A178676 -id:A178676 - cp's OEIS frontend

A242328 a(n) = 5^n + 2.

3, 7, 27, 127, 627, 3127, 15627, 78127, 390627, 1953127, 9765627, 48828127, 244140627, 1220703127, 6103515627, 30517578127, 152587890627, 762939453127, 3814697265627, 19073486328127, 95367431640627, 476837158203127, 2384185791015627, 11920928955078127

Offset: 0

Vincenzo Librandi, May 13 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-5).

Cf. A000351, A003463, A034474, A132079, A178676, A242329.

Magma
```
[5^n+2: n in [0..30]];
```

Table[5^n + 2, {n, 0, 30}] (* or *) CoefficientList[Series[(3 - 11 x)/((1 - x) (1 - 5 x)), {x, 0, 30}], x]
LinearRecurrence[{6,-5},{3,7},30] (* Harvey P. Dale, Jun 30 2022 *)

G.f.: (3-11*x)/((1-x)*(1-5*x)).
a(n) = 6*a(n-1) - 5*a(n-2) for n > 1.
From Elmo R. Oliveira, Dec 04 2023: (Start)
a(n) = A000351(n) + 2.
a(n) = 5*a(n-1) - 8 with a(0) = 3.
E.g.f.: exp(5*x) + 2*exp(x). (End)

A242329 a(n) = 5^n + 4.

Original entry on oeis.org

5, 9, 29, 129, 629, 3129, 15629, 78129, 390629, 1953129, 9765629, 48828129, 244140629, 1220703129, 6103515629, 30517578129, 152587890629, 762939453129, 3814697265629, 19073486328129, 95367431640629, 476837158203129, 2384185791015629, 11920928955078129

Offset: 0

Vincenzo Librandi, May 13 2014

Subsequence of A226810. - Bruno Berselli, May 13 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-5).

Cf. A000351, A003463, A034474, A132079, A178676, A226810, A242328, A253208 (similar sequence).

Magma
```
[5^n+4: n in [0..30]];
```

Table[5^n + 4, {n, 0, 30}]
LinearRecurrence[{6,-5},{5,9},30] (* Harvey P. Dale, Mar 15 2025 *)

G.f.: (5-21*x)/((1-x)*(1-5*x)).
a(n) = 6*a(n-1) - 5*a(n-2) for n > 1.
From Elmo R. Oliveira, Dec 06 2023: (Start)
a(n) = A000351(n)+4 = A034474(n)+3 = A242328(n)+2.
a(n) = 5*a(n-1) - 16 with a(0) = 5.
E.g.f.: exp(5*x) + 4*exp(x). (End)

A178671 a(n) = 5^n - 5.

Original entry on oeis.org

-4, 0, 20, 120, 620, 3120, 15620, 78120, 390620, 1953120, 9765620, 48828120, 244140620, 1220703120, 6103515620, 30517578120, 152587890620, 762939453120, 3814697265620, 19073486328120, 95367431640620, 476837158203120, 2384185791015620, 11920928955078120

Offset: 0

Vincenzo Librandi, Dec 25 2010

			a(n) = A178676(n)-10 = A242329(n)-9 = A242328(n)-7 = A034474(n)-6 = A000351(n)-5. - _Elmo R. Oliveira_, Dec 06 2023

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-5).

Cf. A000351, A034474, A178676, A242328, A242329.

List([0..30], n -> 5^n-5); # G. C. Greubel, Jan 28 2019

Magma
```
[5^n-5: n in [0..25]];
```

5^Range[0,30]-5 (* Vladimir Joseph Stephan Orlovsky, Feb 20 2011 *)
LinearRecurrence[{6,-5},{-4,0},30] (* Harvey P. Dale, Aug 23 2024 *)

vector(30, n, n--; 5^n-5) \\ G. C. Greubel, Jan 28 2019

[5^n-5 for n in range(30)] # G. C. Greubel, Jan 28 2019

a(n) = 5*a(n-1) + 20 with a(0) = -4.
From R. J. Mathar, Jan 03 2011: (Start)
G.f.: 4*(-1+6*x)/((1-5*x)*(1-x)).
a(n) = 4*A104891(n-1), n > 0. (End)
a(n) = 6*a(n-1) - 5*a(n-2) for n > 1. - Vincenzo Librandi, Jan 25 2013
E.g.f.: exp(5*x) - 5*exp(x). - G. C. Greubel, Jan 28 2019

A178675 a(n) = 4^n + 4.

Original entry on oeis.org

5, 8, 20, 68, 260, 1028, 4100, 16388, 65540, 262148, 1048580, 4194308, 16777220, 67108868, 268435460, 1073741828, 4294967300, 17179869188, 68719476740, 274877906948, 1099511627780, 4398046511108, 17592186044420, 70368744177668, 281474976710660, 1125899906842628

Offset: 0

Vincenzo Librandi, Dec 25 2010

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

Cf. A052548, A178674, A178676, A178681.

List([0..30], n -> 4^n + 4); # G. C. Greubel, Jan 27 2019

Magma
```
[4^n+4: n in [0..35]];
```

I:=[5, 8]; [n le 2 select I[n] else 5*Self(n-1)-4*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jun 18 2013

Table[4^n +4, {n, 0, 40}] (* or *) CoefficientList[Series[(5-17x)/((4x - 1)(x-1)), {x, 0, 30}], x] (* Vincenzo Librandi, Jun 18 2013 *)
LinearRecurrence[{5,-4},{5,8},30] (* Harvey P. Dale, Sep 12 2023 *)

vector(40, n, n--; 4^n+4) \\ G. C. Greubel, Jan 27 2019

[4^n+4 for n in range(40)] # G. C. Greubel, Jan 27 2019

a(n) = 4*(a(n-1) - 3) with n > 0, a(0)=5.
G.f.: ( 5-17*x ) / ( (1-4*x)*(1-x) ). - R. J. Mathar, Jan 05 2011
a(n) = 5*a(n-1) - 4*a(n-2). - Vincenzo Librandi, Jun 18 2013
E.g.f.: exp(4*x) + 4*exp(x). - G. C. Greubel, Jan 27 2019

A384853 Squared length of interior diagonal of n-th (U, V)-crossbox, where U = (1, 0, 1) and V = (0, 1, 0), as in Comments.

