A081655 -id:A081655 - cp's OEIS frontend

A057651 a(n) = (3*5^n - 1)/2.

1, 7, 37, 187, 937, 4687, 23437, 117187, 585937, 2929687, 14648437, 73242187, 366210937, 1831054687, 9155273437, 45776367187, 228881835937, 1144409179687, 5722045898437, 28610229492187, 143051147460937, 715255737304687, 3576278686523437, 17881393432617187, 89406967163085937

Offset: 0

N. J. A. Sloane, Oct 13 2000

Sum of n-th row of triangle of powers of 5: 1; 1 5 1; 1 5 25 5 1 ; 1 5 25 125 25 5 1; ... - Philippe Deléham, Feb 23 2014

			a(0) = 1;
a(1) = 1 + 5 + 1 = 7;
a(2) = 1 + 5 + 25 + 5 + 1 = 37;
a(3) = 1 + 5 + 25 + 125 + 25 + 5 + 1 = 187; etc. - _Philippe Deléham_, Feb 23 2014
G.f. = 1 + 7*x + 37*x^2 + 187*x^3 + 937*x^4 + 4687*x^5 + 23437*x^6 + ...

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

Cf. A024049, A081655, A097162, A198762.

Cf. A020989, A061801, A112468, A112739.

[(3*5^n-1)/2: n in [0..30]]; // Vincenzo Librandi, Oct 30 2011

G.f=(1+x)/(1-5*x)/(1-x): gser:=series(g, x=0, 43): seq(coeff(gser, x, n), n=0..30); # Zerinvary Lajos, Jan 11 2009

Table[(3*5^n-1)/2,{n,0,30}] (* Vladimir Joseph Stephan Orlovsky, Jan 29 2012 *)

a(n)=3*5^n\2 \\ Charles R Greathouse IV, Dec 22 2011

G.f.: (1+x)/(1 - 6*x + 5*x^2).
a(0)=1, a(n) = 5*a(n-1) + 2; a(n) = a(n-1) + 6*(5^(n-1)). - Amarnath Murthy, May 27 2001
a(n) = 6*a(n-1) - 5*a(n-2), n > 1. - Vincenzo Librandi, Oct 30 2011
a(n) = Sum_{k=0..n} A112468(n,k)*6^k. - Philippe Deléham, Feb 23 2014
From Elmo R. Oliveira, Mar 29 2025: (Start)
E.g.f.: exp(x)*(3*exp(4*x) - 1)/2.
a(n) = A097162(2*n) = A198762(n)/2. (End)

A198480 a(n) = 2*7^n - 1.

Original entry on oeis.org

1, 13, 97, 685, 4801, 33613, 235297, 1647085, 11529601, 80707213, 564950497, 3954653485, 27682574401, 193778020813, 1356446145697, 9495123019885, 66465861139201, 465261027974413, 3256827195820897, 22797790370746285

Offset: 0

Vincenzo Librandi, Oct 27 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-7).

Cf. A081655, A048473.

Magma
```
[2*7^n-1: n in [0..20]]
```

CoefficientList[Series[(1+5*x)/((1-x)*(1-7*x)),{x,0,40}],x] (* Vincenzo Librandi, Jul 06 2012 *)
2*7^Range[0,20]-1 (* or *) LinearRecurrence[{8,-7},{1,13},20] (* Harvey P. Dale, Aug 19 2022 *)

A198480(n):=2*7^n-1$ makelist(A198480(n),n,0,20); /* Martin Ettl, Nov 09 2012 */

a(n)=2*7^n-1 \\ Charles R Greathouse IV, Dec 28 2011

a(n) = 7*a(n-1)+6, n>0.
G.f.: (1+5*x)/((1-x)*(1-7*x)). - Vincenzo Librandi, Jul 06 2012
a(n) = 8*a(n-1) -7*a(n-2). - Vincenzo Librandi, Jul 06 2012

A081654 a(n) = 2*4^n - 0^n.

