A175655 -id:A175655 - cp's OEIS frontend

A045891 First differences of A045623.

1, 1, 3, 7, 16, 36, 80, 176, 384, 832, 1792, 3840, 8192, 17408, 36864, 77824, 163840, 344064, 720896, 1507328, 3145728, 6553600, 13631488, 28311552, 58720256, 121634816, 251658240, 520093696, 1073741824, 2214592512, 4563402752

Offset: 0

Wolfdieter Lang

Let M_n be the n X n matrix m_(i,j) = 3 + abs(i-j), then det(M_n) =(-1)^(n+1)*a(n+1). - Benoit Cloitre, May 28 2002
If X_1, X_2, ..., X_n are 2-blocks of a (2n+3)-set X then, for n>=1, a(n+2) is the number of (n+1)-subsets of X intersecting each X_i, (i=1..n). - Milan Janjic, Nov 18 2007
Equals row sums of triangle A152194. - Gary W. Adamson, Nov 28 2008
An elephant sequence, see A175655. For the central square 16 A[5] vectors, with decimal values between 19 and 400, lead to this sequence (without the first leading 1). For the corner squares these vectors lead to the companion sequence A045623. - Johannes W. Meijer, Aug 15 2010
a(n) is the total number of runs of 1 in the compositions of n+1. For example, a(3) = A045623(3) - A045623(2) = 12 - 5 = 7 runs of only 1 in the compositions of 4, enumerated "()" as follows: 3,(1); (1),3; 2,(1,1);(1),2,(1); (1,1),2; (1,1,1,1). More generally, the total number of runs of only part k in the compositions of n+k is A045623(n) - A045623(n-k). - Gregory L. Simay, May 02 2017
This is essentially the p-INVERT of (1,1,1,1,1,...) for p(S) = 1 - S - S^2 + S^3; see A291000.  - Clark Kimberling, Aug 24 2017

			G.f. = 1 + x + 3*x^2 + 7*x^3 + 16*x^4 + 36*x^5 + 80*x^6 + ... - _Michael Somos_, Mar 26 2022

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
Frank Ellermann, Illustration of binomial transforms
Milan Janjic, Two Enumerative Functions
Milan Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
Milan Janjic and B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
Thomas Selig and Haoyue Zhu, New combinatorial perspectives on MVP parking functions and their outcome map, arXiv:2309.11788 [math.CO], 2023. See p. 29.
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Cf. A027656, A034007, A045623, A152194, A175655.

[1,1] cat [(n+4)*2^(n-3): n in [2..40]]; // G. C. Greubel, Sep 27 2022

Join[{1,1,a=3,b=7},Table[c=4*b-4*a;a=b;b=c,{n,100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 15 2011 *)
Table[ If[n<2, 1, 2^(n-3)*(n+4)], {n, 0, 30}] (* Jean-François Alcover, Sep 12 2012 *)
LinearRecurrence[{4,-4},{1,1,3,7},40] (* Harvey P. Dale, May 03 2019 *)

v=[1,1,3,7];for(i=1,99,v=concat(v,4*(v[#v]-v[#v-1])));v \\ Charles R Greathouse IV, Jun 01 2011

