ a(i), b(i) : 0 <= i < n}, b(0) = 0, b(n) = 2(b(n-1) - a(n-1)) + a(n) + 1 = a(n) + 2^n - 1.
CROSSREFS
Apart from initial zero, complement of A081691.
Define two sequences by A_n = mex{A_i,B_i : 0 <= i < n}, B_n = B_{n-1} + (A_n-A_{n-1})(A_n-A_{n-1}+1), where the mex of a set is the smallest nonnegative integer not in the set. Sequence gives A_n. B_n is in A081693.
+10
2
0, 1, 3, 4, 5, 6, 7, 9, 11, 13, 15, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 47, 49, 51, 53, 55, 56, 57, 58, 59, 61, 63, 65, 67, 69, 70, 71, 72, 73, 75, 77, 79, 81, 83, 84, 85, 86, 87, 89, 91, 93, 95, 97, 98, 99, 100
COMMENTS
Conjecture: Except for the initial 0, this is the sequence of positions of 0 in the fixed point of the morphism 0->01, 1->0000; see A284683. - Clark Kimberling, Apr 13 2017
FORMULA
Let a(n) = this sequence, b(n) = A081691. Then a(n) = mex{ a(i), b(i) : 0 <= i < n}, b(0) = 0, b(n) = 2(b(n-1) - a(n-1)) + a(n) + 1.
MATHEMATICA
mex[{}]=0; mex[s_] := Complement[Range[0, 1+Max@@s], s][[1]]; A[0]=B[0]=0; A[n_] := A[n]=mex[Flatten[Table[{A[i], B[i]}, {i, 0, n-1}]]]; B[n_] := B[n]=B[n-1]+(A[n]-A[n-1])*(A[n]-A[n-1]+1); a := A
Define two sequences by A_n = mex{A_i,B_i : 0 <= i < n}, B_n = B_{n-1} + (A_n-A_{n-1})(A_n-A_{n-1}+1), where the mex of a set is the smallest nonnegative integer not in the set. Sequence gives B_n. A_n is in A081692.
+10
2
0, 2, 8, 10, 12, 14, 16, 22, 28, 34, 40, 46, 48, 50, 52, 54, 60, 62, 64, 66, 68, 74, 76, 78, 80, 82, 88, 90, 92, 94, 96, 102, 104, 106, 108, 110, 116, 122, 128, 134, 140, 142, 144, 146, 148, 154, 160, 166, 172, 178, 180, 182, 184, 186, 192, 198, 204, 210, 216, 218
COMMENTS
Conjecture: Except for the initial 0, this is the sequence of positions of 1 in the fixed point of the morphism 0->01, 1->0000; see A284683. - Clark Kimberling, April 13 2017
MATHEMATICA
mex[{}]=0; mex[s_] := Complement[Range[0, 1+Max@@s], s][[1]]; A[0]=B[0]=0; A[n_] := A[n]=mex[Flatten[Table[{A[i], B[i]}, {i, 0, n-1}]]]; B[n_] := B[n]=B[n-1]+(A[n]-A[n-1])*(A[n]-A[n-1]+1); a := B
a(n) = 9^n - 8^n - 7^n - 6^n + 3*5^n.
+10
1
1, 3, 7, 33, 643, 11073, 151867, 1816713, 19996963, 208630833, 2099398027, 20597485593, 198424412083, 1885822419393, 17740469253787, 165580566245673, 1535948935336003, 14178113530908753, 130361707324735147, 1194785495130736953, 10921581632007328723, 99616564791408530913
FORMULA
G.f.: -(4182*x^4-2082*x^3+387*x^2-32*x+1)/((5*x-1)*(6*x-1)*(7*x-1)*(8*x-1)*(9*x-1)). [ Colin Barker, Aug 12 2012] E.g.f.: exp(5*x)*(exp(4*x) - exp(3*x) - exp(2*x) - exp(x) + 3).
a(n) = 35*a(n-1) - 485*a(n-2) + 3325*a(n-3) - 11274*a(n-4) + 15120*a(n-5) for n > 4. (End)
MATHEMATICA
LinearRecurrence[{35, -485, 3325, -11274, 15120}, {1, 3, 7, 33, 643}, 30] (* Harvey P. Dale, Jun 26 2017 *)
Losing positions n (P-positions) in the following game: two players take turns dividing the current value of n by either a prime power > 1 or by A007947(n) to obtain the new value of n. The winner is the player whose division results in 1.
+10
0
1, 12, 18, 20, 28, 44, 45, 50, 52, 63, 68, 75, 76, 92, 98, 99, 116, 117, 120, 124, 147, 148, 153, 164, 168, 171, 172, 175, 188, 207, 212, 216, 236, 242, 244, 245, 261, 264, 268, 270, 275, 279, 280, 284, 292, 312, 316, 325, 332, 333, 338, 356, 363, 369, 378, 387, 388
COMMENTS
The game is equivalent to the game of Nim with the additional allowed move consisting of removing one object from each pile.
MATHEMATICA
Clear[moves, los]; A003557[n_]:= {Module[{aux = FactorInteger[n], L=Length[FactorInteger[n]]}, Product[aux[[i, 1]]^(aux[[i, 2]]-1), {i, L}]]}; moves[n_] :=moves[n] = Module[{aux = FactorInteger[n], L=Length[ FactorInteger [n]]}, Union[Flatten[Table[n/aux[[i, 1]]^j, {i, 1, L}, {j, 1, aux[[i, 2]]}], 1], A003557[n]]]; los[1]=True; los[m_] := los[m] = If[PrimeQ[m], False, Union@Flatten@Table[los[moves[m][[i]]], {i, 1, Length[moves[m]]}] == {False}]; Select[Range[400], los]
CROSSREFS
Cf. A003557, A227691, A227763, A227764, A171947, A005240, A081691, A099352, A171945, A171949, A285304, A120442, A137295, A275432.
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