Original entry on oeis.org

1, 8, 32, 128, 512, 2048, 8192, 32768, 131072, 524288, 2097152, 8388608, 33554432, 134217728, 536870912, 2147483648, 8589934592, 34359738368, 137438953472, 549755813888, 2199023255552, 8796093022208, 35184372088832

Offset: 0

Paul Barry, Mar 26 2003

Binomial transform of A081632. Inverse binomial transform of A081655.

			a(0) = 2*4^0 - 0^0 = 2 - 1 = 1 (use 0^0 = 1).

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

Cf. A000244 (3^n), A187093.

Essentially the same as A004171.

[2*4^n-0^n: n in [0..30]]; // Vincenzo Librandi, Aug 10 2013

CoefficientList[Series[(1 + 4 x) / (1 - 4 x), {x, 0, 40}], x] (* Vincenzo Librandi, Aug 10 2013 *)

a(n)=2*4^n-0^n \\ Charles R Greathouse IV, Apr 09 2012

x='x+O('x^100); Vec((1+4*x)/(1-4*x)) \\ Altug Alkan, Dec 14 2015

a(0)=1, a(n) = 2*4^n, n>0
G.f.: (1+4*x)/(1-4*x).
E.g.f. 2*exp(4*x)-1.
With interpolated zeros, this is 2^n - 0^n + (-2)^n. - Paul Barry, Sep 06 2003
a(n) = A081294(n+1), n>0. - R. J. Mathar, Sep 17 2008
For n>0, a(n) = 2 * (1 + 3^(n-1) + Sum{x=1..n-2}Sum{k=0..x-1}(binomial(x-1,k)*(3^(k+1) + 3^(n-x+k)))). - J. Conrad, Dec 10 2015

A081656 a(n) = 2*6^n - 2^n.

Original entry on oeis.org

1, 10, 68, 424, 2576, 15520, 93248, 559744, 3358976, 20154880, 120931328, 725592064, 4353560576, 26121379840, 156728311808, 940369936384, 5642219749376, 33853318758400, 203119913074688, 1218719479496704, 7312316879077376, 43873901278658560, 263243407680339968

Offset: 0

Paul Barry, Mar 26 2003

Binomial transform of A081655.

Inverse binomial transform of A081657.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-12).

Third column of array A094424.

Cf. A081655, A081657.

[2*6^n-2^n: n in [0..30]]; // Vincenzo Librandi, Aug 10 2013

CoefficientList[Series[(1 + 2 x) / ((1 - 6 x) (1 - 2 x)), {x, 0, 40}], x] (* Vincenzo Librandi, Aug 10 2013 *)
Table[2*6^n-2^n,{n,0,20}] (* or *) LinearRecurrence[{8,-12},{1,10},20] (* Harvey P. Dale, Mar 17 2019 *)

a(n)=2*6^n-2^n \\ Charles R Greathouse IV, Oct 07 2015

G.f.: (1+2*x)/((1-6*x)(1-2*x)).
E.g.f.: 2*exp(6*x) - exp(2*x).
a(0)=1, a(2)=10; for n>2, a(n) = 8*a(n-1) - 12*a(n-2). - Vincenzo Librandi, Aug 10 2013

A198688 6*7^n-1.

Original entry on oeis.org

5, 41, 293, 2057, 14405, 100841, 705893, 4941257, 34588805, 242121641, 1694851493, 11863960457, 83047723205, 581334062441, 4069338437093, 28485369059657, 199397583417605, 1395783083923241, 9770481587462693, 68393371112238857

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 (8,-7).

Cf. A048473, A081655, A198480, A198647.

Magma
```
[6*7^n-1: n in [0..30]]
```

a(n) = 7*a(n-1)+6. a(n) = 8*a(n-1)-7*a(n-2), n>1.

G.f. ( 5+x ) / ( (7*x-1)*(x-1) ). - R. J. Mathar, Oct 30 2011

A198762 a(n) = 35^n - 1 = 2A057651(n).

Original entry on oeis.org

2, 14, 74, 374, 1874, 9374, 46874, 234374, 1171874, 5859374, 29296874, 146484374, 732421874, 3662109374, 18310546874, 91552734374, 457763671874, 2288818359374, 11444091796874, 57220458984374, 286102294921874, 1430511474609374, 7152557373046874, 35762786865234374

Offset: 0

Vincenzo Librandi, Oct 30 2011

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

Cf. A024049, A057651, A081655.

Magma
```
[(3*5^n-1): n in [0..30]];
```

CoefficientList[Series[2*(1 + x)/(1 - 6*x + 5*x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jan 04 2013 *)
LinearRecurrence[{6,-5},{2,14},30] (* Harvey P. Dale, Mar 19 2013 *)

a(n) = 5*a(n-1) + 4.
a(n) = 6*a(n-1) - 5*a(n-2), n > 1.
G.f.: 2*(1 + x)/(1 - 6*x + 5*x^2). - Vincenzo Librandi, Jan 04 2013
E.g.f.: exp(x)*(3*exp(4*x) - 1). - Elmo R. Oliveira, Mar 29 2025

A198764 6*5^n-1.

Original entry on oeis.org

5, 29, 149, 749, 3749, 18749, 93749, 468749, 2343749, 11718749, 58593749, 292968749, 1464843749, 7324218749, 36621093749, 183105468749, 915527343749, 4577636718749, 22888183593749, 114440917968749, 572204589843749, 2861022949218749

Offset: 0

Vincenzo Librandi, Oct 30 2011

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

Cf. A024049, A057651, A081655.

Magma
```
[6*5^n-1: n in [0..30]]
```

CoefficientList[Series[(5 - x)/(1 - 6*x + 5*x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jan 04 2013 *)
6*5^Range[0,30]-1 (* or *) LinearRecurrence[{6,-5},{5,29},30] (* Harvey P. Dale, Dec 21 2014 *)

a(n) = 5*a(n-1)+4.
a(n) = 6*a(n-1)-5*a(n-2), n>1.
G.f.: (5 - x)/(1 - 6*x + 5*x^2). - Vincenzo Librandi, Jan 04 2013

A137215 a(n) = 3(10^n) + (n^2 + 1)(10^n - 1)/9.

Original entry on oeis.org

3, 32, 355, 4110, 48887, 588886, 7111107, 85555550, 1022222215, 12111111102, 142222222211, 1655555555542, 19111111111095, 218888888888870, 2488888888888867, 28111111111111086, 315555555555555527, 3522222222222222190, 39111111111111111075, 432222222222222222182

Offset: 0

Ctibor O. Zizka, Mar 06 2008

Sequence generalized: a(n) = a(0)*(B^n) + F(n)* ((B^n)-1)/(B-1); a(0), B integers, F(n) arithmetic function.
Examples:
a(0) = 1, B = 10, F(n) = 1 gives A002275, F(n) = 2 gives A090843, F(n) = 3 gives A097166, F(n) = 4 gives A099914, F(n) = 5 gives A099915.
a(0) = 1, B = 2, F(n) = 1 gives A000225, F(n) = 2 gives A033484, F(n) = 3 gives A036563, F(n) = 4 gives A048487, F(n) = 5 gives A048488, F(n) = 6 gives A048489.
a(0) = 1, B = 3, F(n) = 1 gives A003462, F(n) = 2 gives A048473, F(n) = 3 gives A134931, F(n) = 4 gives A058481, F(n) = 5 gives A116952.
a(0) = 1, B = 4, F(n) = 1 gives A002450, F(n) = 2 gives A020989, F(n) = 3 gives A083420, F(n) = 4 gives A083597, F(n) = 5 gives A083584.
a(0) = 1, B = 5, F(n) = 1 gives A003463, F(n) = 2 gives A057651, F(n) = 3 gives A117617, F(n) = 4 gives A081655.
a(0) = 2, B = 10, F(n) = 1 gives A037559, F(n) = 2 gives A002276.

			a(3) = 3*10^3 + (3*3 + 1)*(10^3 - 1)/9 = 4110.

G. C. Greubel, Table of n, a(n) for n = 0..985
Index entries for linear recurrences with constant coefficients, signature (33,-393,1991,-3930,3300,-1000).

Cf. A002275, A011557.

Table[3*10^n +(n^2 +1)*(10^n -1)/9, {n,0,30}] (* G. C. Greubel, Jan 05 2022 *)

a(n) = 3*(10^n) + (n*n+1)*((10^n)-1)/9; \\ Jinyuan Wang, Feb 27 2020

[3*10^n +(1+n^2)*(10^n -1)/9 for n in (0..30)] # G. C. Greubel, Jan 05 2022

a(n) = 3*(10^n) + (n^2 + 1)*(10^n - 1)/9.

O.g.f.: (3 - 67*x + 478*x^2 - 1002*x^3 + 850*x^4 - 100*x^5)/((1-x)^3 * (1-10*x)^3). - R. J. Mathar, Mar 16 2008

More terms from R. J. Mathar, Mar 16 2008

More terms from Jinyuan Wang, Feb 27 2020

A198686 4*7^n-1.

Original entry on oeis.org

3, 27, 195, 1371, 9603, 67227, 470595, 3294171, 23059203, 161414427, 1129900995, 7909306971, 55365148803, 387556041627, 2712892291395, 18990246039771, 132931722278403, 930522055948827, 6513654391641795, 45595580741492571

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 (8,-7).

Cf. A048473, A081655, A198480, A198647.

Magma
```
[4*7^n-1: n in [0..30]]
```

4*7^Range[0,20]-1 (* or *) LinearRecurrence[{8,-7},{3,27},20] (* Harvey P. Dale, Dec 27 2011 *)

a(n) = 7*a(n-1)+6. a(n) = 8*a(n-1)-7*a(n-2), n>1.

G.f. ( 3+3*x ) / ( (7*x-1)*(x-1) ). - R. J. Mathar, Oct 30 2011

A198687 5*7^n-1.

Original entry on oeis.org

4, 34, 244, 1714, 12004, 84034, 588244, 4117714, 28824004, 201768034, 1412376244, 9886633714, 69206436004, 484445052034, 3391115364244, 23737807549714, 166164652848004, 1163152569936034, 8142067989552244, 56994475926865714

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 (8,-7).

Cf. A048473, A081655, A198480, A198647.

Magma
```
[5*7^n-1: n in [0..30]]
```

CoefficientList[Series[(4+2*x)/((1-x)*(1-7*x)),{x,0,40}],x] (* Vincenzo Librandi, Jul 06 2012 *)
LinearRecurrence[{8,-7},{4,34},20] (* Harvey P. Dale, Jul 23 2024 *)

a(n) = 7*a(n-1)+6 = 8*a(n-1)-7*a(n-2), n>1.

G.f.:(4+2*x)/((1-x)*(1-7*x)). - Vincenzo Librandi, Jul 06 2012

cp's OEIS Frontend

A057651 a(n) = (3*5^n - 1)/2.

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A198480 a(n) = 2*7^n - 1.

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A081654 a(n) = 2*4^n - 0^n.

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A081656 a(n) = 2*6^n - 2^n.

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A198688 6*7^n-1.

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A198762 a(n) = 3*5^n - 1 = 2*A057651(n).

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A198764 6*5^n-1.

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A198762 a(n) = 35^n - 1 = 2A057651(n).

A137215 a(n) = 3(10^n) + (n^2 + 1)(10^n - 1)/9.