A081340 -id:A081340 - cp's OEIS frontend

A081341 Expansion of exp(3x)cosh(3*x).

1, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368, 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568

Offset: 0

Paul Barry, Mar 18 2003

Binomial transform of A081340. 3rd binomial transform of (1,0,9,0,81,0,729,0,...).
For m > 1, n > 0, A166469(A002110(m)*a(n)) = (n+1)*A000045(m+1). For n > 0, A166469(a(n)) = 2n. - Matthew Vandermast, Nov 05 2009
Number of compositions of even natural numbers in n parts <= 5. - Adi Dani, May 29 2011

			From _Adi Dani_, May 29 2011: (Start)
a(2)=18: there are 18 compositions of even natural numbers into 2 parts <= 5:
  for 0: (0,0);
  for 2: (0,2),(2,0),(1,1);
  for 4: (0,4),(4,0),(1,3),(3,1),(2,2);
  for 6: (1,5),(5,1),(2,4),(4,2),(3,3);
  for 8: (3,5),(5,3),(4,4);
  for 10: (5,5).  (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..125
Index entries for linear recurrences with constant coefficients, signature (6).

Cf. A000244, A034494, A081340, A081342.

a:= proc(n) option remember; `if`(n=0, 1,
      add(3^j*a(n-j), j=1..n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 22 2017

Table[Ceiling[1/2(6^n)], {n, 0, 25}]
CoefficientList[Series[-(-1 + 3 x)/(1 - 6 x), {x, 0, 50}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 21 2011 *)
Join[{1},NestList[6#&,3,30]] (* Harvey P. Dale, May 25 2019 *)

x='x+O('x^66); /* that many terms */
Vec((1-3*x)/(1-6*x)) /* show terms */ /* Joerg Arndt, May 29 2011 */

a(0)=1, a(n) = 6^n/2, n > 0.
G.f.: (1-3*x)/(1-6*x).
E.g.f.: exp(3*x)*cosh(3*x).
a(n) = A000244(n)*A011782(n). - Philippe Deléham, Dec 01 2008
a(n) = ((3+sqrt(9))^n + (3-sqrt(9))^n)/2. - Al Hakanson (hawkuu(AT)gmail.com), Dec 08 2008
a(n) = Sum_{k=0..n} A134309(n,k)*3^k = Sum_{k=0..n} A055372(n,k)*2^k. - Philippe Deléham, Feb 04 2012
From Sergei N. Gladkovskii, Jul 19 2012: (Start)
a(n) = ((8*n-4)*a(n-1) - 12*(n-2)*a(n-2))/n, a(0)=1, a(1)=3.
E.g.f. (exp(6*x) + 1)/2 = 1 + 3*x/(G(0) - 6*x) where G(k) = 6*x + 1 + k - 6*x*(k+1)/G(k+1) (continued fraction, Euler's 1st kind, 1-step). (End)
"INVERT" transform of A000244. - Alois P. Heinz, Sep 22 2017

Typo in A-number fixed by Klaus Brockhaus, Apr 04 2010

A201730 Triangle T(n,k), read by rows, given by (2,1/2,3/2,0,0,0,0,0,0,0,...) DELTA (0,1/2,-1/2,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 2, 0, 5, 1, 0, 14, 6, 0, 0, 41, 26, 1, 0, 0, 122, 100, 10, 0, 0, 0, 365, 363, 63, 1, 0, 0, 0, 1094, 1274, 322, 14, 0, 0, 0, 0, 3281, 4372, 1462, 116, 1, 0, 0, 0, 0, 9842, 14760, 6156, 744, 18, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Dec 04 2011

Riordan array ((1-2x)/(1-4x+3x^2),x^2/(1-4x+3x^2)).

A007318*A201701 as lower triangular matrices.

			Triangle begins:
1
2, 0
5, 1, 0
14, 6, 0, 0
41, 26, 1, 0, 0
122, 100, 10, 0, 0, 0
365, 363, 63, 1, 0, 0, 0

Cf. A007051 (1st column), A261064 (2nd column).

