A084214 -id:A084214 - cp's OEIS frontend

A182349 G.f.: exp( Sum_{n>=1} 6 * A084214(n) * x^n/n ) where g.f. of A084214 is (1+x^2)/((1+x)(1-2x)).

1, 6, 30, 120, 435, 1446, 4536, 13560, 39045, 108950, 296178, 787368, 2053335, 5265750, 13306380, 33188040, 81815145, 199585830, 482290630, 1155444120, 2746489851, 6481600326, 15195437280, 35407315800, 82038719565, 189089191926, 433704632346, 990244936520

Offset: 0

Paul D. Hanna, Apr 25 2012

nonn

			G.f.: A(x) = 1 + 6*x + 30*x^2 + 120*x^3 + 435*x^4 + 1446*x^5 + 4536*x^6 +...
such that
log(A(x))/6 = x + 4*x^2/2 + 6*x^3/3 + 14*x^4/4 + 26*x^5/5 + 54*x^6/6 + 106*x^7/7 + 214*x^8/8 +...+ A084214(n) * x^n/n +...

Index entries for linear recurrences with constant coefficients, signature (6,-6,-24,39,42,-72,-48,48,32).

Cf. A084214.

CoefficientList[Series[1/((1+x)^4(1-2x)^5),{x,0,30}],x] (* or *) LinearRecurrence[{6,-6,-24,39,42,-72,-48,48,32},{1,6,30,120,435,1446,4536,13560,39045},30] (* Harvey P. Dale, Aug 11 2021 *)

{A084214(n)=polcoeff((1+x^2)/((1+x)*(1-2*x+x*O(x^n))), n)}
{a(n)=polcoeff(exp(sum(k=1, n, 6*A084214(k)*x^k/k)+x*O(x^n)), n)}
for(n=0, 16, print1(a(n), ", "))

G.f.: 1/((1+x)^4*(1-2*x)^5).

A375476 a(3n)=A001045(n+1), a(3n+1)=A084214(n), a(3*n+2)=A000079(n) for n >= 0.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 4, 4, 5, 6, 8, 11, 14, 16, 21, 26, 32, 43, 54, 64, 85, 106, 128, 171, 214, 256, 341, 426, 512, 683, 854, 1024, 1365, 1706, 2048, 2731, 3414, 4096, 5461, 6826, 8192, 10923, 13654, 16384, 21845, 27306, 32768, 43691, 54614, 65536

Offset: 0

Paul Curtz, Aug 18 2024

Index entries for linear recurrences with constant coefficients, signature (0,0,1,0,0,2).

Cf. A000079, A001045, A084214.

G.f. 1 + x - x^2*(1+x+x^2+x^3+2*x^4+3*x^5) / ( (1+x)*(2*x^3-1)*(x^2-x+1) ).

A048654 a(n) = 2*a(n-1) + a(n-2); a(0)=1, a(1)=4.

Original entry on oeis.org

1, 4, 9, 22, 53, 128, 309, 746, 1801, 4348, 10497, 25342, 61181, 147704, 356589, 860882, 2078353, 5017588, 12113529, 29244646, 70602821, 170450288, 411503397, 993457082, 2398417561, 5790292204

Offset: 0

Barry E. Williams

Generalized Pellian with second term equal to 4.

The generalized Pellian with second term equal to s has the terms a(n) = A000129(n)*s + A000129(n-1). The generating function is -(1+s*x-2*x)/(-1+2*x+x^2). - R. J. Mathar, Nov 22 2007

T. D. Noe, Table of n, a(n) for n = 0..300
Andreas M. Hinz and Paul K. Stockmeyer, Precious Metal Sequences and Sierpinski-Type Graphs, J. Integer Seq., Vol 25 (2022), Article 22.4.8.
A. F. Horadam, Pell Identities, Fib. Quart., Vol. 9, No. 3, 1971, pp. 245-252.
A. F. Horadam, Basic Properties of a Certain Generalized Sequence of Numbers, Fibonacci Quarterly, Vol. 3, No. 3, 1965, pp. 161-176.
A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434.
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (2,1).