[1,1]+[(n+4)*2^(n-3) for n in range(2,40)] # G. C. Greubel, Sep 27 2022

a(n) = Sum_{k=0..n-2} (k+3)*binomial(n-2,k) for n >= 2. - N. J. A. Sloane, Jan 30 2008
a(n) = (n+4)*2^(n-3), n >= 2, with a(0) = a(1) = 1.
G.f.: (1-x)^3/(1-2*x)^2.
Equals binomial transform of A027656.
Starting 1, 3, 7, 16, ... this is ((n+5)*2^n - 0^n)/4, the binomial transform of (1, 2, 2, 3, 3, ...). - Paul Barry, May 20 2003
From Paul Barry, Nov 29 2004: (Start)
a(n) = ((n+4)*2^(n-1) + 3*C(0, n) - C(1, n))/4;
a(n) = Sum_{k=0..floor(n/2)} C(n, 2*k)*(k+1). (End)
a(n) = A045623(n-1) + 2^(n-2) = A034007(n+1) - 2^(n-2) for n>=2. - Philippe Deléham, Apr 20 2009
G.f.: 1 + Q(0)*x/(1-x)^2, where Q(k)= 1 + (k+1)*x/(1 - x - x*(1-x)/(x + (k+1)*(1-x)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Apr 25 2013
a(n) = Sum_{k=0..n} (k+1)*C(n-2,n-k). Peter Luschny, Apr 20 2015
From Amiram Eldar, Jan 13 2021: (Start)
Sum_{n>=0} 1/a(n) = 128*log(2) - 1292/15.
Sum_{n>=0} (-1)^n/a(n) = 782/15 - 128*log(3/2). (End)
E.g.f.: (2 - x + exp(2*x)*(2 + x))/4. - Stefano Spezia, Mar 26 2022

A105476 Number of compositions of n when each even part can be of two kinds.

Original entry on oeis.org

1, 1, 3, 6, 15, 33, 78, 177, 411, 942, 2175, 5001, 11526, 26529, 61107, 140694, 324015, 746097, 1718142, 3956433, 9110859, 20980158, 48312735, 111253209, 256191414, 589951041, 1358525283, 3128378406, 7203954255, 16589089473, 38200952238, 87968220657

Offset: 0

Emeric Deutsch, Apr 09 2005

Row sums of A105475.
Starting (1, 3, 6, 15, ...) = sum of (n-1)-th row terms of triangle A140168. - Gary W. Adamson, May 10 2008
a(n) is also the number of compositions of n using 1's and 2's such that each run of like numbers can be grouped arbitrarily. For example, a(4) = 15 because 4 = (1)+(1)+(1)+(1) = (1+1)+(1)+(1) = (1)+(1+1)+(1) = (1)+(1)+(1+1) = (1+1)+(1+1) = (1+1+1)+(1) = (1)+(1+1+1) = (1+1+1+1) = (2)+(1)+(1) = (1)+(2)+(1) = (1)+(1)+(2) = (2)+(1+1) = (1+1)+(2) = (2)+(2) = (2+2). - Martin J. Erickson (erickson(AT)truman.edu), Dec 09 2008
An elephant sequence, see A175655. For the central square four A[5] vectors, with decimal values 69, 261, 321 and 324, lead to this sequence (without the first leading 1). For the corner squares these vectors lead to the companion sequence A006138. - Johannes W. Meijer, Aug 15 2010
Inverse INVERT transform of the left shifted sequence gives A000034.
Eigensequence of the triangle
  1,
  2, 1,
  1, 2, 1,
  2, 1, 2, 1,
  1, 2, 1, 2, 1,
  2, 1, 2, 1, 2, 1,
  1, 2, 1, 2, 1, 2, 1,
  2, 1, 2, 1, 2, 1, 2, 1 ... - Paul Barry, Feb 10 2011
Pisano period lengths: 1, 3, 1, 6, 24, 3, 24, 6, 1, 24, 120, 6, 156, 24, 24, 12, 16, 3, 90, 24, ... - R. J. Mathar, Aug 10 2012

			a(3)=6 because we have (3),(1,2),(1,2'),(2,1),(2',1) and (1,1,1).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Sean A. Irvine, Walks on Graphs.
Silvana Ramaj, New Results on Cyclic Compositions and Multicompositions, Master's Thesis, Georgia Southern Univ., 2021. See p. 33.
Index entries for linear recurrences with constant coefficients, signature (1,3).

