A048573 -id:A048573 - cp's OEIS frontend

A081254 Numbers k such that A081252(m)/m^2 has a local maximum for m = k.

1, 3, 6, 13, 26, 53, 106, 213, 426, 853, 1706, 3413, 6826, 13653, 27306, 54613, 109226, 218453, 436906, 873813, 1747626, 3495253, 6990506, 13981013, 27962026, 55924053, 111848106, 223696213, 447392426, 894784853, 1789569706, 3579139413

Offset: 1

Klaus Brockhaus, Mar 17 2003

The limit of the local maxima, lim_{m->inf} A081252(m)/m^2 = 1/10. For local minima cf. A081253.

Row sums of the triangle A181971. - Reinhard Zumkeller, Jul 09 2012

			13 is a term since A081252(12)/12^2 = 15/144 = 0.104..., A081252(13)/13^2 = 18/169 = 0.106..., A081252(14)/14^2 = 20/196 = 0.102....

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Thomas Baruchel, Properties of the cumulated deficient binary digit sum, arXiv:1908.02250 [math.NT], 2019.
Klaus Brockhaus, Illustration for A053646, A081252, A081253 and A081254
Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

Cf. A000975, A020714, A020989, A048573, A052549, A053646, A072197, A081252, A081253, A266219 (binary).

[Floor(2^(n-1)*5/3): n in [1..40]]; // Vincenzo Librandi, Apr 04 2012

seq(floor(2^(n-1)*5/3),n=1..35); # Muniru A Asiru, Sep 20 2018

Rest@CoefficientList[Series[-(x^2 - x - 1)*x/((x - 1)*(x + 1)*(2*x - 1)), {x, 0, 32}], x] (* Vincenzo Librandi, Apr 04 2012 *)
a[n_]:=Floor[2^(n-1)*5/3]; Array[a,33,1] (* Stefano Spezia, Sep 01 2018 *)

a(n) = 2^(n-1)*5\3; \\ Altug Alkan, Sep 21 2018

a(n) = floor(2^(n-1)*5/3). [corrected by Michel Marcus, Sep 21 2018]
a(n) = a(n-2) + 5*2^(n-3) for n > 2;
a(n+2) - a(n) = A020714(n-1);
a(n) + a(n-1) = A052549(n-1) for n > 1;
a(2*n+1) = A020989(n); a(2n) = A072197(n-1);
a(n+1) - a(n) = A048573(n-1).
G.f.: -(x^2 - x - 1)*x/((x - 1)*(x + 1)*(2*x - 1)).
a(n) = 5*2^(n-1)/3 + (-1)^n/6-1/2. a(n) = 2*a(n-1) + (1+(-1)^n)/2, a(1)=1. - Paul Barry, Mar 24 2003
a(2n) = 2*a(2*n-1) + 1, a(2*n+1) = 2*a(2*n), a(1)=1. a(n) = A000975(n-1) + 2^(n-1). - Philippe Deléham, Oct 15 2006
a(n) = A005578(n) + A000225(n-1). - Yuchun Ji, Sep 21 2018
a(n) - a(n-2) = 2 * (a(n-1) - a(n-3)), with a(0..2)=[1,3,6]. - Yuchun Ji, Mar 18 2020

A084214 Inverse binomial transform of a math magic problem.

Original entry on oeis.org

1, 1, 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: 0

Paul Barry, May 19 2003

Inverse binomial transform of A060816.

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

Cf. A000975, A001045, A020989, A048654, A060816, A081254.

Cf. A048573.

a084214 n = a084214_list !! n
a084214_list = 1 : xs where
   xs = 1 : 4 : zipWith (+) (map (* 2) xs) (tail xs)
-- Reinhard Zumkeller, Aug 01 2011

[(5*2^n-3*0^n+4*(-1)^n)/6: n in [0..35]]; // Vincenzo Librandi, Jun 15 2011

A084214 := proc(n)
    (5*2^n - 3*0^n + 4*(-1)^n)/6 ;
end proc:
seq(A084214(n),n=0..60) ; # R. J. Mathar, Aug 18 2024

f[n_]:=2/(n+1);x=3;Table[x=f[x];Numerator[x],{n,0,5!}] (* Vladimir Joseph Stephan Orlovsky, Mar 12 2010 *)
LinearRecurrence[{1,2},{1,1,4},50] (* Harvey P. Dale, Mar 05 2021 *)

a(n) = 5<<(n-1)\3 + bitnegimply(1,n); \\ Kevin Ryde, Dec 20 2023

a(n) = (5*2^n - 3*0^n + 4*(-1)^n)/6.
G.f.: (1+x^2)/((1+x)*(1-2*x)).
E.g.f.: (5*exp(2*x) - 3*exp(0) + 4*exp(-x))/6.
From Paul Barry, May 04 2004: (Start)
The binomial transform of a(n+1) is A020989(n).
a(n) = A001045(n-1) + A001045(n+1) - 0^n/2. (End)
a(n) = Sum_{k=0..n} A001045(n+1)*C(1, k/2)*(1+(-1)^k)/2. - Paul Barry, Oct 15 2004
a(n) = a(n-1) + 2*a(n-2) for n > 2. - Klaus Brockhaus, Dec 01 2009
From Yuchun Ji, Mar 18 2019: (Start)
a(n+1) = Sum_{i=0..n} a(i) + 1 - (-1)^n, a(0)=1.
a(n) = A000975(n-3)*10 + 5 + (-1)^(n-3), a(0)=1, a(1)=1, a(2)=4. (End)
a(n) = A081254(n) + (n-1 mod 2). - Kevin Ryde, Dec 20 2023
a(n) = 2*A048573(n-2) for n>=2. - Alois P. Heinz, May 20 2025

A083424 a(n) = (5*4^n + (-2)^n)/6.

Original entry on oeis.org

1, 3, 14, 52, 216, 848, 3424, 13632, 54656, 218368, 873984, 3494912, 13981696, 55922688, 223698944, 894779392, 3579150336, 14316535808, 57266274304, 229064835072, 916259864576, 3665038409728, 14660155736064, 58640618749952

Offset: 0

Paul Barry, Apr 30 2003

Binomial transform of A083423.

			Factorizations of initial terms: 1, (3), (2)*(7), (2)^2*(13), (2)^3*(3)^3, (2)^4*(53), (2)^5*(107), (2)^6*(3)*(71), (2)^7*(7)*(61), (2)^8*(853), (2)^9*(3)*(569), (2)^10*(3413), (2)^11*(6827), (2)^12*(3)^2*(37)*(41), (2)^13*(7)*(47)*(83), (2)^14*(13)*(4201), (2)^15*(3)*(23)*(1583), (2)^16*(218453), ...

Index entries for linear recurrences with constant coefficients, signature (2,8).

Cf. A000079, A048573, A083423.

A083424:=n->(5*4^n+(-2)^n)/6; [seq(A083424(n),n=0..50)]; # N. J. A. Sloane, Jul 18 2014

LinearRecurrence[{2,8},{1,3},30] (* Harvey P. Dale, Apr 21 2019 *)

a(n)=(5*4^n+(-2)^n)/6 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = 2*a(n-1) + 8*a(n-2). - N. J. A. Sloane, Jul 16 2014
G.f.: (1+x)/(1-2*x-8*x^2). [Corrected by N. J. A. Sloane, Jul 16 2014]
E.g.f.: (5*exp(4*x) + exp(-2*x))/6.
From N. J. A. Sloane, Jul 18 2014: (Start)
2^(n-1)|a(n) for n >= 1;
3|a(3n+1). (End)
From Klaus Purath, Oct 15 2020: (Start)
a(n) = A048573(n)*2^(n-1).
a(n) = A048573(n)*(A048573(n+1) - A048573(n-1))/5. (End)
a(n+1) = 4*a(n) - (-1)^n*A000079(n). - Paul Curtz, May 22 2025

A140966 a(n) = (5 + (-2)^n)/3.

