A175654 -id:A175654 - cp's OEIS frontend

A095121 Expansion of (1-x+2x^2)/((1-x)*(1-2x)).

1, 2, 6, 14, 30, 62, 126, 254, 510, 1022, 2046, 4094, 8190, 16382, 32766, 65534, 131070, 262142, 524286, 1048574, 2097150, 4194302, 8388606, 16777214, 33554430, 67108862, 134217726, 268435454, 536870910, 1073741822, 2147483646, 4294967294, 8589934590

Offset: 0

Paul Barry, May 28 2004

a(n+1)/2 = A000225(n). Binomial transform is A002783. Binomial transform of 2 - 2*0^n + (-1)^n or 1,1,3,1,3,1,3,1,...
From Peter C. Heinig (algorithms(AT)gmx.de), Apr 17 2007: (Start)
Number of n-tuples where each entry is chosen from the subsets of {1,2} such that the intersection of all n entries contains exactly one element.
There is the following general formula: The number T(n,k,r) of n-tuples where each entry is chosen from the subsets of {1,2,..,k} such that the intersection of all n entries contains exactly r elements is: T(n,k,r) = binomial(k,r) * (2^n - 1)^(k-r). This may be shown by exhibiting a bijection to a set whose cardinality is obviously binomial(k,r) * (2^n - 1)^(k-r), namely the set of all k-tuples where each entry is chosen from subsets of {1,..,n} in the following way: Exactly r entries must be {1,..,n} itself (there are binomial(k,r) ways to choose them) and the remaining (k-r) entries must be chosen from the 2^n-1 proper subsets of {1,..,n}, i.e., for each of the (k-r) entries, {1,..,n} is forbidden (there are, independent of the choice of the full entries, (2^n - 1)^(k-r) possibilities to do that, hence the formula). The bijection into this set is given by (X_1,..,X_n) |-> (Y_1,..,Y_k) where for each j in {1,..,k} and each i in {1,..,n}, i is in Y_j if and only if j is in X_i.
Examples: a(1)=2 because the two tuples of length one are: ({1}) and ({2}).
a(3)=14 because the fourteen tuples of length three are: ({1},{1},{1}), ({2},{2},{2}), ({1,2},{1},{1}), ({1},{1,2},{1}), ({1},{1},{1,2}), ({1,2},{2},{2}), ({2},{1,2},{2}), ({2},{2},{1,2}), ({1,2},{1,2},{1}), ({1,2},{1},{1,2}), ({1},{1,2},{1,2}), ({1,2}{1,2}{2}), ({1,2}{2}{1,2}), ({2}{1,2}{1,2}).
The image of this set of tuples under the bijection described in the comment is: ({1,2,3},{}), ({},{1,2,3}), ({1,2,3},{1}), ({1,2,3},{2}), ({1,2,3},{3}), ({1},{1,2,3}), ({2},{1,2,3}), ({3},{1,2,3}), ({1,2,3},{1,2}), ({1,2,3},{1,3}), ({1,2,3},{2,3}), ({1,2},{1,2,3}), ({1,3},{1,2,3}), ({2,3},{1,2,3}). Here exactly one entry is {1,..,n}={1,2,3} and the other is a proper subset. (End)
An elephant sequence, see A175654. For the corner squares just one A[5] vector, with decimal value 170, leads to this sequence. For the central square this vector leads to the companion sequence A151821. - Johannes W. Meijer, Aug 15 2010
Conjecture: a(n) is the least m>0 such that A007814(A000108(m)) = n, where A000108 gives the Catalan numbers and A007814(x) is the 2-adic valuation of x (cf. A048881). - L. Edson Jeffery, Nov 26 2015
Also, the decimal representation of the diagonal from the corner to the origin of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 645", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. - Robert Price, Jul 19 2017

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Wolfram Research, Wolfram Atlas of Simple Programs
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A000108, A000225, A000918, A002783, A007814, A048881, A123110, A127509, A151821, A175654.

Row sums of A131108, A132046, A153861, and A193815.

