A084214 -id:A084214 - cp's OEIS frontend

A173078 a(n) = (52^n - 2(-1)^n - 9)/3.

1, 3, 11, 23, 51, 103, 211, 423, 851, 1703, 3411, 6823, 13651, 27303, 54611, 109223, 218451, 436903, 873811, 1747623, 3495251, 6990503, 13981011, 27962023, 55924051, 111848103, 223696211, 447392423, 894784851, 1789569703, 3579139411

Offset: 1

Paul Curtz, Feb 09 2010

The sequence and higher-order differences in subsequent rows are
    1,  3,  11,  23, 51, 103, 211, 423, 851, 1703, 3411, 6823, 13651
    2,  8,  12,  28, 52, 108, 212, 428, 852, 1708, 3412, 6828, 13652
    6,  4,  16,  24, 56, 104, 216, 424, 856, 1704, 3416, 6824, 13656
   -2,  12,  8,  32, 48, 112, 208, 432, 848, 1712, 3408, 6832, 13648
   14,  -4, 24,  16, 64,  96, 224, 416, 864, 1696, 3424, 6816, 13664
  -18,  28, -8,  48, 32, 128, 192, 448, 832, 1728, 3392, 6848,  1363
   46, -36, 56, -16, 96,  64, 256, 384, 896, 1664, 3456, 6784,  1369
The main diagonal 1,8,16,... is essentially A000079.
A subdiagonal is 2, 4, 8, 16, ... A155559.
Other diagonals are 3, 12, 24, 48, ... = 3*A151821, 6, 12, 24, ... = A082505 and -2, -4, -8, -16, ..., a negated variant of A171449.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

List([1..40], n-> (5*2^n - 2*(-1)^n - 9)/3); # G. C. Greubel, Dec 01 2019

[5*2^n/3-2*(-1)^n/3-3: n in [1..40]]; // Vincenzo Librandi, Aug 05 2011

seq( (5*2^n -2*(-1)^n -9)/3, n=1..40); # G. C. Greubel, Dec 01 2019

LinearRecurrence[{2,1,-2},{1,3,11},40] (* Harvey P. Dale, Oct 01 2018 *)

vector(40, n, (5*2^n - 2*(-1)^n - 9)/3) \\ G. C. Greubel, Dec 01 2019

[(5*2^n - 2*(-1)^n - 9)/3 for n in (1..40)] # G. C. Greubel, Dec 01 2019

a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3).
a(n+1) - 2*a(n) = A010686(n-1).
a(n) = A084214(n+1) - 3.
G.f.: x*(1 + x + 4*x^2) / ( (1-x)*(1-2*x)*(1+x) ).
a(2n+3) - a(2n+1) = 10*A000302(n).
E.g.f.: (-2*exp(-x) + 6 - 9*exp(x) + 5*exp(2*x))/3. - G. C. Greubel, Dec 01 2019

A151794 a(1)=2, a(2)=4, a(3)=6; a(n+3) = a(n+2)+ 2*a(n), n>=1.

Original entry on oeis.org

2, 4, 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, 7158278826, 14316557654

Offset: 1

K. S. Bhanu (bhanu_105(AT)yahoo.com), Jun 21 2009

Consider the following coin tossing experiment. Let n >= 1 be a predetermined integer. We toss an unbiased coin sequentially. For each outcome, we score two points for a head (H) and one point for a tail (T). The coin is tossed until the total score reaches n or jumps from n-1 to n+1. The results of the tosses are written in a linear array. Then the probability of non-occurrence of double heads (HH) is given by p(n) = a(n) / 2^n, n>=1.

Bhanu K. S, Deshpande M. N. & Cholkar C. P. (2006): Coin tossing -Some Surprising Results, International Journal of Mathematical Education In Science and Technology, Vol.37, No.1, pp.115-119.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,2).

Join[{2},LinearRecurrence[{1,2},{4,6},40]] (* Harvey P. Dale, Oct 19 2012 *)

Vec(2*x*(-x+x^2-1)/((1+x)*(2*x-1)) + O(x^100)) \\ Colin Barker, Jun 12 2015

G.f.: 2*x*(-x+x^2-1)/((1+x)*(2*x-1)).
a(n) = A084214(n), n>1.
a(n) = A168648(n-2), n>2.
a(n) = 2*A048573(n-2), n>1.
a(n) = (4*(-1)^n+5*2^n)/6 for n>1. - Colin Barker, Jun 12 2015

A275788 a(0) = 0, a(n+1) = 2*a(n) + (-1)^floor(n/3).