Original entry on oeis.org

1, 5, 9, 21, 57, 165, 489, 1461, 4377, 13125, 39369, 118101, 354297, 1062885, 3188649, 9565941, 28697817, 86093445, 258280329, 774840981, 2324522937, 6973568805, 20920706409, 62762119221, 188286357657, 564859072965, 1694577218889, 5083731656661

Offset: 1

Clark Kimberling, Jul 02 2025

nonn

Suppose that U and V are 3-dimensional vectors over the field of real numbers. Define f(1) = U, f(2) = V, f(3) = UxV, where x = cross product, and for n>=2, define f(n) = h(n - 1), g(n) = f(n - 1) + g(n - 1) - h(n - 1), h(n) = f(n) x g(n).
The parallelopiped having edge vectors f(n), g(n), h(n) is the n-th (U,V)-crossbox, with volume |f(n).(g(n) x h(n))|, where . = dot product, and interior diagonal length ||g(n)||. These two sequences, after removal of their first 2 terms, are given for selected U and V by the following table, except for the 3 initial terms:
    U             V         volume   squared diagonal length, ||g(n)||^2
(1, 0, 0)     (0, 1, 0)     A000079           A052548
(1, 0, 0)     (0, 1, 1)     A008776         3*A052919
(1, 0, 0)     (1, 0, 1)     A000351           A178676
(1, 0, 0)     (1, 1, 1)     A167747         5*A204061
(1, 0, 0)     (0, 2, 0)     A005054         4*A199215
(1, 0, 0)     (1, 2, 0)     A013731         8*A199552
(1, 0, 0)     (2, 1, 0)     A011557        10*A000533
(1, 0, 0)     (1, 1, 2)     A067403        18*A135423
(1, 0, 0)     (2, 1, 1)     A334603        11*A199750
(1, 0, 1)     (0, 1, 0)     A008776           this sequence
(1, 1, 0)     (0, 1, 1)     A081341         6*A199318
(1, 1, 0)     (1, 1, 1)     A270369         9*A199559
(1, 2, 3)     (3, 2, 1)   2*A009992   48 + 96*A009992

			Taking U = (1, 0, 1) and V = (0, 1, 0), successive edge vectors are given by
(f(n)) = ( (1, 0, 1), (-1,0,1), (-1,2,-1), (3,0,-3), (3,-6,3), ...)
(g(n)) = ( (0,1,0), (2,1,0), (2,-1,2), (-2,1,4), (-2,7,-2), (10,1,-8), ...)
(h(n)) = ( (-1.0,1), (-1,2,-1), (3,0,-3), (3,-6,3), (-9,0,9),...)
The successive volumes are (2, 6, 18, 54, 162, 486, 1458, 4374, 13122,...).
The lengths of diagonals of the first five crossboxes are 1, sqrt(5), 3, sqrt(21), sqrt(57), so the first five squared lengths are 1, 5, 9, 21, 57.

Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A000079, A052548, A008776.

f[1] = {1, 0, 1}; g[1] = {0, 1, 0}; h[1] = Cross[f[1], g[1]];
f[n_] := f[n] = h[n - 1];
g[n_] := g[n] = f[n - 1] + g[n - 1] - h[n - 1];
h[n_] := h[n] = Cross[f[n], g[n]];
v[n_] := f[n] . Cross[g[n], h[n]] (* signed volume of nth parallelopiped P(n) *)
d[n_] := Norm[g[n]] (* length of interior diagonal of P(n) *)
Column[Table[{f[n], g[n], h[n]}, {n, 1, 16}]]  (* edge vectors of P(n) *)
Table[v[n], {n, 1, 16}]  (* A008776 *)
u = Table[d[n]^2, {n, 1, 30}] (* A384853 *)
Join[{1},Table[1+2*(3^(n-1)+1),{n,40}]] (* or *) LinearRecurrence[{4,-3},{1,5,9},50] (* Harvey P. Dale, Jul 20 2025 *)

a(0) = 1, a(n) = 1 + 2 * (3^(n-1)+1) for n>=1.
a(n) = 4*a(n-1) - 3*a(n-2) for n>=4.
In general, suppose that U = (a,b,c) and V = (s,t,u), and let D = -(a^2 + b^2 + c^2 + s^2 + t^2 + u^2 + 2 (a s + b t + c u)). Then, linear recurrences are given for n>=3 by f(n) = D*f (n - 2), g(n) = g(n - 1) + D*g(n - 2) - D*g(n - 3), h(n) = D*h(n - 2). If w(n) denotes the volume of the n-th (U,V)-crossbox, then w(n) = D*w(n-1) for n>=2.

cp's OEIS Frontend

A242328 a(n) = 5^n + 2.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A242329 a(n) = 5^n + 4.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A178671 a(n) = 5^n - 5.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

Sage

Formula

A178675 a(n) = 4^n + 4.

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Magma

Magma

Mathematica

PARI

Sage

Formula

A384853 Squared length of interior diagonal of n-th (U, V)-crossbox, where U = (1, 0, 1) and V = (0, 1, 0), as in Comments.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula