A257272 -id:A257272 - cp's OEIS frontend

A168415 a(n) = 2^n + 7.

8, 9, 11, 15, 23, 39, 71, 135, 263, 519, 1031, 2055, 4103, 8199, 16391, 32775, 65543, 131079, 262151, 524295, 1048583, 2097159, 4194311, 8388615, 16777223, 33554439, 67108871, 134217735, 268435463, 536870919, 1073741831, 2147483655

Offset: 0

Vincenzo Librandi, Dec 01 2009

a(n) is prime <=> a(n) is in A104066 <=> n is in A057195 <=> 2^(n-1)*a(n) = A257272(n) is in A125247. - M. F. Hasler, Apr 27 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A000079, A057195, A104066, A125247, A257272.

[2^n+7: n in [0..40]]; // Vincenzo Librandi, Sep 19 2013

a[n_]:=2^n+7; a[Range[0,200]] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2011*)
CoefficientList[Series[(8 - 15 x)/((2 x - 1) (x - 1)), {x, 0, 200}], x] (* Vincenzo Librandi, Sep 19 2013 *)
LinearRecurrence[{3,-2},{8,9},40] (* Harvey P. Dale, Mar 03 2014 *)

a(n)=1<Charles R Greathouse IV, Sep 20 2011

a(n) = 2*a(n-1) - 7, n > 1.
G.f.: (8 - 15*x)/((2*x - 1)*(x - 1)). - R. J. Mathar, Jul 10 2011
a(n) = A000079(n) + 7. - Omar E. Pol, Sep 20 2011
E.g.f.: exp(2*x) + 7*exp(x). - G. C. Greubel, Jul 22 2016
a(n) = 3*a(n-1) - 2*a(n-2) for n > 1. - Elmo R. Oliveira, Nov 11 2023

A257273 a(n) = 2^(n-1)*(2^n+3).

Original entry on oeis.org

2, 5, 14, 44, 152, 560, 2144, 8384, 33152, 131840, 525824, 2100224, 8394752, 33566720, 134242304, 536920064, 2147581952, 8590131200, 34360131584, 137439739904, 549757386752, 2199026401280, 8796099313664, 35184384671744, 140737513521152, 562950003752960, 2251799914348544, 9007199456067584

Offset: 0

M. F. Hasler, Apr 27 2015

a(n) is in A125246 <=> n is in A057732 <=> A062709(n) is in A057733.

These are also the row sum of the triangle A146769: For n>=1, a(n-1) is the sum of row n of A146769.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-8).

Cf. A000079, A007582, A028403, A256873, A256871, A257272.

Cf. A062709, A057732, A057733.

[2^(n-1)*(2^n+3): n in [0..35]]; // Vincenzo Librandi, Apr 27 2015

Table[2^(n - 1) (2^n + 3), {n, 0, 30}] (* Bruno Berselli, Apr 27 2015 *)
CoefficientList[Series[(2 - 7 x)/((1 - 4 x) (1 - 2 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Apr 27 2015 *)
LinearRecurrence[{6,-8},{2,5},30] (* Harvey P. Dale, Dec 21 2024 *)

PARI
```
a(n)=2^(n-1)*(2^n+3)
```

Vec((2-7*x)/((1-4*x)*(1-2*x)) + O(x^100)) \\ Colin Barker, Apr 27 2015

G.f.: (2-7*x)/((1-4*x)*(1-2*x)). - Vincenzo Librandi, Apr 27 2015

a(n) = 6*a(n-1)-8*a(n-2). - Colin Barker, Apr 27 2015

A181444 G.f. satisfies: A(x) = 1 + xSum_{n>=0} log( A(2^nx)^2 )^n/n!.

Original entry on oeis.org

1, 1, 4, 60, 2480, 242296, 53763904, 28363717952, 41396018951936, 215328934357721024, 4740698193856769942528, 430771050114778618253200384, 151994706469390446336698323709952

Offset: 0

Paul D. Hanna, Oct 22 2010

nonn

			G.f.: A(x) = 1 + x + 4*x^2 + 60*x^3 + 2480*x^4 + 242296*x^5 +...
A(x) = 1 + x*[1 + log(A(2x)^2) + log(A(4x)^2)^2/2! + log(A(8x)^2)^3/3! + log(A(16x)^2)^4/4! + log(A(32x)^2)^5/5! +...].
Coefficients in the 2^n-th powers of A(x) begin:
A^(2^0)=[1, 1, 4, 60, 2480, 242296, 53763904, 28363717952,...];
A^(2^1)=[(1), 2, 9, 128, 5096, 490032, 108035840, 56837199680,...];
A^(2^2)=[1,(4), 22, 292, 10785, 1002752, 218139920, 114116667872,...];
A^(2^3)=[1, 8,(60), 760, 24390, 2104632, 444861660, 230028874632,...];
A^(2^4)=[1, 16, 184,(2480), 64540, 4690704, 926901832,...];
A^(2^5)=[1, 32, 624, 10848,(242296), 12359328, 2033807312,...];
A^(2^6)=[1, 64, 2272, 61632, 1568240,(53763904), 5278676128,...];
A^(2^7)=[1, 128, 8640, 414080, 16187360, 588318336,(28363717952),...];
A^(2^8)=[1, 256, 33664, 3040000, 213028800, 12475903232, 658516757120,(41396018951936),...]; ...
where the diagonal terms in parenthesis form this sequence (shift left).
The third column of this table is given by A257272 (observation by _Bruno Berselli_). - _M. F. Hasler_, Apr 27 2015

Cf. A132695, A181445.

{a(n)=local(A=1+sum(m=1,n-1,a(m)*x^m));polcoeff(1+x*sum(m=0,n,log(subst(A^2,x,2^m*x)+x*O(x^n))^m/m!),n)}

a(n+1) = [x^n] A(x)^(2^(n+1)) for n>=0, with a(0)=1, where A(x) = Sum_{n>=0} a(n)*x^n.

cp's OEIS Frontend

A168415 a(n) = 2^n + 7.

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A257273 a(n) = 2^(n-1)*(2^n+3).

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Magma

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PARI

Formula

A181444 G.f. satisfies: A(x) = 1 + xSum_{n>=0} log( A(2^nx)^2 )^n/n!.

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PARI

Formula

cp's OEIS Frontend

A168415 a(n) = 2^n + 7.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A257273 a(n) = 2^(n-1)*(2^n+3).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

A181444 G.f. satisfies: A(x) = 1 + x*Sum_{n>=0} log( A(2^n*x)^2 )^n/n!.

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Author

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PARI

Formula

A181444 G.f. satisfies: A(x) = 1 + xSum_{n>=0} log( A(2^nx)^2 )^n/n!.