Cf. A006130, A105475, A105963, A274977.

a:=[1,3];; for n in [3..40] do a[n]:=a[n-1]+3*a[n-2]; od; Concatenation([1], a); # G. C. Greubel, Jan 15 2020

I:=[1,1,3]; [n le 3 select I[n] else Self(n-1)+3*Self(n-2): n in [1..35]]; // Vincenzo Librandi, Jul 21 2013

R:=PowerSeriesRing(Integers(), 33); Coefficients(R!( 1/(1-(x/(1-x))-x^2/(1-x^2)))); // Marius A. Burtea, Jan 15 2020

G:=(1-z^2)/(1-z-3*z^2): Gser:=series(G,z=0,35): 1,seq(coeff(Gser,z^n),n=1..33);

CoefficientList[Series[(1-x^2)/(1-x-3x^2), {x,0,35}], x] (* or *) Join[{1}, LinearRecurrence[{1, 3}, {1, 3}, 50]] (* Vladimir Joseph Stephan Orlovsky, Jul 17 2011; typo fixed by Vincenzo Librandi, Jul 21 2013 *)
Table[Round[Sqrt[3]^(n-3)*(2*Sqrt[3]*Fibonacci[n+1, 1/Sqrt[3]] +Fibonacci[n, 1/Sqrt[3]])], {n, 0, 40}] (* G. C. Greubel, Jan 15 2020 *)

Vec((1-x^2)/(1-x-3*x^2)+O(x^40)) \\ Charles R Greathouse IV, Jun 13 2013

def A105476_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x^2)/(1-x-3*x^2) ).list()
A105476_list(40) # G. C. Greubel, Jan 15 2020

G.f.: (1-x^2) / (1-x-3*x^2).
a(n) = a(n-1) + 3*a(n-2) for n>=3.
a(n) = 3*A006138(n-2), n>=2.
a(n) = ((2+sqrt(13))*(1+sqrt(13))^n - (2-sqrt(13))*(1-sqrt(13))^n)/(3*2^n*sqrt(13)) for n>0. - Bruno Berselli, May 24 2011
G.f.: 1/(1 - Sum_{k>=1} x^k*(1+x^k) ). - Joerg Arndt, Mar 09 2014
G.f.: 1/(1 - (x/(1-x)) - x^2/(1-x^2)) = 1/(1 - (x+2*x^2+x^3+2*x^4+x^5+2*x^6+...) ); in general 1/(1 - Sum_{j>=1} m(j)*x^j ) is the g.f. for compositions with m(k) sorts of part k. - Joerg Arndt, Feb 16 2015
a(n) = 3^((n-1)/2)*( 2*sqrt(3)*Fibonacci(n, 1/sqrt(3)) + Fibonacci(n, 1/sqrt(3)) ). - G. C. Greubel, Jan 15 2020
E.g.f.: 1/3 + (2/39)*exp(x/2)*(13*cosh((sqrt(13)*x)/2) + 2*sqrt(13)*sinh((sqrt(13)*x)/2)). - Stefano Spezia, Jan 15 2020

A175654 Eight bishops and one elephant on a 3 X 3 chessboard. G.f.: (1 - x - x^2)/(1 - 3x - x^2 + 6x^3).

Original entry on oeis.org

1, 2, 6, 14, 36, 86, 210, 500, 1194, 2822, 6660, 15638, 36642, 85604, 199626, 464630, 1079892, 2506550, 5811762, 13462484, 31159914, 72071654, 166599972, 384912086, 888906306, 2052031172, 4735527306, 10925175254, 25198866036, 58108609526, 133973643090

Offset: 0

Johannes W. Meijer, Aug 06 2010; edited Jun 21 2013

a(n) represents the number of n-move routes of a fairy chess piece starting in a given corner square (m = 1, 3, 7 or 9) on a 3 X 3 chessboard. This fairy chess piece behaves like a bishop on the eight side and corner squares but on the center square the bishop flies into a rage and turns into a raging elephant.
In chaturanga, the old Indian version of chess, one of the pieces was called gaja, elephant in Sanskrit. The Arabs called the game shatranj and the elephant became el fil in Arabic. In Spain chess became chess as we know it today but surprisingly in Spanish a bishop isn't a Christian bishop but a Moorish elephant and it still goes by its original name of el alfil.
On a 3 X 3 chessboard there are 2^9 = 512 ways for an elephant to fly into a rage on the central square (off the center the piece behaves like a normal bishop). The elephant is represented by the A[5] vector in the fifth row of the adjacency matrix A, see the Maple program and A180140. For the corner squares the 512 elephants lead to 46 different elephant sequences, see the overview of elephant sequences and the crossreferences.
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 central square to A175655.