Original entry on oeis.org

2, 1, 3, -1, 7, -9, 23, -41, 87, -169, 343, -681, 1367, -2729, 5463, -10921, 21847, -43689, 87383, -174761, 349527, -699049, 1398103, -2796201, 5592407, -11184809, 22369623, -44739241, 89478487, -178956969, 357913943, -715827881, 1431655767, -2863311529, 5726623063

Offset: 0

Paul Curtz, Jul 27 2008

Inverse binomial transform of A048573.
This is an example of the case k=-1 of sequences with recurrences a(n) = k*a(n-1) + (k+3)*a(n-2) - (2*k+2)*a(n-3).
The case k=1 is covered, for example, by A097163, A135520, A136326, A136336, or A137208.
Sequences with k=2 are A094554 and A094555.
Sequences with k=3 are A084175, A108924, and A139818.

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

Cf. A048573.
Cf. A097163, A135520, A136326, A136336, A137208.
Cf. A094554, A094555.
Cf. A084175, A108924, A139818.
Cf. A000079, A001045, A078008, A083582, A083584, A163834.

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

(5+(-2)^Range[0,30])/3 (* or *) LinearRecurrence[{-1,2},{2,1},40] (* Harvey P. Dale, Apr 23 2019 *)

a(n)=(5+(-2)^n)/3 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = -a(n-1) + 2*a(n-2).
G.f.: (2+3*x)/((1-x)*(1+2*x)).
a(n+1) - a(n) = (-1)^(n+1)*A000079(n).
a(n+3) = (-1)^n*A083582(n).
a(n+1) - 2*a(n) = -a(n+2).
a(n+1) - 3*a(n) = 5*(-1)^(n+1)*A078008(n) = (-1)^(n+1)*A001045(n-1).
a(2n+3) = -A083584(n), a(2n) = A163834(n). - Philippe Deléham, Feb 24 2014
E.g.f.: (5*exp(x) + exp(-2*x))/3. - Stefano Spezia, Jul 27 2024

Definition simplified by R. J. Mathar, Sep 11 2009

A135318 The Kentucky-2 sequence: a(n) = a(n-2) + 2*a(n-4), with a[0..3] = [1, 1, 1, 2].

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 8, 11, 16, 21, 32, 43, 64, 85, 128, 171, 256, 341, 512, 683, 1024, 1365, 2048, 2731, 4096, 5461, 8192, 10923, 16384, 21845, 32768, 43691, 65536, 87381, 131072, 174763, 262144, 349525, 524288, 699051, 1048576, 1398101, 2097152, 2796203

Offset: 0

Paul Curtz, Feb 16 2008

Shifted Jacobsthal recurrence.
From L. Edson Jeffery, Apr 21 2011: (Start)
Let U be the unit-primitive matrix (see [Jeffery])
U=U_(6,2)=
(0 0 1)
(0 2 0)
(2 0 1),
let i in {0,1}, m>=0 an integer and n=2*m+i. Then a(n)=a(2*m+i)=Sum_{j=0..2} (U^m)_(i,j). (End)
a(n) is also the pebbling number of the cycle graph C_{n+1} for n > 1. - Eric W. Weisstein, Jan 07 2021
From Greg Dresden and Ziyi Xie, Aug 25 2023: (Start)
a(n) is the number of ways to tile a zig-zag strip of n cells using squares (of 1 cell) and triangles (of 3 cells). Here is the zig-zag strip corresponding to n=11, with 11 cells:
       _     _
   _|   |_|   |_
  |   |_|   |_|   |_
  |_|   |_|   |_|   |
  |   |_|   |_|   |_|
  |_|   |_|   |_|,
and here are the two types of triangles (where one is just a reflection of the other):
   _               _
  |   |_       _|   |
  |       |     |       |
  |    _| and |_    |
  |_|             |_|.
As an example, here is one of the a(11) = 32 ways to tile the zig-zag strip of 11 cells:
       _     _
   _|   |_|   |_
  |   |_|       |   |_
  |       |_    |       |
  |    _|   |_|    _|
  |_|   |_|   |_|.   (End)

			Let i=0 and m=3. Then U^3 = (2,0,3;0,8,0;6,0,5), and the first-row sum (corresponding to i=0) is 2 + 0 + 3 = 5. Hence a(n) = a(2*m+i) = a(2*3+0) = a(6) = 2 + 3 = 5.

