A111317 -id:A111317 - cp's OEIS frontend

A049347 Period 3: repeat [1, -1, 0].

1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0

Offset: 0

Wolfdieter Lang

G.f. 1/cyclotomic(3, x) (the third cyclotomic polynomial).
Self-convolution yields (-1)^n*A099254(n). - R. J. Mathar, Apr 06 2008
Hankel transform of A099324. - Paul Barry, Aug 10 2009
A057083(n) = p(-1)  where p(x) is the unique degree-n polynomial such that p(k) = a(k) for k = 0..n. - Michael Somos, Apr 29 2012
a(n) appears, together with b(n) = A099837(n+3) in the formula 2*exp(2*Pi*n*I/3) = b(n) + a(n)*sqrt(3)*I, n >= 0, with I = sqrt(-1). See A164116 for the case N=5. - Wolfdieter Lang, Feb 27 2014
The binomial transform is 1, 0, -1, -1, 0, 1, 1, 0, -1, -1.. (see A010891). The inverse binom. transform is 1, -2, 3, -3, 0, 9, -27, 54, -81.. (see A057682). - R. J. Mathar, Feb 25 2023

			G.f. = 1 - x + x^3 - x^4 + x^6 - x^7 + x^9 - x^10 + x^12 - x^13 + x^15 + ...

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 175.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
J.-P. Allouche and M. Mendès France, Stern-Brocot polynomials and power series, arXiv preprint arXiv:1202.0211 [math.NT], 2012.
Elena Barcucci, Antonio Bernini, Stefano Bilotta and Renzo Pinzani, Non-overlapping matrices, arXiv:1601.07723 [cs.DM], 2016.
George Beck and Karl Dilcher, A Matrix Related to Stern Polynomials and the Prouhet-Thue-Morse Sequence, arXiv:2106.10400 [math.CO], 2021.
Index entries for linear recurrences with constant coefficients, signature (-1,-1).
Index entries for sequences related to Chebyshev polynomials.
Index to sequences related to inverse of cyclotomic polynomials

Cf. A001006, A010892, A057078, A057083, A102283, A106510, A130716, A143818, A194770.

Alternating row sums of A049310 (Chebyshev-S). [Wolfdieter Lang, Nov 04 2011]

&cat[[1,-1,0]: n in [0..90]]; // Vincenzo Librandi, Apr 03 2014

A049347 := proc(n)
    op(modp(n,3)+1,[1,-1,0]) ;
end proc:
seq(A049347(n),n=0..100) ; # R. J. Mathar, Aug 06 2016

Flatten[Table[{1, -1, 0}, {27}]] (* Alonso del Arte, Feb 07 2013 *)
CoefficientList[Series[1/Cyclotomic[3, x], {x, 0, 100}], x] (* Vincenzo Librandi, Apr 03 2014 *)
LinearRecurrence[{-1, -1}, {1, -1}, 90] (* Ray Chandler, Sep 15 2015 *)
Table[DirichletCharacter[3, 2, n + 1], {n, 0, 29}] (* Steven Foster Clark, May 29 2019 *)
Table[Mod[n + 2, 3] - 1, {n, 0, 20}] (* Michael Somos, Sep 24 2019 *)
Table[ChebyshevU[n, -1/2], {n, 0, 20}] (* Eric W. Weisstein, Jan 09 2024 *)
ChebyshevU[Range[0, 20], -1/2] (* Eric W. Weisstein, Jan 09 2024 *)

A049347(n) := block(
        [1,-1,0][1+mod(n,3)]
)$ /* R. J. Mathar, Mar 19 2012 */

{a(n) = n++; kronecker( -3, n)} /* Michael Somos, Oct 03 2006 */

{a(n) = [1, -1, 0][n%3 + 1]} /* Michael Somos, Oct 15 2008 */

a(n)=(n+2)%3-1 /* Jaume Oliver Lafont, Mar 24 2009 */

def A049347():
    x, y = 1, -1
    while True:
        yield x
        x, y = y, -x - y
a = A049347(); [next(a) for i in range(40)] # Peter Luschny, Jul 11 2013

