A024056 -id:A024056 - cp's OEIS frontend

A024017 a(n) = 2^n - n^7.

1, 1, -124, -2179, -16368, -78093, -279872, -823415, -2096896, -4782457, -9998976, -19485123, -35827712, -62740325, -105397120, -170826607, -268369920, -410207601, -611957888, -893347451, -1278951424, -1798991389, -2490163584, -3396436839

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..239
Index entries for linear recurrences with constant coefficients, signature (10,-44,112,-182,196,-140,64,-17,2).

Cf. sequences of the form k^n-n^7: this sequence (k=2), A024030 (k=3), A024043 (k=4), A024056 (k=5), A024069 (k=6), A024082 (k=7), A024095 (k=8), A024108 (k=9), A024121 (k=10), A024134 (k=11), A024147 (k=12).

[2^n-n^7: n in [0..25]]; // Vincenzo Librandi, Apr 30 2011

I:=[1,1,-124,-2179,-16368,-78093,-279872,-823415,-2096896]; [n le 9 select I[n] else 10*Self(n-1)-44*Self(n-2)+112*Self(n-3)-182*Self(n-4)+196*Self(n-5)-140*Self(n-6)+64*Self(n-7)-17*Self(n-8)+2*Self(n-9): n in [1..35]]; // Vincenzo Librandi, Oct 07 2014

Table[2^n - n^7, {n, 0, 30}] (* or *) CoefficientList[Series[(1 - 9 x - 90 x^2 - 1007 x^3 + 36 x^4 + 3585 x^5 + 2290 x^6 + 231 x^7 + 3 x^8)/((1 - 2 x) (1 - x)^8), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 07 2014 *)
LinearRecurrence[{10,-44,112,-182,196,-140,64,-17,2},{1,1,-124,-2179,-16368,-78093,-279872,-823415,-2096896},30] (* Harvey P. Dale, Feb 28 2023 *)

G.f.: (1-9*x-90*x^2-1007*x^3+36*x^4+3585*x^5+ 2290*x^6 +231*x^7+3*x^8)/((1-2*x)*(1-x)^8). - Vincenzo Librandi, Oct 07 2014

a(n) = 10*a(n-1)-44*a(n-2)+112*a(n-3)-182*a(n-4)+196*a(n-5)-140*a(n-6)+64*a(n-7)-17*a(n-8)+2*a(n-9). - Vincenzo Librandi, Oct 07 2014

A198695 a(n) = 11*4^n - 1.

Original entry on oeis.org

10, 43, 175, 703, 2815, 11263, 45055, 180223, 720895, 2883583, 11534335, 46137343, 184549375, 738197503, 2952790015, 11811160063, 47244640255, 188978561023, 755914244095, 3023656976383, 12094627905535, 48378511622143, 193514046488575, 774056185954303, 3096224743817215

Offset: 0

Vincenzo Librandi, Oct 29 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A024056, A097743, A114569, A156760, A199211.

Magma
```
[11*4^n-1: n in [0..30]];
```

11*4^Range[0,30]-1 (* or *) NestList[4#+3&,10,30] (* or *) LinearRecurrence[ {5,-4},{10,43},30] (* Harvey P. Dale, Aug 07 2021 *)

a(n) = 4*a(n-1) + 3.
a(n) = 5*a(n-1) - 4*a(n-2), n > 1.
G.f.: (10-7*x)/((4*x-1)*(x-1)). - R. J. Mathar, Oct 30 2011
From Elmo R. Oliveira, May 07 2025: (Start)
E.g.f.: exp(x)*(11*exp(3*x) - 1).
a(n) = A199211(n) - 2. (End)

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

cp's OEIS Frontend

A024017 a(n) = 2^n - n^7.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Magma

Mathematica

Formula

A198695 a(n) = 11*4^n - 1.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula