A057651 -id:A057651 - cp's OEIS frontend

A238303 Triangle T(n,k), 0<=k<=n, read by rows given by T(n,0) = 1, T(n,k) = 2 if k>0.

1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 0

Philippe Deléham, Feb 24 2014

Row sums are A005408(n).
Diagonals sums are A109613(n).
Sum_{k=0..n} T(n,k)*x^k = A033999(n), A000012(n), A005408(n), A036563(n+2), A058481(n+1), A083584(n), A137410(n), A233325(n), A233326(n), A233328(n), A211866(n+1), A165402(n+1) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 respectively.
Sum_{k=0..n} T(n,k)*x^(n-k) = A151575(n), A000012(n), A040000(n), A005408(n), A033484(n), A048473(n), A020989(n), A057651(n), A061801(n), A238275(n), A238276(n), A138894(n), A090843(n), A199023(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 respectively.
Sum_{k=0..n} T(n,k)^x = A000027(n+1), A005408(n), A016813(n), A017077(n) for x = 0, 1, 2, 3 respectively.
Sum_{k=0..n} k*T(n,k) = A002378(n).
Sum_{k=0..n} A000045(k)*T(n,k) = A019274(n+2).
Sum_{k=0..n} A000142(k)*T(n,k) = A066237(n+1).

			Triangle begins:
1;
1, 2;
1, 2, 2;
1, 2, 2, 2;
1, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2;
...

Antti Karttunen, Table of n, a(n) for n = 0..22154; the first 210 rows of triangle

Cf. Diagonals: A040000.
Cf. Columns: A000012, A007395.
First differences of A001614.

A238303(n) = (2-ispolygonal(n,3)); \\ Antti Karttunen, Jan 19 2025

T(n,0) = A000012(n) = 1, T(n+k,k) = A007395(n) = 2 for k>0.

Data section extended to a(104) by Antti Karttunen, Jan 19 2025

A165755 a(n) = (5-3*5^n)/2.

Original entry on oeis.org

1, -5, -35, -185, -935, -4685, -23435, -117185, -585935, -2929685, -14648435, -73242185, -366210935, -1831054685, -9155273435, -45776367185, -228881835935, -1144409179685, -5722045898435, -28610229492185

Offset: 0

Philippe Deléham, Sep 26 2009

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (6, -5).

(5-3*5^Range[0,20])/2 (* or *) LinearRecurrence[{6,-5},{1,-5},20] (* Harvey P. Dale, Apr 18 2013 *)

x='x+O('x^99); Vec((1-11*x)/(1-6*x+5*x^2)) \\ Altug Alkan, Apr 07 2016

a(n) = 5*a(n-1) - 10, a(0)=1.
a(n) = 6*a(n-1)-5*a(n-2), a(0)= 1, a(1)= -5, for n>1.
G.f.: (1-11x)/(1-6x+5x^2).
a(n) = Sum_{0<=k<=n} A112555(n,k)*(-6)^(n-k).
a(n) = (-5)*A057651(n-1).
E.g.f.: (1/2)*(5*exp(x) - 3*exp(5*x)). - G. C. Greubel, Apr 07 2016

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

A238366 a(n) = 5*a(n-2) + 2, a(0) = 1, a(1) = 2.

Original entry on oeis.org

1, 2, 7, 12, 37, 62, 187, 312, 937, 1562, 4687, 7812, 23437, 39062, 117187, 195312, 585937, 976562, 2929687, 4882812, 14648437, 24414062, 73242187, 122070312, 366210937, 610351562, 1831054687, 3051757812, 9155273437, 15258789062, 45776367187, 76293945312

Offset: 0

Philippe Deléham, Feb 25 2014

Row sums of triangle in A152717.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,5,-5).

Cf. A057651, A125831, A152717, A198306.

LinearRecurrence[{1,5,-5},{1,2,7},40] (* Harvey P. Dale, Jul 18 2024 *)

G.f.: (1+x)/((1-x)*(1-5*x^2)).
a(n) = Sum_{k=0..n} A152717(n,k).
a(2*n) = A057651(n).
a(2*n+1) = A125831(n+1) = 2*A003463(n+1).
a(n) = a(n-1) + 5*a(n-2) - 5*a(n-3), a(0) = 1, a(1) = 2, a(2) = 7.
a(n) = A198306(n+1) for n > 1. - Georg Fischer, Oct 23 2018

A090842 Square array of numbers read by antidiagonals where T(n,k) = ((k+3)*(k+2)^n-2)/(k+1).

Original entry on oeis.org

1, 1, 4, 1, 5, 10, 1, 6, 17, 22, 1, 7, 26, 53, 46, 1, 8, 37, 106, 161, 94, 1, 9, 50, 187, 426, 485, 190, 1, 10, 65, 302, 937, 1706, 1457, 382, 1, 11, 82, 457, 1814, 4687, 6826, 4373, 766, 1, 12, 101, 658, 3201, 10886, 23437, 27306, 13121, 1534, 1, 13, 122, 911, 5266

Offset: 0

Paul Barry, Dec 09 2003

Nodes on a tree with degree k interior nodes and degree 1 boundary nodes.