Cf. A000129, A001333, A048655, A038761, A084214, A100525.

a048654 n = a048654_list !! n
a048654_list =
   1 : 4 : zipWith (+) a048654_list (map (* 2) $ tail a048654_list)
-- Reinhard Zumkeller, Aug 01 2011

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!((1+2*x)/(1-2*x-x^2))); // G. C. Greubel, Jul 26 2018

LinearRecurrence[{2,1},{1,4},30] (* Harvey P. Dale, Jul 27 2011 *)

a[0]:1$
a[1]:4$
a[n]:=2*a[n-1]+a[n-2]$
A048654(n):=a[n]$
makelist(A048654(n),n,0,30); /* Martin Ettl, Nov 03 2012 */

a(n)=(([0, 1; 1,2]^n)*[1,4]~)[1] \\ Charles R Greathouse IV, May 18 2015

[lucas_number1(n+1,2,-1) +2*lucas_number1(n,2,-1) for n in (0..40)] # G. C. Greubel, Aug 09 2022

a(n) = ((3+sqrt(2))*(1+sqrt(2))^n - (3-sqrt(2))*(1-sqrt(2))^n)/2*sqrt(2).
a(n) = 2*A000129(n+2) - 3*A000129(n+1). - Creighton Dement, Oct 27 2004
G.f.: (1+2*x)/(1-2*x-x^2). - Philippe Deléham, Nov 03 2008
a(n) = binomial transform of 1, 3, 2, 6, 4, 12, ... . - Al Hakanson (hawkuu(AT)gmail.com), Aug 08 2009
E.g.f.: exp(x)*cosh(sqrt(2)*x) + 3*exp(x)*sinh(sqrt(2)*x)/sqrt(2). - Vaclav Kotesovec, Feb 16 2015
a(n) is the denominator of the continued fraction [4, 2, ..., 2, 4] with n-1 2's in the middle. For the numerators, see A221174. - Greg Dresden and Tongjia Rao, Sep 02 2021
a(n) = A001333(n) + A000129(n). - G. C. Greubel, Aug 09 2022

A048573 a(n) = a(n-1) + 2*a(n-2), a(0)=2, a(1)=3.

Original entry on oeis.org

2, 3, 7, 13, 27, 53, 107, 213, 427, 853, 1707, 3413, 6827, 13653, 27307, 54613, 109227, 218453, 436907, 873813, 1747627, 3495253, 6990507, 13981013, 27962027, 55924053, 111848107, 223696213, 447392427, 894784853, 1789569707, 3579139413, 7158278827, 14316557653

Offset: 0

Michael Somos, Jun 17 1999

Number of positive integers requiring exactly n signed bits in the modified non-adjacent form representation. - Ralf Stephan, Aug 02 2003
The n-th entry (n>1) of the sequence is equal to the 1,1-entry of the n-th power of the unnormalized 4 X 4 Haar matrix: [1 1 1 0 / 1 1 -1 0 / 1 1 0 1 / 1 1 0 -1]. - Simone Severini, Oct 27 2004
Pisano period lengths:  1, 1, 6, 2, 2, 6, 6, 2, 18, 2, 10, 6, 12, 6, 6, 2, 8, 18, 18, 2, ... - R. J. Mathar, Aug 10 2012
For n >= 1, a(n) is the number of ways to tile a strip of length n+2 with blue squares and blue and red dominos, with the restriction that the first two tiles must be the same color. - Guanji Chen and Greg Dresden, Jul 15 2024

			G.f. = 2 + 3*x + 7*x^2 + 13*x^3 + 27*x^4 + 53*x^5 + 107*x^6 + 213*x^7 + 427*x^8 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Wieb Bosma, Signed bits and fast exponentiation, Journal de théorie des nombres de Bordeaux, Vol. 13, No. 1 (2001), pp. 27-41.
Karl Dilcher and Larry Ericksen, Continued fractions and Stern polynomials, Ramanujan Journal 45.3 (2018): 659-681. See Table 2.
Karl Dilcher and Hayley Tomkins, Square classes and divisibility properties of Stern polynomials, Integers, Vol. 18 (2018), Article #A29.
Petro Kosobutskyy, The Collatz problem as a reverse n->0 problem on a graph tree formed from theta*2^n Jacobsthal-type numbers, arXiv:2306.14635 [math.GM], 2023.
Petro Kosobutskyy and Dariia Rebot, Collatz conjecture 3n+/-1 as a Newton binomial problem, Comp. Des. Sys. Theor. Prac., Lviv Nat'l Polytech. Univ. (Ukraine 2023) Vol. 5, No. 1, 137-145. See p. 140.
Saad Mneimneh, Simple Variations on the Tower of Hanoi to Guide the Study of Recurrences and Proofs by Induction, Department of Computer Science, Hunter College, CUNY, 2019.
Sam Northshield, Stern's diatomic sequence 0, 1, 1, 2, 1, 3, 2, 3, 1, 4, ..., Amer. Math. Monthly, Vol. 117, No. 7 (2010), pp. 581-598.
Index entries for linear recurrences with constant coefficients, signature (1,2).