[-2+4*2^(n-1)+(Binomial(2*n,n) mod 2): n in [0..40]]; // Vincenzo Librandi, Aug 14 2015

ZL := [S, {S=Prod(B,B), B=Set(Z, 1 <= card)}, labeled]: seq(combstruct[ count](ZL, size=n), n=1..31); # Zerinvary Lajos, Mar 13 2007
for k from 1 to 31 do 2*(2^k-1); od;

Join[{1}, LinearRecurrence[{3, -2}, {2, 6}, 50]] (* Vladimir Joseph Stephan Orlovsky, Feb 24 2012 *)
Join[{1},NestList[2#+2&,2,40]] (* Harvey P. Dale, Dec 25 2013 *)

Vec((1-x+2*x^2)/((1-x)*(1-2*x)) + O(x^40)) \\ Michel Marcus, Aug 14 2015

vector(100, n, n--; if(n==0, 1, 2*2^n-2)) \\ Altug Alkan, Nov 26 2015

G.f.: (1-x+2*x^2)/((1-x)*(1-2*x)).
a(n) = A000918(n+1), n >= 1.
a(n) = 2*2^n - 2 + 0^n; a(n) = 3*a(n-1) - 2*a(n-2).
a(0)=1, a(1)=2, a(n) = 2*a(n-1) + 2 for n>1. - Philippe Deléham, Sep 28 2006
a(n) = Sum_{k=0..n} 2^k*A123110(n,k). - Philippe Deléham, Feb 09 2007
a(n) = 5*a(n-2) - 4*a(n-4) for n>4 [Because x(n)=f*x(n-1)+g*x(n-2) => x(n)=(f^2+2*g)*x(n-2)-g^2*x(n-4), here with f=3 and g=-2]. - Hermann Stamm-Wilbrandt, Aug 13 2015
E.g.f.: 1 + 2*exp(x)*(exp(x) - 1). - Stefano Spezia, Feb 25 2022

Edited by N. J. A. Sloane, Apr 25 2007

A175655 Eight bishops and one elephant on a 3 X 3 chessboard. G.f.: (1+x-5x^2)/(1-3x-x^2+6*x^3).

Original entry on oeis.org

1, 4, 8, 22, 50, 124, 290, 694, 1628, 3838, 8978, 21004, 48962, 114022, 265004, 615262, 1426658, 3305212, 7650722, 17697430, 40911740, 94528318, 218312114, 503994220, 1163124866, 2683496134, 6189647948, 14273690782

Offset: 0

Johannes W. Meijer, Aug 06 2010, Aug 10 2010

a(n) represents the number of n-move routes of a fairy chess piece starting in the central square (m = 5) on a 3 X 3 chessboard. This fairy chess piece behaves like a bishop on the eight side and corner squares but on the central square the bishop turns into a raging elephant, see A175654.
For the central square the 512 elephants lead to 46 different elephant sequences, see the cross-references for examples.
The sequence above corresponds to 16 A[5] vectors with decimal values 71, 77, 101, 197, 263, 269, 293, 323, 326, 329, 332, 353, 356, 389, 449 and 452. These vectors lead for the side squares to A000079 and for the corner squares to A175654.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,1,-6).

Cf. Elephant sequences central square [decimal value A[5]]: A000007 [0], A000012 [16], A000045 [1], A011782 [2], A000079 [3], A003945 [42], A099036 [11], A175656 [7], A105476 [69], A168604 [26], A045891 [19], A078057 [21], A151821 [170], A175657 [43], 4*A172481 [15; n>=-1], A175655 [71, this sequence], 4*A026597 [325; n>=-1], A033484 [58], A087447 [27], A175658 [23], A026150 [85], A175661 [171], A036563 [186], A098156 [59], A046717 [341], 2*A001792 [187; n>=1 with a(0)=1], A175659 [343].

Cf. A000244, A006130, A075118, A175654.

I:=[1, 4, 8]; [n le 3 select I[n] else 3*Self(n-1)+Self(n-2)-6*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jul 21 2013

with(LinearAlgebra): nmax:=27; m:=5; A[5]:= [0,0,1,0,0,0,1,1,1]: A:=Matrix([[0,0,0,0,1,0,0,0,1], [0,0,0,1,0,1,0,0,0], [0,0,0,0,1,0,1,0,0], [0,1,0,0,0,0,0,1,0], A[5], [0,1,0,0,0,0,0,1,0], [0,0,1,0,1,0,0,0,0], [0,0,0,1,0,1,0,0,0], [1,0,0,0,1,0,0,0,0]]): for n from 0 to nmax do B(n):=A^n: a(n):= add(B(n)[m,k],k=1..9): od: seq(a(n), n=0..nmax);

CoefficientList[Series[(1 + x - 5 x^2) / (1 - 3 x - x^2 + 6 x^3), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 21 2013 *)
LinearRecurrence[{3,1,-6},{1,4,8},40] (* Harvey P. Dale, Dec 25 2024 *)

a(n)=([0,1,0; 0,0,1; -6,1,3]^n*[1;4;8])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