			Rows begin:
  1 4 10 22 ...
  1 5 17 53 ...
  1 6 26 106 ...
  1 7 37 187 ...

L. He, X. Liu and G. Strang, Trees with Cantor Eigenvalue Distribution, Studies in Applied Mathematics 110 (2), 123-138, 2003.

Rows include A033484, A048473, A020989, A057651, A061801, A090843.

The total number of nodes on a tree with degree k interior nodes and degree 1 boundary nodes is given by N(k, r) = (k*(k-1)^r-2)/(k-2).

G.f.: Sum_{k>=0} (1+x*y)/(1-x*y)/(1-(k+2)*x*y)*y^k. - Vladeta Jovovic, Dec 12 2003

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

A198763 a(n) = 4*5^n-1.

Original entry on oeis.org

3, 19, 99, 499, 2499, 12499, 62499, 312499, 1562499, 7812499, 39062499, 195312499, 976562499, 4882812499, 24414062499, 122070312499, 610351562499, 3051757812499, 15258789062499, 76293945312499, 381469726562499, 1907348632812499

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
```
[4*5^n-1: n in [0..30]]
```

CoefficientList[Series[(3 + x)/(1 - 6*x + 5*x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jan 04 2013 *)
NestList[5#+4&,3,30] (* or *) LinearRecurrence[{6,-5},{3,19},30] (* Harvey P. Dale, Jul 03 2021 *)

a(n) = 5*a(n-1)+4.
a(n) = 6*a(n-1)-5*a(n-2), n>1.
G.f.: (3 + x)/(1 - 6*x + 5*x^2). - Vincenzo Librandi, Jan 04 2013

A198765 7*5^n-1.

Original entry on oeis.org

6, 34, 174, 874, 4374, 21874, 109374, 546874, 2734374, 13671874, 68359374, 341796874, 1708984374, 8544921874, 42724609374, 213623046874, 1068115234374, 5340576171874, 26702880859374, 133514404296874

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

CoefficientList[Series[2*(3 - x)/(1 - 6*x + 5*x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jan 04 2013 *)
7 * 5^Range[0, 19] - 1 (* Alonso del Arte, Dec 05 2013 *)

a(n) = 5*a(n-1) + 4.
a(n) = 6*a(n-1) - 5*a(n-2), n > 1.
G.f.: 2*(3 - x)/(1 - 6*x + 5*x^2). - Vincenzo Librandi, Jan 04 2013

A198766 a(n) = (7*5^n - 1)/2.

Original entry on oeis.org

3, 17, 87, 437, 2187, 10937, 54687, 273437, 1367187, 6835937, 34179687, 170898437, 854492187, 4272460937, 21362304687, 106811523437, 534057617187, 2670288085937, 13351440429687, 66757202148437, 333786010742187, 1668930053710937

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
```
[(7*5^n-1)/2: n in [0..30]];
```

CoefficientList[Series[(3 - x)/(1 - 6*x + 5*x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jan 04 2013 *)
LinearRecurrence[{6,-5},{3,17},30] (* Harvey P. Dale, Jan 23 2015 *)

a(n) = 5*a(n-1)+2.
a(n) = 6*a(n-1)-5*a(n-2), n>1.
G.f.: (3 - x)/(1 - 6*x + 5*x^2). - Vincenzo Librandi, Jan 04 2013
E.g.f.: exp(x)*(7*exp(4*x) - 1)/2. - Stefano Spezia, Mar 08 2025

A198767 8*5^n-1.

Original entry on oeis.org

7, 39, 199, 999, 4999, 24999, 124999, 624999, 3124999, 15624999, 78124999, 390624999, 1953124999, 9765624999, 48828124999, 244140624999, 1220703124999, 6103515624999, 30517578124999, 152587890624999, 762939453124999

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
```
[8*5^n-1: n in [0..30]];
```

CoefficientList[Series[(7 - 3*x)/(1 - 6*x + 5*x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jan 04 2013 *)

a(n) = 5*a(n-1)+4.
a(n) = 6*a(n-1)-5*a(n-2), n>1.
G.f.: (7 - 3*x)/(1 - 6*x + 5*x^2). - Vincenzo Librandi, Jan 04 2013

cp's OEIS Frontend

A238303 Triangle T(n,k), 0<=k<=n, read by rows given by T(n,0) = 1, T(n,k) = 2 if k>0.

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A165755 a(n) = (5-3*5^n)/2.

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A198764 6*5^n-1.

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A238366 a(n) = 5*a(n-2) + 2, a(0) = 1, a(1) = 2.

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A090842 Square array of numbers read by antidiagonals where T(n,k) = ((k+3)*(k+2)^n-2)/(k+1).

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

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Sage

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A198763 a(n) = 4*5^n-1.

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A198765 7*5^n-1.

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