cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A081690 From P-positions in a certain game.

Original entry on oeis.org

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
Offset: 0

Views

Author

N. J. A. Sloane, Apr 02 2003

Keywords

Links

Crossrefs

Apart from initial zero, complement of A081691.

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.

Extensions

Corrected and extended by Vladeta Jovovic, Apr 04 2003

A081692 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.

Original entry on oeis.org

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
Offset: 0

Views

Author

N. J. A. Sloane, Apr 02 2003

Keywords

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

Links

Crossrefs

Apart from initial zero, complement of A081693. Cf. A081691.

Programs

  • 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

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.

Extensions

More terms from Vladeta Jovovic, Apr 04 2003

A081693 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.

Original entry on oeis.org

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
Offset: 0

Views

Author

N. J. A. Sloane, Apr 02 2003

Keywords

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, Apr 13 2017

Links

Crossrefs

Apart from initial terms, complement of A081692. Cf. A081691.

Programs

  • 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

Extensions

More terms from Vladeta Jovovic, Apr 04 2003

A081890 a(n) = 9^n - 8^n - 7^n - 6^n + 3*5^n.

Original entry on oeis.org

1, 3, 7, 33, 643, 11073, 151867, 1816713, 19996963, 208630833, 2099398027, 20597485593, 198424412083, 1885822419393, 17740469253787, 165580566245673, 1535948935336003, 14178113530908753, 130361707324735147, 1194785495130736953, 10921581632007328723, 99616564791408530913
Offset: 0

Views

Author

Paul Barry, Mar 30 2003

Keywords

Comments

Binomial transform of A081687.

Links

Crossrefs

Cf. A081687, A081691.

Programs

  • Mathematica
    LinearRecurrence[{35,-485,3325,-11274,15120},{1,3,7,33,643},30] (* Harvey P. Dale, Jun 26 2017 *)

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]
From Elmo R. Oliveira, Sep 12 2024: (Start)
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)

Extensions

a(19)-a(21) from Elmo R. Oliveira, Sep 12 2024

A340780 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.

Original entry on oeis.org

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
Offset: 1

Views

Author

José María Grau Ribas, Jan 21 2021

Keywords

Comments

The game is equivalent to the game of Nim with the additional allowed move consisting of removing one object from each pile.

Crossrefs

Cf. A003557, A227691, A227763, A227764, A171947, A005240, A081691, A099352, A171945, A171949, A285304, A120442, A137295, A275432.

Programs

  • 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]
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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