G.f.: (1+x-5*x^2)/(1-3*x-x^2+6*x^3).
a(n) = 3*a(n-1) + a(n-2) - 6*a(n-3) with a(0)=1, a(1)=4 and a(2)=8.
a(n) = ((10+8*A)*A^(-n-1) + (10+8*B)*B^(-n-1))/13 - 2^n with A = (-1+sqrt(13))/6 and B = (-1-sqrt(13))/6.
Limit_{k->oo} a(n+k)/a(k) = (-1)^(n)*2*A000244(n)/(A075118(n)-A006130(n-1)*sqrt(13)).
E.g.f.: 2*exp(x/2)*(13*cosh(sqrt(13)*x/2) + 5*sqrt(13)*sinh(sqrt(13)*x/2))/13 - cosh(2*x) - sinh(2*x). - Stefano Spezia, Jan 31 2023

A027934 a(0)=0, a(1)=1, a(2)=2; for n > 2, a(n) = 3a(n-1) - a(n-2) - 2a(n-3).

Original entry on oeis.org

0, 1, 2, 5, 11, 24, 51, 107, 222, 457, 935, 1904, 3863, 7815, 15774, 31781, 63939, 128488, 257963, 517523, 1037630, 2079441, 4165647, 8342240, 16702191, 33433039, 66912446, 133899917, 267921227, 536038872, 1072395555, 2145305339

Offset: 0

Clark Kimberling

Number of compositions of n with at least one even part (offset 2). - Vladeta Jovovic, Dec 29 2004
First differences of A008466. a(n) = A008466(n+2) - A008466(n+1). - Alexander Adamchuk, Apr 06 2006
Starting with "1" = eigensequence of a triangle with the Fibonacci series as the left border and the rest 1's. - Gary W. Adamson, Jul 24 2010
An elephant sequence, see A175654. For the corner squares 24 A[5] vectors, with decimal values between 11 and 416, lead to this sequence (without the leading 0). For the central square these vectors lead to the companion sequence A099036 (without the first leading 1). - Johannes W. Meijer, Aug 15 2010
a(n) = Sum_{k=1..n} A108617(n,k) / 2. - Reinhard Zumkeller, Oct 07 2012
a(n) is the number of binary strings that contain the substring 11 or end in 1. a(3) = 5 because we have: 001, 011, 101, 110, 111. - Geoffrey Critzer, Jan 04 2014
a(n-1), n >= 1, is the number of nonexisting (due to the maturation delay) "[male-female] pairs of Fibonacci rabbits" at the beginning of the n-th month. - Daniel Forgues, May 06 2015
a(n-1) is the number of subsets of {1,2,..,n} that contain n that have at least one pair of consecutive integers. For example, for n=5, a(4) = 11 and the 11 subsets are {4,5}, {1,2,5}, {1,4,5}, {2,3,5}, {2,4,5}, {3,4,5}, {1,2,3,5}, {1,2,4,5}, {1,3,4,5}, {2,3,4,5}, {1,2,3,4,5}.  Note that A008466(n) is the number of all subsets of {1,2,..,n} that have at least one pair of consecutive integers. - Enrique Navarrete, Aug 15 2020

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Patrick Letendre, Polynomials with integer roots, arXiv:1911.00480 [math.NT], 2019. See p. 4.
Toufik Mansour and Mark Shattuck, Pattern avoidance in flattened derangements, Disc. Math. Lett. (2025) Vol. 15, 67-74. See p. 74.
Augustine O. Munagi, Integer Compositions and Higher-Order Conjugation, J. Int. Seq., Vol. 21 (2018), Article 18.8.5.
Denis Neiter and Amsha Proag, Links Between Sums Over Paths in Bernoulli's Triangles and the Fibonacci Numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.8.3.
OEIS Wiki, Fibonacci rabbits
Index entries for linear recurrences with constant coefficients, signature (3,-1,-2).

Row sums of triangle A131767. - Gary W. Adamson, Jul 13 2007
a(n) = A101220(1, 2, n+1).
T(n, n) + T(n, n+1) + ... + T(n, 2n), T given by A027926.
Diagonal sums of A055248.
Cf. A000045, A000079, A008466, A059570, A099036, A047967, A000032.

