A024070 -id:A024070 - cp's OEIS frontend

A024018 2^n-n^8.

1, 1, -252, -6553, -65520, -390593, -1679552, -5764673, -16776960, -43046209, -99998976, -214356833, -429977600, -815722529, -1475772672, -2562857857, -4294901760, -6975626369, -11019698432, -16983038753, -25598951424

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..244
Index entries for linear recurrences with constant coefficients, signature (11,-54,156,-294,378,-336,204,-81,19,-2).

Cf. sequences of the form k^n-n^8: this sequence (k=2), A024031 (k=3), A024044 (k=4), A024057 (k=5), A024070 (k=6), A024083 (k=7), A024096 (k=8), A024109 (k=9), A024122 (k=10), A024135 (k=11), A024148 (k=12).

[2^n-n^8: n in [0..25]]; // Vincenzo Librandi, Apr 30 2011

I:=[1,1,-252,-6553,-65520,-390593,-1679552,-5764673,-16776960, -43046209]; [n le 10 select I[n] else 11*Self(n-1)-54*Self(n-2) +156*Self(n-3)-294*Self(n-4)+378*Self(n-5)-336*Self(n-6)+204*Self(n-7) -81*Self(n-8)+19*Self(n-9)-2*Self(n-10): n in [1..35]]; // Vincenzo Librandi, Oct 08 2014

Table[2^n - n^8, {n, 0, 30}] (* or *) CoefficientList[Series[(1 - 10 x - 209 x^2 - 3883 x^3 - 6907 x^4 + 15493 x^5 + 27029 x^6 + 8303 x^7 + 502 x^8 + x^9)/((1 - 2 x) (1 - x)^9), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 08 2014 *)

G.f.: (1 -10*x -209*x^2 -3883*x^3 -6907*x^4 +15493*x^5 +27029*x^6 +8303*x^7 +502*x^8 +x^9) / ((1-2*x)*(1-x)^9). - Vincenzo Librandi, Oct 08 2014

a(n) = 11*a(n-1) -54*a(n-2) +156*a(n-3) -294*a(n-4) +378*a(n-5) -336*a(n-6)+204*a(n-7) -81*a(n-8) +19*a(n-9) -2*a(n-10) for n>9. - Vincenzo Librandi, Oct 08 2014

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

Original entry on oeis.org

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

Juri-Stepan Gerasimov, Jun 12 2014

Sequences of the form (k-1)^m - m^(k+1):
k\m | 0 |  1 |     2 |       3 |         4 |          5 |           6 |
-----------------------------------------------------------------------
 | 1 | -2 |    -1 |      -4 |        -3 |         -6 |          -5 |
 | 1 | -1 |    -4 |      -9 |       -16 |        -25 |         -36 |
 | 1 |  0 |    -7 |     -26 |       -63 |       -124 |        -215 |
 | 1 |  1 |   -12 |     -73 |      -240 |       -593 |       -1232 |
 | 1 |  2 |   -23 |    -216 |      -943 |      -2882 |       -7047 |
 | 1 |  3 |   -43 |    -665 |     -3840 |     -14601 |      -42560 |
 | 1 |  4 |  -103 |   -2062 |    -15759 |     -75000 |     -264311 |
 | 1 |  5 |  -220 |   -6345 |    -64240 |    -382849 |    -1632960 |
 | 1 |  6 |  -463 |  -19340 |   -259743 |   -1936318 |    -9960047 |
 | 1 |  7 |  -960 |  -58537 |  -1044480 |   -9732857 |   -60204032 |
| 1 |  8 | -1967 | -176418 |  -4187743 |  -48769076 |  -362265615 |
| 1 |  9 | -3996 | -530441 | -16767216 | -244040625 | -2175782336 |

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

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

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).

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

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

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 *)

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

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

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A024018 2^n-n^8.

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

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