A243860 a(n) = 2^(n+1) - (n-1)^2.

1, 4, 7, 12, 23, 48, 103, 220, 463, 960, 1967, 3996, 8071, 16240, 32599, 65340, 130847, 261888, 523999, 1048252, 2096791, 4193904, 8388167, 16776732, 33553903, 67108288, 134217103, 268434780, 536870183, 1073741040, 2147482807, 4294966396, 8589933631, 17179868160, 34359737279, 68719475580

Offset: 0

Author

Juri-Stepan Gerasimov, Jun 12 2014

Keywords

• nonn
• easy

Comments

Sequences of the form (k-1)^m - m^(k+1):

k\m | 0 | 1 | 2 | 3 | 4 | 5 | 6 |

-----------------------------------------------------------------------

0 | 1 | -2 | -1 | -4 | -3 | -6 | -5 |

1 | 1 | -1 | -4 | -9 | -16 | -25 | -36 |

2 | 1 | 0 | -7 | -26 | -63 | -124 | -215 |

3 | 1 | 1 | -12 | -73 | -240 | -593 | -1232 |

4 | 1 | 2 | -23 | -216 | -943 | -2882 | -7047 |

5 | 1 | 3 | -43 | -665 | -3840 | -14601 | -42560 |

6 | 1 | 4 | -103 | -2062 | -15759 | -75000 | -264311 |

7 | 1 | 5 | -220 | -6345 | -64240 | -382849 | -1632960 |

8 | 1 | 6 | -463 | -19340 | -259743 | -1936318 | -9960047 |

9 | 1 | 7 | -960 | -58537 | -1044480 | -9732857 | -60204032 |

10 | 1 | 8 | -1967 | -176418 | -4187743 | -48769076 | -362265615 |

11 | 1 | 9 | -3996 | -530441 | -16767216 | -244040625 | -2175782336 |

Examples

			1 = 2^(0+1) - (0-1)^2, 4 = 2^(1+1) - (1-1)^2, 7 = 2^(2+1) - (2-1)^2.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (5,-9,7,-2).

Crossrefs

Sequences of the form (k-1)^m - m^(k+1): A000012 (m = 0), A023444 (m = 1), (-1)*(this sequence) for m = 2, A114285 (k = 0),(A000007-A000290) for k = 1, A024001 (k = 2), A024014 (k = 3), A024028 (k = 4), A024042 (k = 5), A024056 (k = 6), A024070 (k = 7), A024084 (k = 8), A024098 (k = 9), A024112 (k = 10), A024126 (k = 11).

Programs

Magma

[2^(n+1) - (n-1)^2: n in [0..35]];

Maple

A243860:=n->2^(n + 1) - (n - 1)^2; seq(A243860(n), n=0..30); # Wesley Ivan Hurt, Jun 12 2014

Mathematica

Table[2^(n + 1) - (n - 1)^2, {n, 0, 30}] (* Wesley Ivan Hurt, Jun 12 2014 *)
LinearRecurrence[{5,-9,7,-2},{1,4,7,12},40] (* Harvey P. Dale, Nov 29 2015 *)

PARI

Vec((6*x^3-4*x^2-x+1)/((x-1)^3*(2*x-1)) + O(x^100)) \\ Colin Barker, Jun 12 2014

Formula

a(n) = 5*a(n-1)-9*a(n-2)+7*a(n-3)-2*a(n-4). - Colin Barker, Jun 12 2014

G.f.: (6*x^3-4*x^2-x+1) / ((x-1)^3*(2*x-1)). - Colin Barker, Jun 12 2014