List([0..35], n-> 2^n - Fibonacci(n+1) ); # G. C. Greubel, Sep 27 2019

a027934 n = a027934_list !! n
a027934_list = 0 : 1 : 2 : zipWith3 (\x y z -> 3 * x - y - 2 * z)
               (drop 2 a027934_list) (tail a027934_list) a027934_list
-- Reinhard Zumkeller, Oct 07 2012

[2^n - Fibonacci(n+1): n in [0..35]]; // G. C. Greubel, Sep 27 2019

A027934:= proc(n) local K; K:= Matrix ([[2,0,0], [0,1,1], [0,1,0]])^n; K[1,1]-K[2,2] end: seq (A027934(n), n=0..31); # Alois P. Heinz, Jul 28 2008
a := n -> 2^n - combinat:-fibonacci(n+1): seq(a(n),n=0..31); # Peter Luschny, May 09 2015

nn=31; a:=1/(1-x-x^2); b:=1/(1-2x); CoefficientList[Series[a*x*(1+x*b), {x,0,nn}], x] (* Geoffrey Critzer, Jan 04 2014 *)
LinearRecurrence[{3,-1,-2}, {0,1,2}, 32] (* Jean-François Alcover, Jan 09 2016 *)
nxt[{a_,b_,c_}]:={b,c,3c-b-2a}; NestList[nxt,{0,1,2},40][[;;,1]] (* Harvey P. Dale, Feb 02 2025 *)

a(n)=2^n-fibonacci(n+1) \\ Charles R Greathouse IV, Jun 11 2015

[2^n - fibonacci(n+1) for n in (0..35)] # G. C. Greubel, Sep 27 2019

a(n) = Sum_{j=0..floor(n/2)} Sum_{k=0..n-2*j} binomial(n-j, n-2*j-k). - Paul Barry, Feb 07 2003
From Paul Barry, Jan 23 2004: (Start)
Row sums of A105809.
G.f.: x*(1-x)/((1-2*x)*(1-x-x^2)).
a(n) = 2^n - Fibonacci(n+1). (End) - corrected Apr 06 2006 and Oct 05 2012
a(n) = Sum_{j=0..n} Sum_{k=0..n} binomial(n-k, k+j). - Paul Barry, Aug 29 2004
a(n) = (Sum of (n+1)-th row of the triangle in A108617) / 2. - Reinhard Zumkeller, Jun 12 2005
a(n) = term (1,1) - term (2,2) in the 3 X 3 matrix [2,0,0; 0,1,1; 0,1,0]^n. - Alois P. Heinz, Jul 28 2008
a(n) = 2^n - A000045(n+1). - Geoffrey Critzer, Jan 04 2014
a(n) ~ 2^n. - Daniel Forgues, May 06 2015
From Bob Selcoe, Mar 29 2016: (Start)
a(n) = 2*a(n-1) + A000045(n-2).
a(n) = 4*a(n-2) + A000032(n-2). (End)
a(n) = 2^(n-1) - ( ((1+sqrt(5))/2)^n - ((1-sqrt(5))/2)^n)/sqrt(5). - Haider Ali Abdel-Abbas, Aug 17 2019

Simpler definition from Miklos Kristof, Nov 24 2003
Initial zero added by N. J. A. Sloane, Feb 13 2008
Definition fixed by Reinhard Zumkeller, Oct 07 2012

A006138 a(n) = a(n-1) + 3*a(n-2).

Original entry on oeis.org

1, 2, 5, 11, 26, 59, 137, 314, 725, 1667, 3842, 8843, 20369, 46898, 108005, 248699, 572714, 1318811, 3036953, 6993386, 16104245, 37084403, 85397138, 196650347, 452841761, 1042792802, 2401318085, 5529696491, 12733650746, 29322740219, 67523692457, 155491913114

Offset: 0

N. J. A. Sloane

The binomial transform of a(n) is b(n) = A006190(n+1), which satisfies b(n) = 3*b(n-1) + b(n-2). - Paul Barry, May 21 2006
Partial sums of A105476. - Paul Barry, Feb 02 2007
An elephant sequence, see A175654. For the corner squares four A[5] vectors, with decimal values 69, 261, 321 and 324, lead to this sequence. For the central square these vectors lead to the companion sequence A105476 (without the first leading 1). - Johannes W. Meijer, Aug 15 2010
Equals the INVERTi transform of A063782: (1, 3, 10, 32, 104, ...). - Gary W. Adamson, Aug 14 2010
Pisano period lengths: 1, 3, 1, 6, 24, 3, 24, 6, 3, 24, 120, 6, 156, 24, 24, 12, 16, 3, 90, 24, ... - R. J. Mathar, Aug 10 2012
The sequence is the INVERT transform of A016116: (1, 1, 2, 2, 4, 4, 8, 8, ...). - Gary W. Adamson, Jul 17 2015

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
N. T. Gridgeman, A new look at Fibonacci generalization, Fib,. Quart., 11 (1973), 40-55.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (1,3).