Cf. A084214 (first differences).
Cf. A001045, A014551, A024493, A062510.
Cf. A007283, A007679, A078008, A081254.
Cf. A083322, A083944, A083581, A084170.
Cf. A102713, A127978, A130755, A139763.
Cf. A140360, A140966, A166249, A168642.
Cf. A280560, A290604, A340627.
Cf. A112387, A135318.

[(5*2^n+(-1)^n)/3: n in [0..35]]; // Vincenzo Librandi, Jul 05 2011

LinearRecurrence[{1,2},{2,3},40] (* Harvey P. Dale, Dec 11 2017 *)

{a(n) = if( n<0, 0, (5*2^n + (-1)^n) / 3)};

{a(n) = if (n<0 ,0, if( n<2, n+2, a(n-1) + 2*a(n-2)))};

[(5*2^n+(-1)^n)/3 for n in range(35)] # G. C. Greubel, Apr 10 2019

G.f.: (2 + x) / (1 - x - 2*x^2).
a(n) = (5*2^n + (-1)^n) / 3.
a(n) = 2^(n+1) - A001045(n).
a(n) = A084170(n)+1 = abs(A083581(n)-3) = A081254(n+1) - A081254(n) = A084214(n+2)/2.
a(n) = 2*A001045(n+1) + A001045(n) (note that 2 is the limit of A001045(n+1)/A001045(n)). - Paul Barry, Sep 14 2009
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=-3, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=-charpoly(A,-1). - Milan Janjic, Jan 27 2010
Equivalently, with different offset, a(n) = b(n+1) with b(0)=1 and b(n) = Sum_{i=0..n-1} (-1)^i (1 + (-1)^i b(i)). - Olivier Gérard, Jul 30 2012
a(n) = A000975(n-2)*10 + 5 + 2*(-1)^(n-2), a(0)=2, a(1)=3. - Yuchun Ji, Mar 18 2019
a(n+1) = Sum_{i=0..n} a(i) + 1 + (1-(-1)^n)/2, a(0)=2. - Yuchun Ji, Apr 10 2019
a(n) = 2^n + J(n+1) = J(n+2) + J(n+1) - J(n), where J is A001045. - Yuchun Ji, Apr 10 2019
a(n) = A001045(n+2) + A078008(n) = A062510(n+1) - A078008(n+1) = (A001045(n+2) + A062510(n+1))/2 = A014551(n) + 2*A001045(n). - Paul Curtz, Jul 14 2021
From Thomas Scheuerle, Jul 14 2021: (Start)
a(n) = A083322(n) + A024493(n).
a(n) = A127978(n) - A102713(n).
a(n) = A130755(n) - A166249(n).
a(n) = A007679(n) + A139763(n).
a(n) = A168642(n) XOR A007283(n).
a(n) = A290604(n) + A083944(n). (End)
From Paul Curtz, Jul 21 2021: (Start)
a(n) = 5*A001045(n) - A280560(n+1) = abs(A140360(n+1)) - A280560(n+1).
a(n) = 2^n + A001045(n+1) = A001045(n+3) - A000079(n).
a(n) = A001045(n+4) - A340627(n). (End)
a(n) = A001045(n+5) - A005010(n).
a(n+1) + a(n) = a(n+2) - a(n) = 5*2^n. - Michael Somos, Feb 22 2023
a(n) = A135318(2*n) + A135318(2*n+1) = A112387(2*n) + A112387(2*n+1). - Paul Curtz, Jun 26 2024
E.g.f.: (cosh(x) + 5*cosh(2*x) - sinh(x) + 5*sinh(2*x))/3. - Stefano Spezia, May 18 2025

Formula of Milan Janjic moved here from wrong sequence by Paul D. Hanna, May 29 2010

A084215 Expansion of g.f.: (1+x^2)/(1-2*x).