G.f.: 1/(1+x+x^2).
a(n) = +1 if n mod 3 = 0, a(n) = -1 if n mod 3 = 1, else 0.
a(n) = S(n, -1) = U(n, -1/2) (Chebyshev's U(n, x) polynomials.)
a(n) = 2*sqrt(3)*cos(2*Pi*n/3 + Pi/6)/3. - Paul Barry, Mar 15 2004
a(n) = Sum_{k >= 0} (-1)^(n-k)*C(n-k, k).
Given g.f. A(x), then B(x) = x * A(x) satisfies 0 = f(B(x), B(x^2)) where f(u, v) = u^2 - v + 2*u*v. - Michael Somos, Oct 03 2006
Euler transform of length 3 sequence [-1, 0, 1]. - Michael Somos, Oct 03 2006
a(n) = b(n+1) where b(n) is multiplicative with b(3^e) = 0^e, b(p^e) = 1 if p == 1 (mod 3), b(p^e) = (-1)^e if p == 2 (mod 3). - Michael Somos, Oct 03 2006
From Michael Somos, Oct 03 2006: (Start)
G.f.: (1 - x) /(1 - x^3).
a(n) = -a(1-n) = -a(n-1) - a(n-2) = a(n-3). (End)
From Michael Somos, Apr 29 2012: (Start)
G.f.: 1 / (1 + x / ( 1 - x / (1 + x))).
a(n) = (-1)^n * A010892(n).
a(n) * n! = A194770(n+1).
Revert transform of A001006. Convolution inverse of A130716. MOBIUS transform of A002324. EULER transform is A111317. BIN1 transform of itself. STIRLING transform is A143818(n+2). (End)
a(-n) = A057078(n). a(n) = A106510(n+1) unless n=0. - Michael Somos, Oct 15 2008
G.f. A(x) = 1/(1+x+x^2) = Q(0); Q(k) =  1- x/(1 - x^2/(x^2 - 1 + x/(x - 1 + x^2/(x^2 - 1/Q(k+1))))); (continued fraction 3 kind, 5-step ). - Sergei N. Gladkovskii, Jun 19 2012
a(n) = -1 + floor(67/333*10^(n+1)) mod 10. - Hieronymus Fischer, Jan 03 2013
a(n) = -1 + floor(19/26*3^(n+1)) mod 3. - Hieronymus Fischer, Jan 03 2013
a(n) = ceiling((n-1)/3) - ceiling(n/3) + floor(n/3) - floor((n-1)/3). - Wesley Ivan Hurt, Dec 06 2013
a(n) = n + 1 - 3*floor((n+2)/3). - Mircea Merca, Feb 04 2014
a(n) = A102283(n+1) for all n in Z. - Michael Somos, Sep 24 2019
E.g.f.: exp(-x/2)*(3*cos(sqrt(3)*x/2) - sqrt(3)*sin(sqrt(3)*x/2))/3. - Stefano Spezia, Oct 26 2022

Edited by Charles R Greathouse IV, Mar 23 2010

A005169 Number of fountains of n coins.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 9, 15, 26, 45, 78, 135, 234, 406, 704, 1222, 2120, 3679, 6385, 11081, 19232, 33379, 57933, 100550, 174519, 302903, 525734, 912493, 1583775, 2748893, 4771144, 8281088, 14373165, 24946955, 43299485, 75153286, 130440740, 226401112, 392955956, 682038999, 1183789679, 2054659669, 3566196321, 6189714276

Offset: 0

N. J. A. Sloane

A fountain is formed by starting with a row of coins, then stacking additional coins on top so that each new coin touches two in the previous row.
Also the number of Dyck paths for which the sum of the heights of the vertices that terminate an upstep (i.e., peaks and doublerises) is n. Example: a(4)=3 because we have UDUUDD, UUDDUD and UDUDUDUD. - Emeric Deutsch, Mar 22 2008
Also the number of ordered trees with path length n (follows from previous comment via a standard bijection). - Emeric Deutsch, Mar 22 2008
Probably first studied by Jim Propp (unpublished).
Number of compositions of n with c(1) = 1 and c(i+1) <= c(i) + 1. (Slide each row right 1/2 step relative to the row below, and count the columns.) - Franklin T. Adams-Watters, Nov 24 2009
With the additional requirement for weak unimodality one obtains A001524. - Joerg Arndt, Dec 09 2012