Cf. A063782. - Gary W. Adamson, Aug 14 2010

Cf. A006130, A006190, A016116, A105476, A122950, A175654, A274977.

a:=[1,2];; for n in [3..40] do a[n]:=a[n-1]+3*a[n-2]; od; a; # G. C. Greubel, Nov 19 2019

[n le 2 select n else Self(n-1)+3*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Sep 15 2016

A006138:=-(1+z)/(-1+z+3*z**2); # Simon Plouffe in his 1992 dissertation

CoefficientList[Series[(1+z)/(1-z-3*z^2), {z,0,40}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 11 2011 *)
Table[(I*Sqrt[3])^(n-1)*(I*Sqrt[3]*ChebyshevU[n, 1/(2*I*Sqrt[3])] + ChebyshevU[n-1, 1/(2*I*Sqrt[3])]), {n, 0, 40}]//Simplify (* G. C. Greubel, Nov 19 2019 *)
LinearRecurrence[{1,3},{1,2},40] (* Harvey P. Dale, May 29 2025 *)

main(size)={my(v=vector(size),i);v[1]=1;v[2]=2;for(i=3,size,v[i]=v[i-1]+3*v[i-2]);return(v);} /* Anders Hellström, Jul 17 2015 */

def A006138_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+x)/(1-x-3*x^2)).list()
A006138_list(40) # G. C. Greubel, Nov 19 2019

a(n) = Sum_{k=0..n+1} A122950(n+1,k)*2^(n+1-k). - Philippe Deléham, Jan 04 2008
G.f.: (1+x)/(1-x-3*x^2). - Paul Barry, May 21 2006
a(n) = Sum_{k=0..n} C(floor((2n-k)/2),n-k)*3^floor(k/2). - Paul Barry, Feb 02 2007
a(n) = A006130(n) + A006130(n-1). - R. J. Mathar, Jun 22 2011
a(n) = (i*sqrt(3))^(n-1)*(i*sqrt(3)*ChebyshevU(n, 1/(2*i*sqrt(3))) + ChebyshevU(n-1, 1/(2*i*sqrt(3)))), where i=sqrt(-1). - G. C. Greubel, Nov 19 2019

Typo in formula corrected by Johannes W. Meijer, Aug 15 2010

A123203 a(n) = 2^(n+1) - 3*n.

Original entry on oeis.org

1, 2, 7, 20, 49, 110, 235, 488, 997, 2018, 4063, 8156, 16345, 32726, 65491, 131024, 262093, 524234, 1048519, 2097092, 4194241, 8388542, 16777147, 33554360, 67108789, 134217650, 268435375, 536870828, 1073741737, 2147483558

Offset: 1

Gary W. Adamson, Jun 13 2007

An elephant sequence, see A175654. For the corner squares just one A[5] vector, with decimal value 186, leads to this sequence. For the central square this vector leads to the companion sequence A036563. - Johannes W. Meijer, Aug 15 2010

			a(4) = 20, row sums of 4th row of triangle A131062: (1, 9, 9, 1).
a(4) = 20 = (1, 3, 3, 1) dot (1, 1, 4, 4) = (1 + 3 + 12 + 4).

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Joseph Breen and Emma Copeland, Non-orientable Nurikabe, arXiv:2506.12612 [math.CO], 2025. See pp. 1, 4.
Tamas Lengyel, On p-adic properties of the Stirling numbers of the first kind, Journal of Number Theory, 148 (2015) 73-94.
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A000079, A000295, A109128, A131060, A131061, A131063, A131064, A131065, A131066.

[2^(n+1) -3*n: n in [1..40]]; // G. C. Greubel, Sep 14 2024

Table[2^(n+1) - 3*n, {n,40}] (* Vladimir Joseph Stephan Orlovsky, Nov 15 2008 *)
LinearRecurrence[{4,-5,2},{1,2,7},40] (* Harvey P. Dale, Mar 30 2024 *)

def A123203(n): return 2^(n+1) -3*n
[A123203(n) for n in range(1,41)] # G. C. Greubel, Sep 14 2024

Binomial transform of [1, 1, 4, 4, 4, ...].
Equals row sums of triangle A131061.
From Johannes W. Meijer, Aug 15 2010; corrected by Colin Barker, Jul 28 2012: (Start)
a(n) = 2^(1+n) - 3*n.
a(n) = 3*A000295(n-1) + A000079(n-1).
(End)
G.f.: x*(1 - 2*x + 4*x^2)/((1-x)^2*(1-2*x)). - Colin Barker, Jul 28 2012
a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3). - Colin Barker, Jul 29 2012
E.g.f.: 2*exp(2*x) - 3*x*exp(x) - 2. - G. C. Greubel, Sep 14 2024

More terms from Vladimir Joseph Stephan Orlovsky, Nov 15 2008

Title changed by G. C. Greubel, Sep 14 2024

A066373 a(n) = (3n-2)2^(n-3).