Original entry on oeis.org

1, 2, 5, 10, 20, 40, 80, 160, 320, 640, 1280, 2560, 5120, 10240, 20480, 40960, 81920, 163840, 327680, 655360, 1310720, 2621440, 5242880, 10485760, 20971520, 41943040, 83886080, 167772160, 335544320, 671088640, 1342177280, 2684354560, 5368709120, 10737418240

Offset: 0

Paul Barry, May 19 2003

Associated with a math magic problem.

Elements are the sums of consecutive pairs of elements of A084214.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2).

Cf. A020714, A060816, A084214.

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+x^2)/(1-2*x))); // G. C. Greubel, Oct 08 2018

Join[{1, 2, a = 5}, Table[a = 2*a, {n,0,40}]] (* Vladimir Joseph Stephan Orlovsky, Jun 09 2011 *)
Table[Int[2^(n-2)*5],{n,0,40}] (* Taher Jamshidi, Sep 15 2012 *)
CoefficientList[Series[(1 + x^2)/(1 - 2 x), {x, 0, 30}], x] (* G. C. Greubel, Oct 08 2018 *)

x='x+O('x^30); Vec((1+x^2)/(1-2*x)) \\ G. C. Greubel, Oct 08 2018

a(n) = Sum_{k=0..n} 2^(n-k)*binomial(1, k/2)*(1+(-1)^k)/2. - Paul Barry, Oct 15 2004
a(n) = A020714(n-2), n > 1. - R. J. Mathar, Dec 19 2008
From Gary W. Adamson, Aug 26 2011: (Start)
a(n) is the sum of top row terms of M^n, M is an infinite square production matrix as follows:
  1, 1, 0, 0, 0, 0, ...
  1, 1, 1, 0, 0, 0, ...
  0, 0, 0, 0, 0, 0, ...
  0, 0, 0, 0, 0, 0, ...
  ...
E.g.: a(4) = 20 = (8 + 8 + 4) since the top row of M^4 = (8, 8, 4, 0, 0, 0, ...). (End)
a(n) = floor(2^(n-2)*5). - Taher Jamshidi, Sep 15 2012
a(n) = 2*a(n-1) for n >= 3, a(0) = 1, a(1) = 2, a(2) = 5. - Philippe Deléham, Mar 13 2013
E.g.f.: (5*exp(2*x) - 2*x - 1)/4. - Stefano Spezia, Feb 20 2023

A104509 Matrix inverse of triangle A104505, which is the right-hand side of triangle A084610 of coefficients in (1 + x - x^2)^n.

Original entry on oeis.org

1, 1, -1, 3, -2, 1, 4, -6, 3, -1, 7, -12, 10, -4, 1, 11, -25, 25, -15, 5, -1, 18, -48, 60, -44, 21, -6, 1, 29, -91, 133, -119, 70, -28, 7, -1, 47, -168, 284, -296, 210, -104, 36, -8, 1, 76, -306, 585, -699, 576, -342, 147, -45, 9, -1, 123, -550, 1175, -1580, 1485, -1022, 525, -200, 55, -10, 1, 199, -979, 2310, -3454, 3641

Offset: 0

Paul D. Hanna, Mar 11 2005

Riordan array ( (1 + x^2)/(1 - x - x^2), -x/(1 - x - x^2) ) belonging to the hitting time subgroup of the Riordan group (see Peart and Woan). - Peter Bala, Jun 29 2015

The sums of absolute values along steep diagonals in this triangle are: 1, 1, 3, 4 + |-1|, 7 + |-2|, 11 + |-6|, 18 + |-12| + 1, ... and these are the tribonacci numbers A000213 that begin with 1, 1, 1, 3. To see this, replace the y in the g.f. A(x,y) = (1 + x^2)/(1-x-x^2 + x*y) with y=-x^2, multiply by x, and add 1, to obtain the g.f. (1 - x^2)/(1-x-x^2-x^3) for A000213. - Noah Carey and Greg Dresden, Nov 02 2021