A201730 := proc(n,k)
    (1-2*x)/(1-4*x+(3-y)*x^2) ;
    coeftayl(%,y=0,k) ;
    coeftayl(%,x=0,n) ;
end proc:
seq(seq(A201730(n,k),k=0..n),n=0..12) ; # R. J. Mathar, Dec 06 2011

m = 13;
(* DELTA is defined in A084938 *)
DELTA[Join[{2, 1/2, 3/2}, Table[0, {m}]], Join[{0, 1/2, -1/2}, Table[0, {m}]], m] // Flatten (* Jean-François Alcover, Feb 19 2020 *)

G.f.: (1-2x)/(1-4x+(3-y)*x^2).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A139011(n), A000079(n), A007051(n), A006012(n), A001075(n), A081294(n), A001077(n), A084059(n), A108851(n), A084128(n), A081340(n), A084132(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively.
Sum_{k, k>+0} T(n+k,k) = A081704(n) .
T(n,k) = 3*T(n-1,k)+ Sum_{j>0} T(n-1-j,k-1).
T(n,k) = 4*T(n-1,k)+ T(n-2,k-1) - 3*T(n-2,k) with T(0,0)=1, T(1,0)= 2, T(1,1) = 0 and T(n,k) = 0 if k<0 or if n

A003665 a(n) = 2^(n-1)*( 2^n + (-1)^n ).

Original entry on oeis.org

1, 1, 10, 28, 136, 496, 2080, 8128, 32896, 130816, 524800, 2096128, 8390656, 33550336, 134225920, 536854528, 2147516416, 8589869056, 34359869440, 137438691328, 549756338176, 2199022206976, 8796095119360, 35184367894528, 140737496743936, 562949936644096, 2251799847239680

Offset: 0

N. J. A. Sloane

Binomial transform of expansion of cosh(3*x), the aerated version of A001019, 1,0,9,0,81,0,729,... - Paul Barry, Apr 05 2003
Alternatively: start with the fraction 1/1, take the numerators of fractions built according to the rule: add top and bottom to get the new bottom, add top and 9 times the bottom to get the new top. The limit of the sequence of fractions used to generate this sequence is sqrt(9). - Cino Hilliard, Sep 25 2005
This sequence also gives the number of ordered pairs of subsets (A, B) of {1, 2, ..., n} such that |A u B| is even. (Here "u" stands for the set-theoretic union.) The special case n = 13 can be found as in CRUX Problem 3257. - Walther Janous (walther.janous(AT)tirol.com), Mar 01 2008
For n > 0, a(n) is term (1,1) in the n-th power of the 2 X 2 matrix [1,3; 3,1]. - Gary W. Adamson, Aug 06 2010
a(n) is the number of compositions of n when there are 1 type of 1 and 9 types of other natural numbers. - Milan Janjic, Aug 13 2010
a(n) = ((1+3)^n+(1-3)^n)/2. In general, if b(0),b(1),... is the k-th binomial transform of the sequence ((3^n+(-3)^n)/2), then b(n) = ((k+3)^n+(k-3)^n)/2. More generally, if b(0),b(1),... is the k-th binomial transform of the sequence ((m^n+(-m)^n)/2), then b(n) = ((k+m)^n+(k-m)^n)/2. See A034494, A081340-A081342, A034659. - Charlie Marion, Jun 25 2011
Pisano period lengths: 1, 1, 1, 1, 4, 1, 6, 1, 1, 4, 5, 1, 12, 6, 4, 1, 8, 1, 9, 4, ... - R. J. Mathar, Aug 10 2012
Starting with offset 1 the sequence is the INVERT transform of (1, 9, 9, 9, ...). - Gary W. Adamson, Aug 06 2016

John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, p. 16.
M. Gardner, Riddles of Sphinx, M.A.A., 1987, p. 145.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
R. K. Guy, s-Additive sequences, Preprint, 1994. (Annotated scanned copy)
Bill Sands, Problem 3257, Crux Math. 33 (2007), No.5, p. 298.
Index entries for linear recurrences with constant coefficients, signature (2,8).

Cf. A001019, A001045, A014551, A078008, A098158.

Cf. A034494, A081340-A081342, A034659.