Original entry on oeis.org

2, 7, 20, 52, 128, 304, 704, 1600, 3584, 7936, 17408, 37888, 81920, 176128, 376832, 802816, 1703936, 3604480, 7602176, 15990784, 33554432, 70254592, 146800640, 306184192, 637534208, 1325400064, 2751463424, 5704253440, 11811160064, 24427626496, 50465865728, 104152956928

Offset: 2

N. J. A. Sloane, Jan 04 2002

An elephant sequence, see A175654. For the corner squares 16 A[5] vectors, with decimal values between 59 and 440, lead to this sequence (with a leading 1 added). For the central square these vectors lead to the companion sequence A098156 (without a(1)). - Johannes W. Meijer, Aug 15 2010

a(n) is the total number of 1's in runs of 1's of length >= 2 over all binary words with n bits. - Félix Balado, Jan 15 2024

Harry J. Smith, Table of n, a(n) for n = 2..200
M. Azaola and F. Santos, The number of triangulations of the cyclic polytope C(n,n-4), Discrete Comput. Geom., 27 (2002), 29-48.
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Column k=2 of A229079.

Cf. A016789, A098156, A130129, A288834.

seq((3*n-2)*2^(n-3),n=2..30); # Emeric Deutsch, Jul 23 2006

Array[(3 # - 2)*2^(# - 3) &, 28, 2] (* or *)
Drop[CoefficientList[Series[x^2*(2 - x)/(1 - 2 x)^2, {x, 0, 29}], x], 2] (* Michael De Vlieger, Jun 30 2018 *)

a(n) = { (3*n - 2)*2^(n - 3) } /* Harry J. Smith, Feb 11 2010 */

G.f.: x^2*(2-x)/(1-2x)^2. - Emeric Deutsch, Jul 23 2006
a(n) = 2*a(n-1) +3*2^(n-3). - Vincenzo Librandi, Mar 20 2011
a(n+1) - a(n) = A098156(n). - R. J. Mathar, Apr 25 2013
From Paul Curtz, Jun 29 2018: (Start)
a(n) = A130129(n-2) - A130129(n-3) for n >= 2.
Binomial transform of A016789.
Inverse binomial transform of A288834.
Also the main diagonal of the difference table of m -> (-1)^m*(m+2).
    2,  -3,   4,  -5, ...
   -5,   7,  -9,  11, ...
   12, -16,  20, -24, ...
  -28,  36, -44,  52, ... . (End)

A074878 Row sums of triangle in A074829.

Original entry on oeis.org

1, 2, 6, 14, 32, 70, 150, 316, 658, 1358, 2784, 5678, 11534, 23356, 47178, 95110, 191440, 384854, 772902, 1550972, 3110306, 6234142, 12490176, 25015774, 50088862, 100270460, 200690970, 401624726, 803642288, 1607920198, 3216868854, 6435401788, 12873496114, 25751348846

Offset: 1

Joseph L. Pe, Sep 30 2002

An elephant sequence, see A175654. For the corner squares 16 A[5] vectors, with decimal values between 43 and 424, lead to this sequence. For the central square these vectors lead to the companion sequence A175657. - Johannes W. Meijer, Aug 15 2010

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-1,-2).

Cf. A000045.

List([1..40], n-> 3*2^(n-1) - 2*Fibonacci(n+1)); # G. C. Greubel, Jul 12 2019

[3*2^(n-1) - 2*Fibonacci(n+1): n in [1..40]]; // G. C. Greubel, Jul 12 2019

Table[3*2^(n-1) - 2*Fibonacci[n+1], {n, 1, 40}] (* G. C. Greubel, Jul 12 2019 *)

vector(40, n, 3*2^(n-1) -2*fibonacci(n+1)) \\ G. C. Greubel, Jul 12 2019

[3*2^(n-1) - 2*fibonacci(n+1) for n in (1..40)] # G. C. Greubel, Jul 12 2019

From Philippe Deléham, Sep 20 2006: (Start)
a(1)=1, a(2)=2, a(3)=6, a(n) = 3*a(n-1) - a(n-2) - 2*a(n-3) for n>3.
a(n) = 3*2^(n-1) - 2*F(n+1), F(n)=A000045(n).
G.f.: x*(1-x+x^2)/(1-3*x+x^2+2*x^3). (End)
a(1)=1, a(n) = 2*(a(n-1) + F(n-2)) where the Fibonacci number F(n-2) = A000045(n-2). - Anton Vrba (antonvrba(AT)yahoo.com), Feb 06 2007
a(n) = 3*2^n - 2*F(n+2), with offset 0 and F(n)=A000045(n). - Johannes W. Meijer, Aug 15 2010