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Minerva Catral et al., Generalizing Zeckendorf's Theorem: The Kentucky Sequence, arXiv:1409.0488 [math.NT], 2014. See 1.3 p. 2, same sequence without the first 2 terms.
Shaoshi Chen, Hanqian Fang, Sergey Kitaev, and Candice X.T. Zhang, Patterns in Multi-dimensional Permutations, arXiv:2411.02897 [math.CO], 2024. See pp. 2, 17.
L. E. Jeffery, Unit-primitive matrices.
Eric Weisstein's World of Mathematics, Cycle Graph
Eric Weisstein's World of Mathematics, Pebbling Number
Index entries for linear recurrences with constant coefficients, signature (0,1,0,2).

Cf. A000079, A001045, A112387.

Cf. A048573.

[(2^Floor(n/2)*(5-(-1)^n)+(-1)^Floor(n/2)*(1+(-1)^n))/6: n in [0..50]]; // Vincenzo Librandi, Aug 10 2011

a:= n-> (<<0|1>, <2|1>>^(iquo(n, 2, 'm')). <<1, 1+m>>)[1,1]:
seq(a(n), n=0..50);  # Alois P. Heinz, May 30 2022

LinearRecurrence[{0,1,0,2},{1,1,1,2},40] (* Harvey P. Dale, Oct 14 2015 *)

From R. J. Mathar, Feb 19 2008: (Start)
O.g.f.: (1/(1+x^2)+(-2-3*x)/(2*x^2-1))/3.
a(2n) = A001045(n+1).
a(2n+1) = A000079(n). (End)
From L. Edson Jeffery, Apr 21 2011: (Start)
G.f.: (1+x+x^3)/((1+x^2)*(1-2*x^2)).
a(n) = (((-i)^(n+1)-i^(n+1))*2*i*sqrt(2)+3*(1+(-1)^(n+1))*2^((n+2)/2)+(1-(-1)^(n+1))*2^((n+5)/2))/(12*sqrt(2)), where i=sqrt(-1). (End)
a(n) = (2^floor(n/2)*(5-(-1)^n)+(-1)^floor(n/2)*(1+(-1)^n))/6 = (A016116(n)*A010711(n)+2*A056594(n))/6. - Bruno Berselli, Apr 21 2011
a(2n) = 2*a(2n-1) - a(2n-2); a(2n+1) = a(2n) + a(2n-2). - Richard R. Forberg, Aug 19 2013
a(n) = A112387(n + (-1)^n). - Alois P. Heinz, Sep 28 2023
E.g.f.: (2*cos(x) + 4*cosh(sqrt(2)*x) + 3*sqrt(2)*sinh(sqrt(2)*x))/6. - Stefano Spezia, Nov 09 2024
a(2*n) + a(2*n+1) = A048573(n) for n >= 0. - Paul Curtz, May 18 2025

More terms from R. J. Mathar, Feb 19 2008

A251149 T(n,k) = Number of (n+1) X (k+1) 0..2 arrays with every 2 X 2 subblock summing to 4 and no 2 X 2 subblock having exactly two nonzero entries.

Original entry on oeis.org

13, 27, 27, 53, 49, 53, 107, 87, 87, 107, 213, 161, 143, 161, 213, 427, 299, 247, 247, 299, 427, 853, 565, 433, 401, 433, 565, 853, 1707, 1075, 777, 667, 667, 777, 1075, 1707, 3413, 2065, 1413, 1141, 1061, 1141, 1413, 2065, 3413, 6827, 3991, 2607, 1987, 1743

Offset: 1

R. H. Hardin, Nov 30 2014

Table starts:
...13...27...53...107...213...427...853..1707..3413...6827..13653..27307..54613
...27...49...87...161...299...565..1075..2065..3991...7761..15163..29749..58563
...53...87..143...247...433...777..1413..2607..4863...9167..17433..33417..64493
..107..161..247...401...667..1141..1987..3521..6327..11521..21227..39541..74387
..213..299..433...667..1061..1743..2925..5003..8689..15307..27317..49359..90237
..427..565..777..1141..1743..2763..4491..7453.12569..21501..37255..65355.116035
..853.1075.1413..1987..2925..4491..7101.11491.18917..31587..53389..91275.157789
.1707.2065.2607..3521..5003..7453.11491.18193.29359..48081..79675.133405.225555
.3413.3991.4863..6327..8689.12569.18917.29359.46575..75087.122521.201881.335501
.6827.7761.9167.11521.15307.21501.31587.48081.75087.119441.192507.313341.514067