List([0..30], n-> 2^(n-1)*(2^n +(-1)^n)); # G. C. Greubel, Aug 02 2019

[2^(n-1)*( 2^n + (-1)^n ): n in [0..30]]; // Vincenzo Librandi, Aug 19 2011

A003665:=n->2^(n-1)*( 2^n + (-1)^n ): seq(A003665(n), n=0..30); # Wesley Ivan Hurt, Apr 28 2017

CoefficientList[Series[(1+8x)/(1-2x-8x^2), {x,0,30}], x] (* or *)
LinearRecurrence[{2,8}, {1,1}, 30] (* Robert G. Wilson v, Sep 18 2013 *)

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

[2^(n-1)*(2^n +(-1)^n) for n in (0..30)] # G. C. Greubel, Aug 02 2019

From Paul Barry, Mar 01 2003: (Start)
a(n) = 2*a(n-1) + 8*a(n-2), a(0)=a(1)=1.
a(n) = (4^n + (-2)^n)/2.
G.f.: (1-x)/((1+2*x)*(1-4*x)). (End)
From Paul Barry, Apr 05 2003: (Start)
a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2*k)*9^k.
E.g.f. exp(x)*cosh(3*x). (End)
a(n) = (A078008(n) + A001045(n+1))*2^(n-1) = A014551(n)*2^(n-1). - Paul Barry, Feb 12 2004
Given a(0)=1, b(0)=1 then for i=1, 2, ..., a(i)/b(i) = (a(i-1) + 9*b(i-1)) / (a(i-1) + b(i-1)). - Cino Hilliard, Sep 25 2005
a(n) = Sum_{k=0..n} A098158(n,k)*9^(n-k). - Philippe Deléham, Dec 26 2007
a(n) = ((1+sqrt(9))^n + (1-sqrt(9))^n)/2. - Al Hakanson (hawkuu(AT)gmail.com), Dec 08 2008
If p[1]=1, and p[i]=9, (i>1), and if A is Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det(A). - Milan Janjic, Apr 29 2010
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(9*k-1)/(x*(9*k+8) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 28 2013

Entry revised by N. J. A. Sloane, Nov 22 2006

A087404 a(n) = 4a(n-1) + 5a(n-2) for n > 1, with a(0) = 2 and a(1) = 4.

Original entry on oeis.org

2, 4, 26, 124, 626, 3124, 15626, 78124, 390626, 1953124, 9765626, 48828124, 244140626, 1220703124, 6103515626, 30517578124, 152587890626, 762939453124, 3814697265626, 19073486328124, 95367431640626, 476837158203124, 2384185791015626, 11920928955078124, 59604644775390626

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Sep 01 2003

Let F(x) = Product_{n>=0} (1 - x^(3*n+1))/(1 - x^(3*n+2)). This sequence is the simple continued fraction expansion of the real number 1 + F(-1/5) = 2.24761 97788 60361 46849 ... = 2 + 1/(4 + 1/(26 + 1/(124 + 1/(626 + ...)))). See A111317. - Peter Bala, Dec 26 2012

Michael De Vlieger, Table of n, a(n) for n = 0..1430
Weerayuth Nilsrakoo and Achariya Nilsrakoo, On One-Parameter Generalization of Jacobsthal Numbers, WSEAS Trans. Math. (2025) Vol. 24, 51-61. See p. 3.
Index entries for linear recurrences with constant coefficients, signature (4,5).

Cf. A081340, A111317.

CoefficientList[Series[(2 - 4x)/(1 - 4x - 5x^2), {x, 0, 25}], x]
LinearRecurrence[{4,5},{2,4},30] (* Harvey P. Dale, May 13 2022 *)

[lucas_number2(n,4,-5) for n in range(0, 22)] # Zerinvary Lajos, May 14 2009

G.f.: (2 - 4*x)/(1 - 4*x - 5*x^2).
a(n) = 5^n + (-1)^n.
From Elmo R. Oliveira, Aug 23 2024: (Start)
E.g.f.: exp(-x)*(exp(6*x) + 1).
a(n) = 2*A081340(n). (End)

a(22)-a(24) from Elmo R. Oliveira, Aug 23 2024

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cp's OEIS Frontend

A081341 Expansion of exp(3*x)*cosh(3*x).

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A201730 Triangle T(n,k), read by rows, given by (2,1/2,3/2,0,0,0,0,0,0,0,...) DELTA (0,1/2,-1/2,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

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A003665 a(n) = 2^(n-1)*( 2^n + (-1)^n ).

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A087404 a(n) = 4*a(n-1) + 5*a(n-2) for n > 1, with a(0) = 2 and a(1) = 4.

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A081341 Expansion of exp(3x)cosh(3*x).

A087404 a(n) = 4a(n-1) + 5a(n-2) for n > 1, with a(0) = 2 and a(1) = 4.