A176916 -id:A176916 - cp's OEIS frontend

A176972 a(n) = 7^n + 7*n + 1.

2, 15, 64, 365, 2430, 16843, 117692, 823593, 5764858, 40353671, 282475320, 1977326821, 13841287286, 96889010499, 678223072948, 4747561510049, 33232930569714, 232630513987327, 1628413597910576, 11398895185373277, 79792266297612142, 558545864083284155, 3909821048582988204

Offset: 0

Jonathan Vos Post, Apr 29 2010

			a(5) = 7^5 + 7*5 + 1 = 16843 is prime.

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (9,-15,7).

Cf. A000420, A008589, A016993, A176691, A176805, A176916.

Cf. A370664.

[7^n + 7*n + 1: n in [0..25]]; // Vincenzo Librandi, May 06 2011

Table[7^n+7n+1,{n,0,20}] (* or *) LinearRecurrence[{9,-15,7},{2,15,64},20] (* Harvey P. Dale, Apr 17 2014 *)

a(n) = A000420(n) + A008589(n) + 1 = A000420(n) + A016993(n).
a(n) = 7*a(n-1) - 42*(n-1) + 1, with n > 0. For n=5, a(5) = 7*2430 - 42*4 + 1 = 16843. - Bruno Berselli, May 18 2010
From R. J. Mathar, May 22 2010: (Start)
a(n) = 9*a(n-1) - 15*a(n-2) + 7*a(n-3).
G.f.: (-2 + 3*x + 41*x^2) / ((7*x-1)*(x-1)^2). (End)
E.g.f.: exp(x)*(1 + exp(6*x) + 7*x). - Stefano Spezia, Aug 19 2024

A221907 a(n) = 5^n + 5*n.

Original entry on oeis.org

1, 10, 35, 140, 645, 3150, 15655, 78160, 390665, 1953170, 9765675, 48828180, 244140685, 1220703190, 6103515695, 30517578200, 152587890705, 762939453210, 3814697265715, 19073486328220, 95367431640725, 476837158203230, 2384185791015735, 11920928955078240, 59604644775390745

Offset: 0

Vincenzo Librandi, Mar 02 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-11,5).

Cf. A107585, A176916, A362555, A370557.

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

I:=[1, 10, 35]; [n le 3 select I[n] else 7*Self(n-1)-11*Self(n-2)+5*Self(n-3): n in [1..30]];

Table[(5^n + 5 n), {n, 0, 30}] (* or *) CoefficientList[Series[(1 + 3 x - 24 x^2)/((1 - x)^2 (1 - 5 x)), {x, 0, 30}], x]
LinearRecurrence[{7,-11,5},{1,10,35},30] (* Harvey P. Dale, Jul 23 2013 *)

a(n)=5^n+5*n \\ Charles R Greathouse IV, Apr 18 2013

G.f.: (1+3*x-24*x^2)/((1-x)^2*(1-5*x)).
a(n) = 7*a(n-1) - 11*a(n-2) + 5*a(n-3).
a(n) = A176916(n) - 1.
a(n) = 5*A362555(n) for n > 0. - Hugo Pfoertner, Mar 01 2024
E.g.f.: exp(x)*(exp(4*x) + 5*x). - Elmo R. Oliveira, Sep 10 2024

A375577 Array read by ascending antidiagonals: A(n,k) = k^n + k*n + 1.

Original entry on oeis.org

2, 1, 2, 1, 3, 2, 1, 4, 5, 2, 1, 5, 9, 7, 2, 1, 6, 15, 16, 9, 2, 1, 7, 25, 37, 25, 11, 2, 1, 8, 43, 94, 77, 36, 13, 2, 1, 9, 77, 259, 273, 141, 49, 15, 2, 1, 10, 143, 748, 1045, 646, 235, 64, 17, 2, 1, 11, 273, 2209, 4121, 3151, 1321, 365, 81, 19, 2

Offset: 0

Stefano Spezia, Aug 19 2024

			Array begins:
  2, 2,  2,   2,    2,     2, ...
  1, 3,  5,   7,    9,    11, ...
  1, 4,  9,  16,   25,    36, ...
  1, 5, 15,  37,   77,   141, ...
  1, 6, 25,  94,  273,   646, ...
  1, 7, 43, 259, 1045,  3151, ...
  1, 8, 77, 748, 4121, 15656, ...
  ...

Cf. A000290, A004247, A004248, A005408 (n=1), A005491 (n=3), A007395 (n=0), A054977 (k=0), A176691 (k=2), A176805 (k=3), A176916 (k=5), A176972 (k=7), A214647.

Cf. A375578 (antidiagonal sums).

A[0,0]=2; A[n_,k_]:=k^n+k*n+1;Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten

G.f. for the k-th column: (2*x^2 - 3*x - k^2 + k + 1)/((x - 1)^2*(x - k)).
E.g.f. for the k-th column: exp(x)*(1 + exp((k-1)*x) + k*x).
A(n,1) = n + 2.
A(2,n) = A000290(n+1).
A(n,n) = 2*A214647(n) + 1.

A268414 a(n) = 5a(n-1) - 2n for n > 0, a(0) = 1.

Original entry on oeis.org

1, 3, 11, 49, 237, 1175, 5863, 29301, 146489, 732427, 3662115, 18310553, 91552741, 457763679, 2288818367, 11444091805, 57220458993, 286102294931, 1430511474619, 7152557373057, 35762786865245, 178813934326183, 894069671630871, 4470348358154309, 22351741790771497, 111758708953857435

Offset: 0

Ilya Gutkovskiy, Feb 04 2016

In general, the ordinary generating function for the recurrence relation b(n) = k*b(n - 1) - m*n, with n > 0 and b(0)=1, is (1 - (m + 2)*x + x^2)/((1 - x)^2*(1 - k*x)). This recurrence gives the closed form b(n) = ((k^2 - k*(m + 2) + 1)*k^n + m*((k - 1)*n + k))/(k - 1)^2.

Index entries for linear recurrences with constant coefficients, signature (7,-11,5).

Cf. A014827, A024050, A094195, A104745, A107585, A164045, A176916, A221907.

[(4*n + 3*5^n + 5)/8: n in [0..30]]; // Vincenzo Librandi, Feb 06 2016

Table[(4 n + 3 5^n + 5)/8, {n, 0, 23}]
LinearRecurrence[{7, -11, 5}, {1, 3, 11}, 24]

Vec((1-4*x+x^2)/((1-x)^2*(1-5*x)) + O(x^100)) \\ Altug Alkan, Feb 04 2016

G.f.: (1 - 4*x + x^2)/((1 - x)^2*(1 - 5*x)).
a(n) = (4*n + 3*5^n + 5)/8.
Sum_{n>=0} 1/a(n) = 1.449934283402232875...
Lim_{n -> oo} a(n + 1)/a(n) = 5.
From Elmo R. Oliveira, Sep 10 2024: (Start)
E.g.f.: exp(x)*(3*exp(4*x) + 4*x + 5)/8.
a(n) = 7*a(n-1) - 11*a(n-2) + 5*a(n-3) for n > 2. (End)

a(24)-a(25) from Elmo R. Oliveira, Sep 10 2024

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A176972 a(n) = 7^n + 7*n + 1.

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A221907 a(n) = 5^n + 5*n.

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A375577 Array read by ascending antidiagonals: A(n,k) = k^n + k*n + 1.

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A268414 a(n) = 5*a(n-1) - 2*n for n > 0, a(0) = 1.

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A268414 a(n) = 5a(n-1) - 2n for n > 0, a(0) = 1.