Gary Chartrand, Introductory Graph Theory, pp. 217-221, 1984.
David Hooper and Kenneth Whyld, The Oxford Companion to Chess, pp. 74, 366, 1992.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Viswanathan Anand, The Indian Defense, Time, Jun 19 2008.
Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
Johannes W. Meijer, The elephant sequences.
Wikipedia, War Elephant.
Index entries for linear recurrences with constant coefficients, signature (3,1,-6).

Cf. Elephant sequences corner squares [decimal value A[5]]: A040000 [0], A000027 [16], A000045 [1], A094373 [2], A000079 [3], A083329 [42], A027934 [11], A172481 [7], A006138 [69], A000325 [26], A045623 [19], A000129 [21], A095121 [170], A074878 [43], A059570 [15], A175654 [71, this sequence], A026597 [325], A097813 [58], A057711 [27], 2*A094723 [23; n>=-1], A002605 [85], A175660 [171], A123203 [186], A066373 [59], A015518 [341], A134401 [187], A093833 [343].

Cf. A000244, A006130, A006138, A075188, A175655, A180140.

[n le 3 select Factorial(n) else 3*Self(n-1) +Self(n-2) -6*Self(n-3): n in [1..41]]; // G. C. Greubel, Dec 08 2021

nmax:=28; m:=1; A[1]:=[0,0,0,0,1,0,0,0,1]: A[2]:=[0,0,0,1,0,1,0,0,0]: A[3]:=[0,0,0,0,1,0,1,0,0]: A[4]:=[0,1,0,0,0,0,0,1,0]: A[5]:=[0,0,1,0,0,0,1,1,1]: A[6]:=[0,1,0,0,0,0,0,1,0]: A[7]:=[0,0,1,0,1,0,0,0,0]: A[8]:=[0,0,0,1,0,1,0,0,0]: A[9]:=[1,0,0,0,1,0,0,0,0]: A:=Matrix([A[1], A[2], A[3], A[4], A[5], A[6], A[7], A[8], A[9]]): 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);

LinearRecurrence[{3,1,-6}, {1,2,6}, 80] (* Vladimir Joseph Stephan Orlovsky, Feb 21 2012 *)

a(n)=([0,1,0; 0,0,1; -6,1,3]^n*[1;2;6])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

[( (1-x-x^2)/((1-2*x)*(1-x-3*x^2)) ).series(x,n+1).list()[n] for n in (0..40)] # G. C. Greubel, Dec 08 2021

G.f.: (1 - x - 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)=2 and a(2)=6.
a(n) = ((6+10*A)*A^(-n-1) + (6+10*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)).
a(n) = b(n) - b(n-1) - b(n-2), where b(n) = Sum_{k=1..n} Sum_{j=0..k} binomial(j,n-3*k+2*j)*(-6)^(k-j)*binomial(k,j)*3^(3*k-n-j), n>0, b(0)=1, with a(0) = b(0), a(1) = b(1) - b(0). - Vladimir Kruchinin, Aug 20 2010
a(n) = 2*A006138(n) - 2^n = 2*(A006130(n) + A006130(n-1)) - 2^n. - G. C. Greubel, Dec 08 2021
E.g.f.: 2*exp(x/2)*(13*cosh(sqrt(13)*x/2) + 3*sqrt(13)*sinh(sqrt(13)*x/2))/13 - cosh(2*x) - sinh(2*x). - Stefano Spezia, Feb 12 2023

A099036 a(n) = 2^n - Fibonacci(n).