More terms from Philippe Deléham, Sep 20 2006

Terms a(23) onward added by G. C. Greubel, Jul 12 2019

A172481 a(n) = (3n2^n+2^(n+4)+2*(-1)^n)/18.

Original entry on oeis.org

1, 2, 5, 11, 25, 55, 121, 263, 569, 1223, 2617, 5575, 11833, 25031, 52793, 111047, 233017, 487879, 1019449, 2126279, 4427321, 9204167, 19107385, 39612871, 82021945, 169636295, 350457401, 723284423, 1491308089, 3072094663, 6323146297, 13004206535, 26724240953

Offset: 0

Paul Curtz, Feb 04 2010

The binomial transform is in A126184.
An elephant sequence, see A175654 and A175655. There are 24 A[5] vectors, with decimal values between 7 and 448, that lead for the corner squares to this sequence. Its companion sequence for the central square is A175656. Furthermore there are 36 A[5] vectors, with decimal values between 15 and 480, that lead for the central square to four times this sequence for n >= -1. Its companion sequence for the corner squares is A059570. - Johannes W. Meijer, Aug 15 2010
a(n) is also the number of runs of weakly increasing parts in all compositions of n+1. a(2) = 5: (111), (12), (2)(1), (3). - Alois P. Heinz, Apr 30 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,0,-4)

Cf. A059570, A151529, A127984.

[(3*n*2^n+2^(n+4)+2*(-1)^n)/18: n in [0..40]]; // Vincenzo Librandi, Aug 04 2011

Table[(3n 2^n+2^(n+4)+2(-1)^n)/18,{n,0,40}]  (* or *)
CoefficientList[Series[(1-x-x^2)/((1+x)(1-2x)^2), {x,0,40}], x]  (* Harvey P. Dale, Mar 28 2011 *)

a(n)=(3*n*2^n+2^(n+4)+2*(-1)^n)/18 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: (1-x-x^2)/((1+x)*(1-2*x)^2).
a(n) = A001045(n-1)+2*a(n-1), n>0.
a(n)+A139790(n) = 2^(n+1) = A000079(n+1).
a(n) = A139790(n)+A140960(n).
a(n) = A001045(n)+(-1)^n*A084219(n).
a(n) = A127984(n) + 2^(n-1).  Application:  Problem 11623, AMM 119 (2012) 161. - Stephen J. Herschkorn, Feb 11 2012

Definition replaced by explicit formula by R. J. Mathar, Feb 11 2010

A134401 Row sums of triangle A134400.

Original entry on oeis.org

1, 2, 8, 24, 64, 160, 384, 896, 2048, 4608, 10240, 22528, 49152, 106496, 229376, 491520, 1048576, 2228224, 4718592, 9961472, 20971520, 44040192, 92274688, 192937984, 402653184, 838860800, 1744830464, 3623878656, 7516192768

Offset: 0

Gary W. Adamson, Oct 23 2007

Essentially the same sequence as A036289.
An elephant sequence, see A175654. For the corner squares four A[5] vectors, with decimal values 187, 190, 250 and 442, lead to this sequence. For the central square these vectors lead to the companion sequence 2*A001792, for n >= 1 and a(0)=1. - Johannes W. Meijer, Aug 15 2010
Number of vertices on a partially truncated n-cube (column 1 of A271316). - Vincent J. Matsko, Apr 07 2016

			a(3) = 24 = sum of row 3 terms of triangle A134400: (3 + 9 + 9 + 3).
a(3) = 24 = (1, 3, 3, 1) dot (1, 1, 5, 5) = (1 + 3 + 15 + 5).