			Rows begin:
   1;
   1,   -1;
   3,   -2,   1;
   4,   -6,   3,   -1;
   7,  -12,  10,   -4,   1;
  11,  -25,  25,  -15,   5,   -1;
  18,  -48,  60,  -44,  21,   -6,   1;
  29,  -91, 133, -119,  70,  -28,   7,  -1;
  47, -168, 284, -296, 210, -104,  36,  -8, 1;
  76, -306, 585, -699, 576, -342, 147, -45, 9, -1; ...

Robert Israel, Table of n, a(n) for n = 0..10152 (rows 0 to 141, flattened).
P. Peart and W.-J. Woan, A divisibility property for a subgroup of Riordan matrices, Discrete Applied Mathematics, Vol. 98, Issue 3, Jan 2000, 255-263.
Wikipedia, Lucas polynomials.

Leftmost column is A000204 (Lucas numbers). Other columns include: A045925, A067988. Row sums are: {1,0,2,0,2,0,2,...}. Absolute row sums form: A099425. Antidiagonal sums are: {1,1,2,2,2,2,2,...}. Absolute antidiagonal sums are: A084214.

Cf. A104505, A084610, A000213.

S:= series((1 + x^2)/(1-x-x^2 + x*y),x, 20):
for n from 0 to 19 do R[n]:= coeff(S,x,n) od:
seq(seq(coeff(R[n],y,j),j=0..n), n=0..19); # Robert Israel, Jun 30 2015

nmax = 11;
T[n_, k_] := Coefficient[(1 + x - x^2)^n, x, n + k];
M = Table[T[n, k], {n, 0, nmax}, {k, 0, nmax}] // Inverse;
Table[M[[n+1, k+1]], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 27 2019 *)

{ T(n,k) = my(X=x+x*O(x^n),Y=y+y*O(y^k)); polcoeff(polcoeff((1 + X^2)/(1-X-X^2 + X*Y),n,x),k,y); }

{ tabl(nn) = my(m = matrix(nn, nn, n, k, n--; k--; if((nMichel Marcus, Jun 30 2015

{ A104509(n,k) = if(n==0, k==0, (-1)^k * sum(i=0, (n-k)\2, n/(n-i) * binomial(n-k-i,i) * binomial(n-i,k) )); } \\ Max Alekseyev, Oct 11 2021

For n>=1, a(n,k) = (-1)^k * Sum_{i=0..[(n-k)/2]} n/(n-i) * binomial(n-i,i) * binomial(n-2*i,k) = (-1)^k * Sum_{i=0..[(n-k)/2]} n/(n-i) * binomial(n-k-i,i) * binomial(n-i,k). - Max Alekseyev, Oct 11 2021
G.f.: A(x, y) = (1 + x^2)/(1-x-x^2 + x*y).
G.f. for column k: g_k(x) = -(x^2+1)*x^k/(x^2+x-1)^(k+1). - Robert Israel, Jun 30 2015
G.f. for row n>=1 is the Lucas polynomial L_n(1-x). - Max Alekseyev, Oct 11 2021

A084640 Generalized Jacobsthal numbers.

Original entry on oeis.org

0, 1, 5, 11, 25, 51, 105, 211, 425, 851, 1705, 3411, 6825, 13651, 27305, 54611, 109225, 218451, 436905, 873811, 1747625, 3495251, 6990505, 13981011, 27962025, 55924051, 111848105, 223696211, 447392425, 894784851, 1789569705, 3579139411

Offset: 0

Paul Barry, Jun 06 2003

This is the sequence A(0,1;1,2;4) of the family of sequences [a,b:c,d:k] considered by G. Detlefs, and treated as A(a,b;c,d;k) in the W. Lang link given below. - Wolfdieter Lang, Oct 18 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Wolfdieter Lang, Notes on certain inhomogeneous three term recurrences. [From _Wolfdieter Lang_, Oct 18 2010]
Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

Cf. A000975, A000225, A084639.