			An example of a fountain with 19 coins:
... O . O O
.. O O O O O O . O
. O O O O O O O O O
From _Peter Bala_, Dec 26 2012: (Start)
F(1/10) = Sum_{n >= 0} a(n)/10^n has the simple continued fraction expansion 1 + 1/(8 + 1/(1 + 1/(8 + 1/(1 + 1/(98 + 1/(1 + 1/(98 + 1/(1 + 1/(998 + 1/(1 + 1/(998 + 1/(1 + ...)))))))))))).
F(-1/10) = Sum_{n >= 0} (-1)^n*a(n)/10^n has the simple continued fraction expansion 1/(1 + 1/(9 + 1/(1 + 1/(9 + 1/(99 + 1/(1 + 1/(99 + 1/(999 + 1/(1 + 1/(999 + 1/(9999 + 1/(1 + ...)))))))))))).
(End)

S. R. Finch, Mathematical Constants, Cambridge, 2003, p. 381.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..4178 (first 501 terms from T. D. Noe)
Peter Bala, Some simple continued fraction expansions
P. Flajolet, Combinatorial aspects of continued fractions, Discrete Mathematics 32 (1980), pp. 125-161.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 331.
Atli Fannar Franklín, Pattern avoidance enumerated by inversions, arXiv:2410.07467 [math.CO], 2024. See pp. 2, 4.
Atli Fannar Franklín, Anders Claesson, Christian Bean, Henning Úlfarsson, and Jay Pantone, Restricted Permutations Enumerated by Inversions, arXiv:2406.16403 [cs.DM], 2024. See p. 2.
M. L. Glasser, V. Privman, N. M. Svrakic, Temperley's triangular lattice compact cluster model: exact solution in terms of the q series. J. Phys. A 20 (1987), no. 18, L1275-L1280.
H. W. Gould, R. K. Guy, and N. J. A. Sloane, Correspondence, 1987.
R. K. Guy, Letter to N. J. A. Sloane, Sep 25 1986.
R. K. Guy, Letter to N. J. A. Sloane, 1987
R. K. Guy, The strong law of small numbers, Amer. Math. Monthly 95 (1988), no. 8, 697-712.
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
R. K. Guy and N. J. A. Sloane, Correspondence, 1988.
Kival Ngaokrajang, Illustration for initial terms
A. M. Odlyzko and H. S. Wilf, The editor's corner: n coins in a fountain, Amer. Math. Monthly, 95 (1988), 840-843.
A. M. Odlyzko, Asymptotic enumeration methods, pp. 1063-1229 of R. L. Graham et al., eds., Handbook of Combinatorics, 1995; see Example 10.7 (pdf, ps)
Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction.

Cf. A001524, A192728, A192729, A192730, A111317, A143951, A285903, A226999 (inverse Euler transform), A291148 (convolution inverse).
First column of A168396. - Franklin T. Adams-Watters, Nov 24 2009
Diagonal of A185646.
Row sums of A047998. Column sums of A138158. - Emeric Deutsch, Mar 22 2008

a005169 0 = 1
a005169 n = a168396 n 1  -- Reinhard Zumkeller, Sep 13 2013; corrected by R. J. Mathar, Sep 16 2013

