A121213 -id:A121213 - cp's OEIS frontend

A081200 6th binomial transform of (0,1,0,1,0,1,...), A000035.

0, 1, 12, 109, 888, 6841, 51012, 372709, 2687088, 19200241, 136354812, 964249309, 6798573288, 47834153641, 336059778612, 2358521965909, 16540171339488, 115933787267041, 812299450322412, 5689910849522509, 39848449432985688, 279034513462540441, 1953718431395986212

Offset: 0

Paul Barry, Mar 11 2003

Binomial transform of A081199.
Conjecture (verified up to a(9)): Number of collinear 5-tuples of points in a 5 X 5 X 5 X ... n-dimensional cubic grid. - Ron Hardin, May 24 2010
a(n) is also the total number of words of length n, over an alphabet of seven letters, of which one of them appears an odd number of times. See the Lekraj Beedassy, Jul 22 2003, comment on A006516 (4-letter case), and the Balakrishnan reference there. For the 2-, 3-, 5-, 6- and 8-letter case analogs see A131577, A003462, A005059, A081199, A081201 respectively. - Wolfdieter Lang, Jul 17 2017

			The a(2) = 12 words of length 2 over {A, B, C, D, E, F, G} with say, A, appearing an odd number of times (that is once) are: AB, AC, AD, AE, AF, AG; BA, CA, DA, EA, FA, GA. - _Wolfdieter Lang_, Jul 17 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Takao Komatsu, Some recurrence relations of poly-Cauchy numbers, J. Nonlinear Sci. Appl., (2019) Vol. 12, Issue 12, 829-845.
Index entries for linear recurrences with constant coefficients, signature (12,-35).

Cf. A000035, A003462, A005059, A006516, A081199, A081201 (binomial transform, and 8-letter analog), A121213, A131577.

Apart from offset same as A016161.

[7^n/2-5^n/2: n in [0..25]]; // Vincenzo Librandi, Aug 07 2013

CoefficientList[Series[x / ((1 - 5 x) (1 - 7 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Aug 07 2013 *)
LinearRecurrence[{12,-35},{0,1},30] (* Harvey P. Dale, Feb 07 2014 *)

[lucas_number1(n,12,35) for n in range(0, 21)] # Zerinvary Lajos, Apr 27 2009

a(n) = 12*a(n-1) - 35*a(n-2), a(0) = 0, a(1) = 1.
G.f.: x/((1-5*x)*(1-7*x)).
a(n) = 7^n/2 - 5^n/2.
a(n) = Sum_{k=0..n-1} 7^k * 5^(n-k-1), with a(0)=0. - Reinhard Zumkeller, Aug 01 2010
a(n) = A121213(n)/2. - Reinhard Zumkeller, Aug 01 2010
E.g.f.: exp(5*x)*(exp(2*x) - 1)/2. - Stefano Spezia, Jun 19 2021

A190540 a(n) = 7^n - 2^n.

Original entry on oeis.org

0, 5, 45, 335, 2385, 16775, 117585, 823415, 5764545, 40353095, 282474225, 1977324695, 13841283105, 96889002215, 678223056465, 4747561477175, 33232930504065, 232630513856135, 1628413597648305, 11398895184848855, 79792266296563425, 558545864081186855, 3909821048578793745

Offset: 0

Vincenzo Librandi, Jun 02 2011

Length-n words from letters {1,2,...,7} with at least one letter >2. [Joerg Arndt, Jun 02 2011]

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (9,-14).

Cf. A016130, A016169, A121213.

Magma
```
[7^n -2^n: n in [0..30]];
```

CoefficientList[Series[5 x/((1 - 2 x) (1 - 7 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 04 2014 *)

a(n)=7^n-1<Charles R Greathouse IV, Jun 08 2011

a(n) = 9*a(n-1) - 14*a(n-2).
G.f.: 5*x/((1-2*x)*(1-7*x)). - Vincenzo Librandi, Oct 04 2014
a(n) = 5*A016130(n-1). - R. J. Mathar, Mar 10 2022
E.g.f.: exp(2*x)*(exp(5*x) - 1). - Elmo R. Oliveira, Sep 10 2024

A190541 a(n) = 7^n - 3^n.

Original entry on oeis.org

0, 4, 40, 316, 2320, 16564, 116920, 821356, 5758240, 40333924, 282416200, 1977149596, 13840755760, 96887416084, 678218289880, 4747547161036, 33232887522880, 232630384847044, 1628413210489960, 11398894023111676, 79792262810827600, 558545853622930804, 3909821017201928440

Offset: 0

Vincenzo Librandi, Jun 02 2011

Length-n words from letters {1,2,...,7} with at least one letter greater than 3. - Joerg Arndt, Jun 02 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (10,-21).

Cf. A000244, A000420.