Original entry on oeis.org

0, 1, 3, 7, 13, 25, 49, 99, 199, 399, 797, 1593, 3185, 6371, 12743, 25487, 50973, 101945, 203889, 407779, 815559, 1631119, 3262237, 6524473, 13048945, 26097891, 52195783, 104391567, 208783133, 417566265, 835132529, 1670265059, 3340530119, 6681060239

Offset: 0

Paul Curtz, Aug 09 2016

nonn

a(n) and its successive differences:
0,  1,  3,  7, 13, 25, 49, ...
1,  2,  4,  6, 12, 24, 50, 100, ...
1,  2,  2,  6, 12, 26, 50, 100, 198, ...
1,  0,  4,  6, 14, 24, 50,  98, 200, 398, ...
-1, 4,  2,  8, 10, 26, 48, 102, 198, 400, 794, ...
5, -2,  6,  2, 16, 22, 54,  96, 202, 394, 800, 1590, ...
-7, 8, -4, 14,  6, 32, 42, 106, 192, 406, 790, 1600, 3178, ...
... .
Each row has the recurrence a(n) + a(n+3) = 7*2^n.
Main diagonal: 2*A001045(n).
Upper diagonals: A084214(n+1), 3*2^n, ... .
Subdiagonals: 2^n, A078008(n), A084214(n+1), -2^n, ... .
a(-n) = 0, 1/2, 3/4, 7/8, -1/16, -17/32, -49/64, 15/128, ... .
b(n), numerators of a(-n), and first differences:
0, 1, 3,  7,  -1, -17, -49,  15, 143,  399,  -113, -1137, ...
1, 2, 4, -8, -16, -32,  64, 128, 256, -512, -1024, ... = A000079(n)*A130151(n), not in the OEIS.

			a(1)=2*0+1=1, a(2)=2*1+1=3, a(2)=2*3+1=7, a(3)=2*7-1=13, a(4)=2*13-1=25, ... .

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,-1,2).

Cf. A000079, A001045, A002264, A005009, A007283, A078008, A084214, A113405, A130151, A274817.

CoefficientList[Series[x (1 + x + x^2)/((1 + x) (1 - 2 x) (1 - x + x^2)), {x, 0, 33}], x] (* Michael De Vlieger, Aug 11 2016 *)
LinearRecurrence[{2,0,-1,2}, {0, 1, 3, 7}, 25] (* G. C. Greubel, Aug 16 2016 *)

concat(0, Vec(x*(1+x+x^2)/((1+x)*(1-2*x)*(1-x+x^2)) + O(x^40))) \\ Colin Barker, Aug 10 2016

From Colin Barker, Aug 09 2016: (Start)
a(n) = 2*a(n-1) - a(n-3) + 2*a(n-4) for n>3.
G.f.: x*(1 + x + x^2) / ((1+x)*(1-2*x)*(1-x+x^2)).
(End)
a(n+3) = 7*2^n - a(n), a(0)=0, a(1)=1, a(2)=3.

More terms from Colin Barker, Aug 10 2016

A338198 Triangle read by rows, T(n,k) = ((k+1)2^(n-k)-(k-2)(-1)^(n-k))/3 for 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 2, 1, 1, 2, 3, 2, 1, 6, 5, 4, 3, 1, 10, 11, 8, 5, 4, 1, 22, 21, 16, 11, 6, 5, 1, 42, 43, 32, 21, 14, 7, 6, 1, 86, 85, 64, 43, 26, 17, 8, 7, 1, 170, 171, 128, 85, 54, 31, 20, 9, 8, 1, 342, 341, 256, 171, 106, 65, 36, 23, 10, 9, 1, 682, 683, 512, 341, 214, 127, 76, 41, 26, 11, 10, 1

Offset: 0

Werner Schulte, Oct 15 2020

This triangle is related to the Jacobsthal numbers (A001045).

			The triangle T(n,k) for 0 <= k <= n starts:
n\k :    0     1     2    3    4    5    6   7   8   9
======================================================
  0 :    1
  1 :    0     1
  2 :    2     1     1
  3 :    2     3     2    1
  4 :    6     5     4    3    1
  5 :   10    11     8    5    4    1
  6 :   22    21    16   11    6    5    1
  7 :   42    43    32   21   14    7    6   1
  8 :   86    85    64   43   26   17    8   7   1
  9 :  170   171   128   85   54   31   20   9   8   1
etc.

For columns k = 0 to 8 see A078008, A001045, A000079, A001045, A084214, A014551, A083595, A083582, A259713 respectively.

Table[((k + 1)*2^(n - k) - (k - 2)*(-1)^(n - k))/3, {n, 0, 11}, {k, 0, n}] // Flatten (* Michael De Vlieger, Oct 15 2020 *)

T(n,n) = 1 for n >= 0; T(n,n-1) = n-1 for n > 0.
T(n,k) = T(n-1,k) + 2 * T(n-2,k) for 0 <= k <= n-2.
T(n,k) = 2 * T(n-1,k) - (k-2) * (-1)^(n-k) for 0 <= k < n.
T(n,k) = T(n+1-k,1) + (k-1) * T(n-k,1) for 0 <= k < n.
T(n+1,k) * T(n-1,k) - T(n,k+1) * T(n,k-1) = T(n-k,1)^2 for 0 < k < n.
Row sums are A083579(n+1) for n >= 0.
G.f. of column k >= 0: (1+(k-1)*t) * t^k / (1-t-2*t^2).
G.f.: Sum_{n>=0, k=0..n} T(n,k) * x^k * t^n = (1 - (1+x)*t + 2*x*t^2) / ((1 - x*t)^2 * (1 - t - 2*t^2)).
Conjecture: Let M(n,k) be the matrix inverse of T(n,k), seen as a matrix. Then M(i,j) = 0 if j < 0 or j > i, M(n,n) = 1 for n >= 0, M(n,n-1) = 1-n for n > 0, and M(n,k) = (-1)^(n-k) * (k^2-2) * (n-2)! / k! for 0 <= k <= n-2.