Original entry on oeis.org

1, 1, 3, 6, 13, 27, 56, 115, 235, 478, 969, 1959, 3952, 7959, 16007, 32158, 64549, 129475, 259560, 520107, 1041811, 2086206, 4176593, 8359951, 16730848, 33479407, 66987471, 134021310, 268117645, 536356683, 1072909784, 2146137379, 4292788987, 8586410014

Offset: 0

Paul Barry, Sep 23 2004

Binomial transform of (-1)^n*A000045(n) + 1 = (-1)^n*A008346(n).
Number of compositions of n+1 that contain 1 as a part. - Vladeta Jovovic, Sep 26 2004
Generated from iterates of M * [1,1,1,...], where M = a tridiagonal matrix with [0,1,1,1,...] as the main diagonal, [1,1,1,...] as the superdiagonal and [1,0,0,0,...] as the subdiagonal. - Gary W. Adamson, Jan 05 2009
Starting with offset 1, generated from iterates of M * [1,1,1,...], M*ANS -> M*ANS,...; where M = = a tridiagonal matrix with (0,1,1,1,...) in the main diagonal, (1,1,1,...) in the superdiagonal and (1,0,0,0,...) in the subdiagonal. - Gary W. Adamson, Jan 04 2009
An elephant sequence, see A175655. For the central square 24 A[5] vectors, with decimal values between 11 and 416, lead to this sequence (without the first leading 1). For the corner squares these vectors lead to the companion sequence A027934 (without the leading 0). - Johannes W. Meijer, Aug 15 2010
Number of fixed points in all compositions of n+1. - Alois P. Heinz, Jun 18 2020

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..200 from Vincenzo Librandi)
M. Archibald, A. Blecher, and A. Knopfmacher, Fixed Points in Compositions and Words, J. Int. Seq., Vol. 23 (2020), Article 20.11.1.
Index entries for linear recurrences with constant coefficients, signature (3,-1,-2).

Cf. A000045, A000079, A003056, A008346, A027934, A101220, A117591, A175655, A212323, A238350.

a099036 n = a099036_list !! n
a099036_list = zipWith (-) a000079_list a000045_list
-- Reinhard Zumkeller, Aug 15 2013

[2^n-Fibonacci(n): n in [0..35]]; // Vincenzo Librandi, May 03 2011

Table[2^n-Fibonacci[n],{n,0,100}] (* Vladimir Joseph Stephan Orlovsky, May 02 2011 *)

a(n)=2^n-fibonacci(n) \\ Charles R Greathouse IV, Sep 24 2015

def A099036(n): return 2**n -fibonacci(n) # G. C. Greubel, Jun 05 2025

G.f.: (1 - x)^2/((1 - 2*x)*(1 - x - x^2)).
a(n) = 3*a(n-1) - a(n-2) - 2*a(n-3).
a(n) = A101220(1,2,n+1) - A101220(1,2,n). - Ross La Haye, Aug 05 2005
a(n) = A000079(n+1) - A117591(n) = A117591(n) - 2 * A000045(n). - Reinhard Zumkeller, Aug 15 2013
a(n) = Sum_{t_1+2*t_2+...+n*t_n = n} multinomial(1+t_1+t_2+...+t_n, 1+t_1, t_2, ..., t_n). - Mircea Merca, Oct 09 2013
a(n) = Sum_{k=0..A003056(n+1)} k * A238350(n+1,k). - Alois P. Heinz, Jun 18 2020
E.g.f.: cosh(2*x) + sinh(2*x) - 2*exp(x/2)*sinh(sqrt(5)*x/2)/sqrt(5). - Stefano Spezia, Jan 31 2023

More terms from Ross La Haye, Aug 05 2005

A087447 a(0) = a(1) = 1; for n > 1, a(n) = (n+2)*2^(n-2).