Similar sequences: A121213, A016169.

Magma
```
[7^n - 3^n: n in [0..30]];
```

A190541:=n->7^n-3^n: seq(A190541(n), n=0..25); # Wesley Ivan Hurt, Oct 04 2014

Table[7^n - 3^n, {n, 0, 25}] (* or *) CoefficientList[Series[4 x /((1 - 3 x) (1 - 7 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 04 2014 *)
LinearRecurrence[{10,-21},{0,4},20] (* Harvey P. Dale, Mar 30 2015 *)

a(n)=7^n-3^n \\ Charles R Greathouse IV, Jun 02 2011

a(n) = 10*a(n-1) - 21*a(n-2).
G.f.: 4*x/((1-3*x)*(1-7*x)). - Vincenzo Librandi, Oct 04 2014
a(n) = A000420(n) - A000244(n). - Wesley Ivan Hurt, Oct 04 2014
E.g.f.: 2*exp(5*x)*sinh(2*x). - Elmo R. Oliveira, Mar 31 2025
a(n) = 4*A016138(n-1). - R. J. Mathar, Jun 07 2025

A190542 a(n) = 7^n - 4^n.

Original entry on oeis.org

0, 3, 33, 279, 2145, 15783, 113553, 807159, 5699265, 40091463, 281426673, 1973132439, 13824509985, 96821901543, 677954637393, 4746487768119, 33228635602305, 232613334118023, 1628344878433713, 11398620307466199, 79791166785984225, 558541466036772903, 3909803456396943633, 27368676971336738679, 191580949905589703745

Offset: 0

Vincenzo Librandi, Jun 02 2011

Length-n words from letters {1,2,...,7} with at least one letter greater than 4. - Joerg Arndt, Jun 02 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (11,-28).

Cf. A000302, A000420, A016169, A121213.

Magma
```
[7^n -4^n: n in [0..30]];
```

A190542:=n->7^n - 4^n; seq(A190542(n), n=0..30); # Wesley Ivan Hurt, Feb 26 2014

Table[7^n - 4^n, {n, 0, 30}] (* Wesley Ivan Hurt, Feb 26 2014 *)
LinearRecurrence[{11,-28},{0,3},30] (* Harvey P. Dale, Dec 21 2019 *)

a(n)=7^n-4^n \\ Charles R Greathouse IV, Jun 02 2011

def A190542(n): return pow(7,n) - pow(4,n)
print([A190542(n) for n in range(31)]) # G. C. Greubel, Nov 13 2024

a(n) = 11*a(n-1) - 28*a(n-2).
a(n) = A000420(n) - A000302(n). - Michel Marcus, Feb 26 2014
From G. C. Greubel, Nov 13 2024: (Start)
G.f.: 3*x/((1-4*x)*(1-7*x)).
E.g.f.: 2*exp(11*x/2)*sinh(3*x/2). (End)

A248340 a(n) = 10^n - 5^n.

Original entry on oeis.org

0, 5, 75, 875, 9375, 96875, 984375, 9921875, 99609375, 998046875, 9990234375, 99951171875, 999755859375, 9998779296875, 99993896484375, 999969482421875, 9999847412109375, 99999237060546875, 999996185302734375, 9999980926513671875

Offset: 0

Vincenzo Librandi, Oct 05 2014

G. C. Greubel, Table of n, a(n) for n = 0..995
Index entries for linear recurrences with constant coefficients, signature (15,-50).

Cf. sequences of the form k^n-5^n: A005062 (k=6), A121213 (k=7), A191468 (k=8), A191466 (k=9), this sequence (k=10), A139743 (k=11).

Cf. A000225, A000351, A011557, A016164.

Magma
```
[10^n-5^n: n in [0..30]];
```

Table[10^n - 5^n, {n,0,30}]
CoefficientList[Series[5 x/((1-5 x)(1-10 x)), {x, 0, 30}], x]

def A248340(n): return pow(10,n) - pow(5,n)
print([A248340(n) for n in range(41)]) # G. C. Greubel, Nov 13 2024

G.f.: 5*x/((1-5*x)*(1-10*x)).
a(n) = 15*a(n-1) - 50*a(n-2).
a(n) = 5^n*(2^n-1) = A000351(n) * A000225(n) = A011557(n) - A000351(n).
a(n) = 5*A016164(n-1).
a(n) = A016164(n) - A011557(n).
E.g.f.: exp(10*x) - exp(5*x). - G. C. Greubel, Nov 13 2024

cp's OEIS Frontend

A081200 6th binomial transform of (0,1,0,1,0,1,...), A000035.

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A190540 a(n) = 7^n - 2^n.

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A190541 a(n) = 7^n - 3^n.

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A190542 a(n) = 7^n - 4^n.

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A248340 a(n) = 10^n - 5^n.

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