			Some solutions for n=4, k=4:
..0..1..0..2..1....1..0..1..1..1....1..0..1..1..2....2..1..2..1..1
..1..2..1..1..0....1..2..1..1..1....1..2..1..1..0....0..1..0..1..1
..1..0..1..1..2....1..0..1..1..1....0..1..0..2..1....1..2..1..2..0
..1..2..1..1..0....1..2..1..1..1....1..2..1..1..0....0..1..0..1..1
..1..0..1..1..2....1..0..1..1..1....0..1..0..2..1....2..1..2..1..1

R. H. Hardin, Table of n, a(n) for n = 1..364

Column 1 is A048573(n+2).

Empirical for column k:
k=1: a(n) = a(n-1) +2*a(n-2)
k=2: a(n) = 3*a(n-1) -5*a(n-3) +a(n-4) +2*a(n-5)
k=3: a(n) = 3*a(n-1) -5*a(n-3) +a(n-4) +2*a(n-5)
k=4: a(n) = 3*a(n-1) -5*a(n-3) +a(n-4) +2*a(n-5)
k=5: a(n) = 3*a(n-1) -5*a(n-3) +a(n-4) +2*a(n-5)
k=6: a(n) = 3*a(n-1) -5*a(n-3) +a(n-4) +2*a(n-5)
k=7: a(n) = 3*a(n-1) -5*a(n-3) +a(n-4) +2*a(n-5)

A112387 a(n) = 2^(n/2) if n is even and a(n-1) - a(n-2) if n is odd, a(1) = 1.

Original entry on oeis.org

1, 1, 2, 1, 4, 3, 8, 5, 16, 11, 32, 21, 64, 43, 128, 85, 256, 171, 512, 341, 1024, 683, 2048, 1365, 4096, 2731, 8192, 5461, 16384, 10923, 32768, 21845, 65536, 43691, 131072, 87381, 262144, 174763, 524288, 349525, 1048576, 699051, 2097152, 1398101, 4194304

Offset: 0

Edwin F. Sampang, Dec 05 2005

This sequence originated from the Fibonacci sequence, but instead of adding the last two terms, you get the average. Example, if you have the initial condition a(1)=x and a(2)=y, a(3)=(x+y)/2, a(4)=(x+3y)/4, a(5)=(3x+5y)/8, a(6)=(5x+11y)/16 and so on and so forth. I used the coefficients of x and y as well as the denominator.

As n approaches infinity a(n)/a(n+1) oscillates between the values 3/2 and 1/3.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,1,0,2).

Cf. A000079, A001045, A048573, A135318.

a:= proc(n) option remember;
      `if`(n::even, 2^(n/2), a(n-1)-a(n-2))
    end: a(1):=1:
seq(a(n), n=0..50);  # Alois P. Heinz, Sep 27 2023

a[1] = 1; a[2] = 2; a[n_] := a[n] = If[ EvenQ[n], 2^(n/2), a[n - 1] - a[n - 2]]; Array[a, 43] (* Robert G. Wilson v, Dec 05 2005 *)
nxt[{n_,a_,b_}]:={n+1,b,If[OddQ[n],2^((n+1)/2),b-a]}; NestList[nxt,{2,1,2},50][[All,2]] (* Harvey P. Dale, Jul 08 2019 *)

a(n) = 2^(n/2) if n is even, a(n) = a(n-1) - a(n-2) if n is odd, and a(1) = 1.
a(2n) = A000079(n), a(2n-1) = A001045(n).
G.f.: (1+x+x^2)/((1+x^2)*(1-2*x^2)). - Joerg Arndt, Apr 25 2021
a(n) = A135318(n + (-1)^n). - Paul Curtz, Sep 27 2023
E.g.f.: (3*cosh(sqrt(2)*x) + sin(x) + sqrt(2)*sinh(sqrt(2)*x))/3. - Stefano Spezia, Jun 30 2024
a(2*n) + a(2*n+1) = A048573(n); a(2*n+1) + a(2*n+2) = A001045(n+3). - Paul Curtz, Jan 03 2025

Edited and extended by Robert G. Wilson v, Dec 05 2005

a(0)=1 prepended by Alois P. Heinz, Sep 27 2023

A081374 Size of "uniform" Hamming covers of distance 1, that is, Hamming covers in which all vectors of equal weight are treated the same, included or excluded from the cover together.