Original entry on oeis.org

1, 1, 4, 10, 24, 56, 128, 288, 640, 1408, 3072, 6656, 14336, 30720, 65536, 139264, 294912, 622592, 1310720, 2752512, 5767168, 12058624, 25165824, 52428800, 109051904, 226492416, 469762048, 973078528, 2013265920, 4160749568

Offset: 0

Paul Barry, Sep 05 2003

Binomial transform of A005408 (with interpolated zeros). Binomial transform is A087448. a(n+2) = 2*A045623(n+1); a(n+1) = A001792(n) + (0^n - (-2)^n)/2. The sequence 1,4,10,... given by 2^n*(n+3)/2 - 0^n/2 is the binomial transform of 1,3,3,5,5,...
Equals real part of binomial transform of [1, 2*i, 3, 4*i, 5, 6*i, ...]; i=sqrt(-1). - Gary W. Adamson, Sep 21 2008
An elephant sequence, see A175655. For the central square 24 A[5] vectors, with decimal values between 27 and 432, lead to this sequence (without the first leading 1). For the corner squares these vectors lead to the companion sequence A057711 (without the leading 0). - Johannes W. Meijer, Aug 15 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..3000
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Essentially same as A079859.

Cf. A001792, A005408, A045623, A057711, A087448, A175655.

Join[{1, 1}, Table[(n + 2) 2^(n - 2), {n, 2, 30}]]  (* Harvey P. Dale, Feb 22 2011 *)

def A087447(n): return n+2<1 else 1 # Chai Wah Wu, Oct 03 2024

a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2k)*(2k+1). - Paul Barry, Nov 29 2004
From Colin Barker, Mar 23 2012: (Start)
G.f.: (1-x)*(1-2*x+2*x^2)/(1-2*x)^2.
a(n) = 4*a(n-1) - 4*a(n-2) for n > 3. (End)
E.g.f.: (1 - x + exp(2*x)*(1 + x))/2. - Stefano Spezia, Jan 31 2023

Definition corrected (by a factor of 2) by R. J. Mathar, Feb 21 2009

A168604 a(n) = 2^(n-2) - 1.

Original entry on oeis.org

1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 4095, 8191, 16383, 32767, 65535, 131071, 262143, 524287, 1048575, 2097151, 4194303, 8388607, 16777215, 33554431, 67108863, 134217727, 268435455, 536870911, 1073741823, 2147483647, 4294967295, 8589934591

Offset: 3

Martin Griffiths, Dec 01 2009

Number of ways of partitioning the multiset {1,1,1,2,3,...,n-2} into exactly two nonempty parts.

An elephant sequence, see A175655. For the central square six A[5] vectors, with decimal values between 26 and 176, lead to this sequence. For the corner squares these vectors lead to the companion sequence A000325 (without the first leading 1). - Johannes W. Meijer, Aug 15 2010

			The partitions of {1,1,1,2,3} into exactly two nonempty parts are {{1},{1,1,2,3}}, {{2},{1,1,1,3}}, {{3},{1,1,1,2}}, {{1,1},{1,2,3}}, {{1,2},{1,1,3}}, {{1,3},{1,1,2}} and {{2,3},{1,1,1}}.

Vincenzo Librandi, Table of n, a(n) for n = 3..1000
M. Griffiths, I. Mezo, A generalization of Stirling Numbers of the Second Kind via a special multiset, JIS 13 (2010) #10.2.5
Index entries for linear recurrences with constant coefficients, signature (3,-2).

The number of ways of partitioning the multiset {1, 1, 1, 2, 3, ..., n-1} into exactly three and four nonempty parts are given in A168605 and A168606, respectively.