Muniru A Asiru, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Cf. A001792, A036289, A097064, A134400, A175654, A271316.

a:=Concatenation([1],List([1..30],n->n*2^n)); # Muniru A Asiru, Oct 28 2018

1,seq(n*2^n,n=1..30); # Muniru A Asiru, Oct 28 2018

F = Function[x, x*2^x];F[Range[1, 10]] (* Eugeny Yakimovitch (Eugeny.Yakimovitch(AT)gmail.com), Jan 08 2008 *)
{1}~Join~Table[n 2^n, {n, 28}] (* or *) Total /@ Join[{{1}}, Table[n Binomial[n, k], {n, 28}, {k, 0, n}]] (* Michael De Vlieger, Apr 07 2016 *)

x='x+O('x^99); Vec((1-2*x+4*x^2)/(1-2*x)^2) \\ Altug Alkan, Apr 07 2016

Binomial transform of repeats of (4n+1): [1, 1, 5, 5, 9, 9, 13, 13, ...].
a(n) = n*2^n, n > 1. - Eugeny Yakimovitch (Eugeny.Yakimovitch(AT)gmail.com), Jan 08 2008
From Colin Barker, Jul 29 2012: (Start)
a(n) = 4*a(n-1) - 4*a(n-2) for n > 2.
G.f.: (1 - 2*x + 4*x^2)/(1-2*x)^2. (End)
E.g.f.: 1-E(0) where E(k)=1 - (k+1)/(1 - 2*x/(2*x - (k+1)^2/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Dec 07 2012
a(n) = A097064(n+1) for n >= 1. - Georg Fischer, Oct 28 2018
E.g.f.: 1 + 2*exp(2*x)*x. - Stefano Spezia, Feb 12 2023

More terms from Johannes W. Meijer, Aug 15 2010

A097813 a(n) = 32^n - 2n - 2.

Original entry on oeis.org

1, 2, 6, 16, 38, 84, 178, 368, 750, 1516, 3050, 6120, 12262, 24548, 49122, 98272, 196574, 393180, 786394, 1572824, 3145686, 6291412, 12582866, 25165776, 50331598, 100663244, 201326538, 402653128, 805306310, 1610612676, 3221225410, 6442450880, 12884901822, 25769803708

Offset: 0

Paul Barry, Aug 25 2004

An elephant sequence, see A175654. For the corner squares four A[5] vectors, with decimal values 58, 154, 178 and 184, lead to this sequence. For the central square these vectors lead to the companion sequence A033484. - Johannes W. Meijer, Aug 15 2010

a(n) is also the number of order-preserving partial isometries of an n-chain, i.e., the row sums of A183153 and A183154. - Abdullahi Umar, Dec 28 2010

Harvey P. Dale, Table of n, a(n) for n = 0..1000
F. Al-Kharousi, R. Kehinde and A. Umar, Combinatorial results for certain semigroups of partial isometries of a finite chain, The Australasian Journal of Combinatorics, Volume 58 (3) (2014), 363-375.
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A000295, A033484, A079583, A175654, A183153, A183154.

[3*2^n -2*(n+1): n in [0..40]]; // G. C. Greubel, Dec 30 2021

Table[3 2^n-2n-2,{n,0,40}] (* or *) LinearRecurrence[{4,-5,2},{1,2,6},40] (* Harvey P. Dale, Oct 25 2011 *)

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

[3*2^n -2*(n+1) for n in (0..40)] # G. C. Greubel, Dec 30 2021

G.f.: (1 - 2*x + 3*x^2)/((1-x)^2*(1-2*x)).
a(n) = 2*a(n-1) + 2*n - 2, for n>0, with a(0)=1.
a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3).
From G. C. Greubel, Dec 30 2021: (Start)
a(n) = 2^n + 2*A000295(n).
E.g.f.: 3*exp(2*x) - 2*(1 + x)*exp(x). (End)

cp's OEIS Frontend

A095121 Expansion of (1-x+2x^2)/((1-x)*(1-2x)).

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A175655 Eight bishops and one elephant on a 3 X 3 chessboard. G.f.: (1+x-5*x^2)/(1-3*x-x^2+6*x^3).

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A027934 a(0)=0, a(1)=1, a(2)=2; for n > 2, a(n) = 3*a(n-1) - a(n-2) - 2*a(n-3).

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A006138 a(n) = a(n-1) + 3*a(n-2).

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A123203 a(n) = 2^(n+1) - 3*n.

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A066373 a(n) = (3*n-2)*2^(n-3).

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A175655 Eight bishops and one elephant on a 3 X 3 chessboard. G.f.: (1+x-5x^2)/(1-3x-x^2+6*x^3).

A027934 a(0)=0, a(1)=1, a(2)=2; for n > 2, a(n) = 3a(n-1) - a(n-2) - 2a(n-3).

A066373 a(n) = (3n-2)2^(n-3).

A172481 a(n) = (3n2^n+2^(n+4)+2*(-1)^n)/18.

A097813 a(n) = 32^n - 2n - 2.