Original entry on oeis.org

1, 2, 2, 5, 10, 22, 43, 86, 170, 341, 682, 1366, 2731, 5462, 10922, 21845, 43690, 87382, 174763, 349526, 699050, 1398101, 2796202, 5592406, 11184811, 22369622, 44739242, 89478485, 178956970, 357913942, 715827883, 1431655766, 2863311530, 5726623061

Offset: 1

David Applegate, Aug 22 2003

nonn

Motivation: consideration of the "hats" problem (which boils down to normal hamming covering codes) in the case when the people are indistinguishable or unlabeled.
From Paul Curtz, May 26 2011: (Start)
If we add a(0)=1 in front and build the table of a(n) and iterated differences in further rows we get:
1,    1,  2,  2,  5, 10,
0,    1,  0,  3,  5, 12,
1,   -1,  3,  2,  7,  9,
-2,   4, -1,  5,  2, 13,
6,   -5,  6, -3, 11,  6
-11, 11, -9, 14, -5, 21.
The first column is the inverse binomial transform, which is 1,0 followed by (-1)^n*A083322(n-1), n>=2.
The main diagonal in the table above is A001045, the adjacent upper diagonals are A078008, A048573 and A062092. (End)

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

Cf. A083322.

I:=[1,2,2,5]; [n le 4 select I[n] else 2*Self(n-1)-Self(n-3)+2*Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jul 08 2016

hatwork := proc(n,i,covered) local val, val2; options remember;
# computes the minimum cover of the i-bit through n-bit words.
# if covered is true the i-bit words are already covered (by the (i-1)-bit words)
if (i>n or (i = n and covered)) then 0; elif (i = n and not covered) then 1; else
# one choice is to include the i-bit words in the cover
val := hatwork(n, i+1, true) + binomial(n,i);
# the other choice is not to include the i-bit words in the cover
if (covered) then val2 := hatwork (n, i+1, false); if (val2 < val) then val := val2; fi; else
# if the i-bit words were not covered by (i-1), they must be covered by the (i+1)-bit words
if (i <= n) then val2 := hatwork (n, i+2, true) + binomial(n,i+1); if (val2 < val) then val := val2; fi; fi; fi; val; fi; end proc;
A081374 := proc (n) hatwork(n, 0, false); end proc;

LinearRecurrence[{2,0,-1,2},{1,2,2,5},40] (* Harvey P. Dale, Feb 11 2015 *)

If (n mod 6 = 5) then sum(binomial(n, 3*i+1), i=0..n/3); elif (n mod 6 = 2) then sum(binomial(n, 3*i), i=0..n/3)+1; else sum(binomial(n, 3*i), i=0..n/3); fi;
G.f.: x*(2*x^3-2*x^2+1)/( (1-2*x)*(1+x)*(1-x+x^2) ).
a(n)=2*a(n-1)-a(n-3)+2*a(n-4).
From Paul Curtz, May 26 2011: (Start)
a(n+1) - 2*a(n) has period length 6: repeat 0, -2, 1, 0, 2, -1 (see A080425).
a(n) - A083322(n-1) = A010892(n-1) has period length 6.
a(n) + a(n+3) = 3*2^n = A007283(n).
a(n+6)-a(n) = 21*2^n = A175805(n).
a(n) - A131708(n) = -A131531(n).  (End)

A169969 Locations of row maxima in "crushed" version of Stern's diatomic array.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 21, 27, 43, 53, 85, 107, 171, 213, 341, 427, 683, 853, 1365, 1707, 2731, 3413, 5461, 6827, 10923, 13653, 21845, 27307, 43691, 54613, 87381, 109227, 174763, 218453, 349525, 436907, 699051, 873813, 1398101, 1747627, 2796203, 3495253, 5592405