[2^(n-2)-1 : n in [3..35]]; // Vincenzo Librandi, May 13 2011

f4[n_] := 2^(n - 2) - 1; Table[f4[n], {n, 3, 30}]
LinearRecurrence[{3,-2},{1,3},40] (* Harvey P. Dale, Oct 20 2013 *)

a(n)=2^(n-2)-1 \\ Charles R Greathouse IV, Oct 07 2015

E.g.f.: 2*exp(2*x)-exp(x).
a(n) = A000225(n-2).
G.f.: x^3/((1-x)*(1-2*x))
a(n) = A126646(n-3). - R. J. Mathar, Dec 11 2009
a(n) = 3*a(n-1) - 2*a(n-2). - Arkadiusz Wesolowski, Jun 14 2013
a(n) = A000918(n-2) + 1. - Miquel Cerda, Aug 09 2016

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

A098156 Interleave n+1 and 2n+1 and take binomial transform.

Original entry on oeis.org

1, 2, 5, 13, 32, 76, 176, 400, 896, 1984, 4352, 9472, 20480, 44032, 94208, 200704, 425984, 901120, 1900544, 3997696, 8388608, 17563648, 36700160, 76546048, 159383552, 331350016, 687865856, 1426063360, 2952790016, 6106906624

Offset: 0

Paul Barry, Aug 29 2004

Binomial transform of A029579.

An elephant sequence, see A175655. For the central square 16 A[5] vectors, with decimal values between 59 and 440, lead to this sequence (without a(1)). For the corner squares these vectors lead to the companion sequence A066373 (with a leading 1 added). - Johannes W. Meijer, Aug 15 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
David Anderson, E. S. Egge, M. Riehl, L. Ryan, R. Steinke, Y. Vaughan, Pattern Avoiding Linear Extensions of Rectangular Posets, arXiv:1605.06825 [math.CO], 2016.
Colin Defant, Proofs of Conjectures about Pattern-Avoiding Linear Extensions, arXiv:1905.02309 [math.CO], 2019.
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Concatenation([1,2], List([2..40], n-> 2^(n-3)*(3*n+4))); # G. C. Greubel, May 08 2019

[1,2] cat [2^(n-3)*(3*n+4): n in [2..40]]; // G. C. Greubel, May 08 2019

CoefficientList[Series[(1-2x+x^2+x^3)/(1-2x)^2, {x, 0, 40}], x] (* Vincenzo Librandi, Jul 21 2013 *)
LinearRecurrence[{4,-4},{1,2,5,13},50] (* Harvey P. Dale, Dec 03 2023 *)

{a(n) = if(n==0,1, if(n==1,2, 2^(n-3)*(3*n+4)))}; \\ G. C. Greubel, May 08 2019

[1,2]+[2^(n-3)*(3*n+4) for n in (2..40)] # G. C. Greubel, May 08 2019

G.f.: (1-2*x+x^2+x^3)/(1-2*x)^2.
a(n) = (2 * 0^n + Sum_{k=0..n} (-1)^(n-k)*k*binomial(n,k) + 2^(n+1) + 3*n*2^(n-1) )/4.
a(n) = Sum_{j=0..n} Sum_{k=0..n} binomial(n, 2*(k-j)).
a(n) = Sum_{k=0..n} Sum_{j=0..k} C(n, 2*j). - Paul Barry, Jan 13 2005
a(n) = 2^(n-3)*(3*n+4) for n>=2. - Philip B. Zhang, May 25 2016
E.g.f.: (2 + x + (2 + 3*x)*exp(2*x))/4. - Ilya Gutkovskiy, May 31 2016

A175656 Eight bishops and one elephant on a 3 X 3 chessboard. G.f.: (1-3x^2)/(1-3x+4*x^3).

Original entry on oeis.org

1, 3, 6, 14, 30, 66, 142, 306, 654, 1394, 2958, 6258, 13198, 27762, 58254, 121970, 254862, 531570, 1106830, 2301042, 4776846, 9903218, 20505486, 42409074, 87614350, 180821106, 372827022, 768023666, 1580786574, 3251051634

Offset: 0

Johannes W. Meijer, Aug 06 2010

The a(n) represent 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.