a084640 n = a084640_list !! n
a084640_list = 0 : 1 : (map (+ 4) $
   zipWith (+) (map (* 2) a084640_list) (tail a084640_list))
-- Reinhard Zumkeller, May 23 2013

[5*2^n/3+(-1)^n/3-2: n in [0..35]]; // Vincenzo Librandi, Jun 15 2011

LinearRecurrence[{2,1,-2},{0,1,5},40] (* Harvey P. Dale, Oct 27 2015 *)

x='x+O('x^50); Vec(x*(1+3*x)/((1-x^2)*(1-2*x))) \\ G. C. Greubel, Sep 26 2017

G.f.: x*(1+3*x)/((1-x^2)*(1-2*x)).
a(n) = a(n-1) + 2a(n-2) + 4, a(0)=0, a(1)=1.
a(n) = (5*2^n + (-1)^n - 6)/3.
a(n) = A001045(n+2) + 4*A000975(n-3).
a(n+1) - 2*a(n) = period 2: repeat 1, 3. - Paul Curtz, Apr 03 2008
Contribution from Paul Curtz, Dec 10 2009: (Start)
a(n+2) - a(n) = A020714(n).
Le the array D(n,k) of the first differences be defined via D(0,k) = a(k); D(n+1,k) = D(n,k+1)-D(n,k).
Then D(n,n) = 4*A131577(n); D(1,k) = A084214(k+1); D(2,k) = A115102(k-1) for k>0; D(3,k) = (-1)^(k+1)*A083581(k). (End)
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3), a(0)=0, a(1)=1, a(2)=5. Observed by G. Detlefs. See the W. Lang link. - Wolfdieter Lang, Oct 18 2010

A168648 a(n) = (102^n + 2(-1)^n)/3 for n > 0; a(0) = 1.

Original entry on oeis.org

1, 6, 14, 26, 54, 106, 214, 426, 854, 1706, 3414, 6826, 13654, 27306, 54614, 109226, 218454, 436906, 873814, 1747626, 3495254, 6990506, 13981014, 27962026, 55924054, 111848106, 223696214, 447392426, 894784854, 1789569706, 3579139414

Offset: 0

Klaus Brockhaus, Dec 01 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (1,2).

Cf. A084214 ((5*2^n -3*0^n +4*(-1)^n)/6).

[1] cat [ (10*2^n+2*(-1)^n)/3: n in [1..30] ];

{1}~Join~Table[(10*2^n + 2*(-1)^n)/3, {n,40}] (* or *)
{1}~Join~LinearRecurrence[{1,2}, {6,14}, 40] (* G. C. Greubel, Jul 28 2016 *)

a(n) = if(n, (10<Charles R Greathouse IV, Jul 29 2016

[1]+[(10*2^n +2*(-1)^n)/3 for n in (1..40)] # G. C. Greubel, Feb 05 2021

a(n) = A084214(n+2) for n > 0.
a(n) = a(n-1) + 2*a(n-2) for n > 2; a(0) = 1, a(1) = 6, a(2) = 14.
G.f.: (1+2*x)*(1+3*x)/((1+x)*(1-2*x)).
E.g.f.: (1/3)*(10*exp(2*x) - 9 + 2*exp(-x)). - G. C. Greubel, Jul 28 2016

A340627 a(n) = (112^n - 2(-1)^n)/3 for n >= 0.

Original entry on oeis.org

3, 8, 14, 30, 58, 118, 234, 470, 938, 1878, 3754, 7510, 15018, 30038, 60074, 120150, 240298, 480598, 961194, 1922390, 3844778, 7689558, 15379114, 30758230, 61516458, 123032918, 246065834, 492131670, 984263338, 1968526678, 3937053354, 7874106710, 15748213418, 31496426838

Offset: 0

Paul Curtz, Apr 25 2021

Based on A112387.
Prepended with 0, 1, its difference table is
0,  1,  1,  2,  1,   4,  3,   8, ... = mix A001045(n), 2^n.
1,  0,  1, -1,  3,  -1,  5,  -3, ... = mix A001045(n+1), -A001045(n).
-1, 1, -2,  4, -4,   6, -8,  14, ... = mix -2^n, A084214(n+1).
2, -3,  6, -8, 10, -14, 22, -30, ... = mix 2*A001045(n+2), -a(n).