Offset: 1

N. J. A. Sloane, Aug 08 2010

From Michel Marcus, Jan 22 2015: (Start)
The Stern's diatomic array begins (see A049456).
  1...............................1
  1...............2...............1
  1.......3.......2.......3.......1
  1...4...3...5...2...5...3...4...1
  1.5.4.7.3.8.5.7.2.7.5.8.3.7.4.5.1
  ...
The "crushed" version is obtained by removing the right column, and then squeezing everything to the left.
  1;
  1, 2;
  1, 3, 2, 3;
  1, 4, 3, 5, 2, 5, 3, 4;
  1, 5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 4, 5;
  ...
This gives sequence 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, ... (cf. A002487).
The "crushed" array row maxima are: 1, 2, 3, 5, 8, ... (cf. A000045).
The indices of these values in A002487 are 1, 3, 5, 7, 11, ... : this sequence.
Note, for instance, that for 3rd row, the maximum which is 3, appears twice, at indices 5 and 7, giving 2 terms for this sequence.
(End)

			G.f. = x + 3*x^2 + 5*x^3 + 7*x^4 + 11*x^5 + 13*x^6 + 21*x^7 + 27*x^8 + 43*x^9 + ...

S. Northshield, Stern's diatomic sequence 0, 1, 1, 2, 1, 3, 2, 3, 1, 4, ..., Amer. Math. Monthly, 117 (2010), 581-598.
Index entries for linear recurrences with constant coefficients, signature (0,1,0,2).

Cf. A000079, A026644, A048573, A049456, A084170, A168642.

a[n_] := a[n] = If[n <= 5, {1, 3, 5, 7, 11}[[n]], a[n-2] + 2a[n-4]]; Array[a, 42] (* Jean-François Alcover, Dec 11 2016 *)

fusc(n)=local(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); b; \\ from A002487
lista(nn) = {nb = 2^(nn+1)-1; vall = vector(nb, n, fusc(n)); for (n=1, nn, vmax = 0; for (j=2^(n-1), 2^n-1, if (vall[j] > vmax, vmax = vall[j]);); for (j=2^(n-1), 2^n-1, if (vall[j] == vmax, print1(j, ", "));););} \\ Michel Marcus, Jan 22 2015

a(2n+1) + a(2n+2) = 3*2^(n+1), n>0 . - Yosu Yurramendi, Jun 29 2016
a(2n+3) = 3*2^(n+1) - a(n); a(2n+4) = 3*2^(n+1) + a(n), n>=0, a(0)=0 (new term), a(1)=1, a(2)=3 . - Yosu Yurramendi, Jun 29 2016
G.f.: x*(1 + 3*x + 4*x^2 + 4*x^3 + 4*x^4)/((1 + x^2)*(1 - 2*x^2)). - Ilya Gutkovskiy, Jun 29 2016
For n>1, a(n) = (2^(n/2 - 1)*(5 + 4*sqrt(2) + (-1)^n*(5 - 4*sqrt(2))) + cos(Pi*n/2) + sin(Pi*n/2))/3. - Vaclav Kotesovec, Jun 30 2016
a(2n) = a(2n-7) + 3*2^(n-1); a(2n-1) = a(2n-7) - 3*2^(n-1), n>=5 . - Yosu Yurramendi, Jul 06 2016
a(2n-1) = A168642(n), n>0; a(2n) = A048573(n), n>0; a(2n-1) = A026644(n) + 1, n>1; a(2n) = A084170(n) + 1, n>0 . - Yosu Yurramendi, Dec 11 2016

More terms from Michel Marcus, Jan 22 2015

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

cp's OEIS Frontend

A081254 Numbers k such that A081252(m)/m^2 has a local maximum for m = k.

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A084214 Inverse binomial transform of a math magic problem.

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A083424 a(n) = (5*4^n + (-2)^n)/6.

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A140966 a(n) = (5 + (-2)^n)/3.

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A135318 The Kentucky-2 sequence: a(n) = a(n-2) + 2*a(n-4), with a[0..3] = [1, 1, 1, 2].

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A251149 T(n,k) = Number of (n+1) X (k+1) 0..2 arrays with every 2 X 2 subblock summing to 4 and no 2 X 2 subblock having exactly two nonzero entries.

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A112387 a(n) = 2^(n/2) if n is even and a(n-1) - a(n-2) if n is odd, a(1) = 1.

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