The sequence above corresponds to 24 A[5] vectors with decimal values 7, 13, 37, 67, 70, 73, 76, 97, 100, 133, 193, 196, 259, 262, 265, 268, 289, 292, 322, 328, 352, 385, 388 and 448. These vectors lead for the side squares to A000079 and for the corner squares to A172481.

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

Cf. A175655 (central square).

[((3*n+22)*2^n-4*(-1)^n)/18: n in [0..40]]; // Vincenzo Librandi, Aug 04 2011

with(LinearAlgebra): nmax:=29; m:=5; A[5]:= [0,0,0,0,0,0,1,1,1]: 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 - 3 x^2)/(1 - 3 x + 4 x^3), {x, 0, 29}], x] (* Michael De Vlieger, Nov 02 2018 *)
LinearRecurrence[{3,0,-4},{1,3,6},30] (* Harvey P. Dale, Aug 12 2020 *)

vector(40, n, n--; ((3*n+22)*2^n - 4*(-1)^n)/18) \\ G. C. Greubel, Nov 03 2018

G.f.: (1-3*x^2)/(1 - 3*x + 4*x^3).
a(n) = 3*a(n-1) - 4*a(n-3) with a(0)=1, a(1)=3 and a(2)=6.
a(n) = ((3*n+22)*2^n - 4*(-1)^n)/18.

A175657 Eight bishops and one elephant on a 3 X 3 chessboard: a(n) = 32^n - 2F(n+1), with F(n) = A000045(n).

Original entry on oeis.org

1, 4, 8, 18, 38, 80, 166, 342, 700, 1426, 2894, 5856, 11822, 23822, 47932, 96330, 193414, 388048, 778070, 1559334, 3123836, 6256034, 12525598, 25073088, 50181598, 100420510, 200933756, 402017562, 804277910, 1608948656, 3218532934

Offset: 0

Johannes W. Meijer, Aug 06 2010

The a(n) represent 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.

The sequence above corresponds to 16 A[5] vectors with decimal values 43, 46, 106, 139, 142, 163, 166, 169, 172, 202, 226, 232, 298, 394, 418 and 424. These vectors lead for the side squares to A000079 and for the corner squares to A074878 (a(n)=3*2^n-2*F(n+2)).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-1,-2).

Cf. A000045, A000079, A074878, A175654, A175655 (central square).

I:=[1,4,8]; [n le 3 select I[n] else 3*Self(n-1)-Self(n-2)-2*Self(n-3): n in [1..35]]; // Vincenzo Librandi, Jul 21 2013

with(LinearAlgebra): nmax:=30; m:=5; A[5]:= [0,0,0,1,0,1,0,1,1]: 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);

LinearRecurrence[{3,-1,-2},{1,4,8},40] (* Harvey P. Dale, Aug 12 2012 *)
CoefficientList[Series[(1 + x - 3 x^2) / (1 - 3 x + x^2 + 2 x^3), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 21 2013 *)

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

a(n) = 3*a(n-1)-a(n-2)-2*a(n-3) with a(0)=1, a(1)=4 and a(2)=8.

cp's OEIS Frontend

A045891 First differences of A045623.

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A105476 Number of compositions of n when each even part can be of two kinds.

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A175654 Eight bishops and one elephant on a 3 X 3 chessboard. G.f.: (1 - x - x^2)/(1 - 3*x - x^2 + 6*x^3).

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A099036 a(n) = 2^n - Fibonacci(n).

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A087447 a(0) = a(1) = 1; for n > 1, a(n) = (n+2)*2^(n-2).

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A168604 a(n) = 2^(n-2) - 1.

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A175654 Eight bishops and one elephant on a 3 X 3 chessboard. G.f.: (1 - x - x^2)/(1 - 3x - x^2 + 6x^3).

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

A175656 Eight bishops and one elephant on a 3 X 3 chessboard. G.f.: (1-3x^2)/(1-3x+4*x^3).

A175657 Eight bishops and one elephant on a 3 X 3 chessboard: a(n) = 32^n - 2F(n+1), with F(n) = A000045(n).