Index entries for linear recurrences with constant coefficients, signature (1,2).

Cf. A000079, A001045, A062510, A078008, A112387.

Cf. also A052997.

LinearRecurrence[{1, 2}, {3, 8}, 35] (* Amiram Eldar, Apr 25 2021 *)

a(n) = (11*2^n - 2*(-1)^n)/3 \\ Felix Fröhlich, Apr 25 2021

a(n) = 2^(n+2) - A078008(n), n>=0.
a(n) = (A062510(n) = 3*A001045(n)) + A001045(n+3), n>=0.
a(0)=3, a(2*n+1) = 2*a(2*n) + 2, a(2*n+2) = 2*a(2*n+1) - 2, n>=0.
a(n) = 4*A052997(n-1) + 2, n>=2. - Hugo Pfoertner, Apr 25 2021
a(n+1) = 11*2^n - a(n) for n>=0.
a(n+3) = 33*2^n - a(n) for n>=0.
a(n+5) = 121*2^n - a(n) for n>=0.
etc.
a(n+2) = a(n) + 11*2^n for n>=0.
a(n+4) = a(n) + 55*2^n for n>=0.
a(n+6) = a(n) + 231*2^n for n>=0.
etc.
G.f.: (3 + 5*x)/(1 - x - 2*x^2). - Stefano Spezia, Apr 26 2021
E.g.f: (11*exp(2*x) - 2*exp(-x))/3. - Jianing Song, Apr 26 2021

More terms from Michel Marcus, Apr 25 2021

New name from Jianing Song, Apr 25 2021

A115102 a(0)=2, a(1)=8, a(n) = a(n-1) + 2*a(n-2).

Original entry on oeis.org

2, 8, 12, 28, 52, 108, 212, 428, 852, 1708, 3412, 6828, 13652, 27308, 54612, 109228, 218452, 436908, 873812, 1747628, 3495252, 6990508, 13981012, 27962028, 55924052, 111848108, 223696212, 447392428, 894784852, 1789569708, 3579139412, 7158278828

Offset: 0

Roger L. Bagula, Mar 02 2006

Essentially 2 * A084214.

Index entries for linear recurrences with constant coefficients, signature (1,2).

Cf. A084214.

LinearRecurrence[{1,2},{2,8},40] (* or *) Table[(4(-1)^x+5*2^x)/3,{x,40}] (* Harvey P. Dale, Sep 02 2016 *)

From R. J. Mathar, Jun 14 2011: (Start)
G.f.: (2+6*x)/( (1+x)*(1-2*x) ).
a(n) = 2*A084214(n+1). (End)
a(n) = (4*(-1)^n + 5*2^n)/3. - Harvey P. Dale, Sep 02 2016

Edited by N. J. A. Sloane, Dec 04 2006

cp's OEIS Frontend

A182349 G.f.: exp( Sum_{n>=1} 6 * A084214(n) * x^n/n ) where g.f. of A084214 is (1+x^2)/((1+x)*(1-2*x)).

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A375476 a(3*n)=A001045(n+1), a(3*n+1)=A084214(n), a(3*n+2)=A000079(n) for n >= 0.

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A048654 a(n) = 2*a(n-1) + a(n-2); a(0)=1, a(1)=4.

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A048573 a(n) = a(n-1) + 2*a(n-2), a(0)=2, a(1)=3.

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A084215 Expansion of g.f.: (1+x^2)/(1-2*x).

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A104509 Matrix inverse of triangle A104505, which is the right-hand side of triangle A084610 of coefficients in (1 + x - x^2)^n.

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A084640 Generalized Jacobsthal numbers.

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A182349 G.f.: exp( Sum_{n>=1} 6 * A084214(n) * x^n/n ) where g.f. of A084214 is (1+x^2)/((1+x)(1-2x)).

A375476 a(3n)=A001045(n+1), a(3n+1)=A084214(n), a(3*n+2)=A000079(n) for n >= 0.

A168648 a(n) = (102^n + 2(-1)^n)/3 for n > 0; a(0) = 1.

A340627 a(n) = (112^n - 2(-1)^n)/3 for n >= 0.