A344109 a(n) = (52^n + 7(-1)^n)/3.

Original entry on oeis.org

4, 1, 9, 11, 29, 51, 109, 211, 429, 851, 1709, 3411, 6829, 13651, 27309, 54611, 109229, 218451, 436909, 873811, 1747629, 3495251, 6990509, 13981011, 27962029, 55924051, 111848109, 223696211, 447392429, 894784851, 1789569709, 3579139411, 7158278829, 14316557651, 28633115309, 57266230611, 114532461229, 229064922451

Offset: 0

Paul Curtz, May 09 2021

Élis Gardel da Costa Mesquita, Eudes Antonio Costa, Paula M. M. C. Catarino, and Francisco R. V. Alves, Jacobsthal-Mulatu Numbers, Latin Amer. J. Math. (2025) Vol. 4, No. 1, 23-45. See pp. 24, 26, 43.
Index entries for linear recurrences with constant coefficients, signature (1,2).

Cf. A001045, A020714, A110286, A154879, A321421.

Cf. A014551, A156550, A084214.

LinearRecurrence[{1,2}, {4,1}, 28] (* Amiram Eldar, May 10 2021 *)

a(n+1) = 5*2^n - a(n) for n >= 0, with a(0) = 4.
a(n+2) = 5*2^n + a(n) for n >= 0, with a(0) = 4, a(1) = 1.
a(n+3) = 15*2^n - a(n) for n >= 0, with a(0) = 4, a(1) = 1, a(2) = 9.
a(n) = A001045(n+2) + A154879(n).
a(2*n+1) = A321421(n).
a(n) = a(n-1) + 2*a(n-2) for n >= 2. - Pontus von Brömssen, May 09 2021
G.f.: (4 - 3*x)/(1 - x - 2*x^2). - Stefano Spezia, May 10 2021
a(n) = 2*A014551(n) - A001045(n).
a(n) = abs(A156550(n)) - (-1)^n.
a(n+3) = a(n) + 7*A084214(n+1) for n >= 0, with a(0) = 4.
a(n) = 5*A001045(n+1) - A084214(n+1) for n >= 0.
a(n) = A084214(n+1) + 3*(-1)^n for n >= 0.

A356050 a(n) = 2*A135318(n+1) - A135318(n).

Original entry on oeis.org

1, 1, 3, 4, 5, 6, 11, 14, 21, 26, 43, 54, 85, 106, 171, 214, 341, 426, 683, 854, 1365, 1706, 2731, 3414, 5461, 6826, 10923, 13654, 21845, 27306, 43691, 54614, 87381, 109226, 174763, 218454, 349525, 436906, 699051, 873814, 1398101, 1747626, 2796203, 3495254, 5592405

Offset: 0

Paul Curtz, Aug 19 2022

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

Cf. A001045, A084214, A135318, A230096.

Cf. A094958.

LinearRecurrence[{0, 1, 0, 2}, {1, 1, 3, 4}, 50] (* Amiram Eldar, Aug 19 2022 *)

a(n) = A135318(n) + A230096(n+1).
a(n) = a(n-8) + 5*A094958(n-5).
a(2*n) = A001045(n+2).
a(2*n+1) = A084214(n+1).
From Stefano Spezia, Aug 20 2022: (Start)
O.g.f.: (1 + x + 2*x^2 + 3*x^3)/((1 + x^2)*(1 - 2*x^2)).
E.g.f.: (8*cosh(sqrt(2)*x) - 2*cos(x) + 5*sqrt(2)*sinh(sqrt(2)*x) - 4*sin(x))/6. (End)
3*a(n) = A228826(n+1) +A094958(n+3). - R. J. Mathar, Jan 25 2023

cp's OEIS Frontend

A173078 a(n) = (5*2^n - 2*(-1)^n - 9)/3.

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A151794 a(1)=2, a(2)=4, a(3)=6; a(n+3) = a(n+2)+ 2*a(n), n>=1.

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A275788 a(0) = 0, a(n+1) = 2*a(n) + (-1)^floor(n/3).

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A338198 Triangle read by rows, T(n,k) = ((k+1)*2^(n-k)-(k-2)*(-1)^(n-k))/3 for 0 <= k <= n.

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A344109 a(n) = (5*2^n + 7*(-1)^n)/3.

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A356050 a(n) = 2*A135318(n+1) - A135318(n).

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A173078 a(n) = (52^n - 2(-1)^n - 9)/3.

A338198 Triangle read by rows, T(n,k) = ((k+1)2^(n-k)-(k-2)(-1)^(n-k))/3 for 0 <= k <= n.

A344109 a(n) = (52^n + 7(-1)^n)/3.