P[0]:=1: for n to 40 do P[n]:=sort(expand(t*(sum(P[j]*P[n-j-1]*t^(n-j-1),j= 0..n-1)))) end do: F:=sort(sum(P[k],k=0..40)): seq(coeff(F,t,j),j=0..36); # Emeric Deutsch, Mar 22 2008
# second Maple program:
A005169_G:= proc(x,NK); Digits:=250; Q2:=1;
        for k from NK by -1 to 0 do  Q1:=1-x^k/Q2; Q2:=Q1; od;
        Q3:=Q2; S:=1-Q3;
end:
series(A005169_G(x, 20), x, 21); # Sergei N. Gladkovskii, Dec 18 2011

m = 36; p[0] = 1; p[n_] := p[n] = Expand[t*Sum[p[j]*p[n-j-1]*t^(n-j-1), {j, 0, n-1}]]; f[t_] = Sum[p[k], {k, 0, m}]; CoefficientList[Series[f[t], {t, 0, m}], t] (* Jean-François Alcover, Jun 21 2011, after Emeric Deutsch *)
max = 43; Series[1-Fold[Function[1-x^#2/#1], 1, Range[max, 0, -1]], {x, 0, max}] // CoefficientList[#, x]& (* Jean-François Alcover, Sep 16 2014 *)
b[n_, i_] := b[n, i] = If[n==0, 1, Sum[b[n-j, j], {j, 1, Min[i+1, n]}]];
c[n_] :=  b[n, 0] - b[n-1, 0];
c /@ Range[0, 50] // Accumulate  (* Jean-François Alcover, Nov 14 2020, after Alois P. Heinz in A289080 *)

/* using the g.f. from p. L1278 of the Glasser, Privman, Svrakic paper */
N=30;  x='x+O('x^N);
P(k)=sum(n=0,N, (-1)^n*x^(n*(n+1+k))/prod(j=1,n,1-x^j));
G=1+x*P(1)/( (1-x)*P(1)-x^2*P(2) );
Vec(G) /* Joerg Arndt, Feb 10 2011 */

/* As a continued fraction: */
{a(n)=local(A=1+x,CF);CF=1+x;for(k=0,n,CF=1/(1-x^(n-k+1)*CF+x*O(x^n));A=CF);polcoeff(A,n)} /* Paul D. Hanna */

/* By the Rogers-Ramanujan continued fraction identity: */
{a(n)=local(A=1+x,P,Q);
P=sum(m=0,sqrtint(n),(-1)^m*x^(m*(m+1))/prod(k=1,m,1-x^k));
Q=sum(m=0,sqrtint(n),(-1)^m*x^(m^2)/prod(k=1,m,1-x^k));
A=P/(Q+x*O(x^n));polcoeff(A,n)}  /* Paul D. Hanna */

A005169(n) = f(n, 1), where f(n, p) = 0 if p > n, 1 if p = n, Sum(1 <= q <= p+1; f(n-p, q)) if p < n. f=A168396.
G.f.: F(t) = Sum_{k>=0} P[k], where P[0]=1, P[n] = t*Sum_{j= 0..n-1} P[j]*P[n-j-1]*t^(n-j-1) for n >= 1. - Emeric Deutsch, Mar 22 2008
G.f.: 1/(1-x/(1-x^2/(1-x^3/(1-x^4/(1-x^5/(...)))))) [given on the first page of the Odlyzko/Wilf reference]. - Joerg Arndt, Mar 08 2011
G.f.: 1/G(0), where G(k)= 1 - x^(k+1)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Jun 29 2013
G.f.: A(x) = P(x)/Q(x) where
  P(x) = Sum_{n>=0} (-1)^n* x^(n*(n+1)) / Product_{k=1..n} (1-x^k),
  Q(x) = Sum_{n>=0} (-1)^n* x^(n^2) / Product_{k=1..n} (1-x^k),
due to the Rogers-Ramanujan continued fraction identity. - Paul D. Hanna, Jul 08 2011
From Peter Bala, Dec 26 2012: (Start)
Let F(x) denote the o.g.f. of this sequence. For positive integer n >= 3, the real number F(1/n) has the simple continued fraction expansion 1 + 1/(n-2 + 1/(1 + 1/(n-2 + 1/(1 + 1/(n^2-2 + 1/(1 + 1/(n^2-2 + 1/(1 + ...)))))))), while for n >= 2, F(-1/n) has the simple continued fraction expansion 1/(1 + 1/(n-1 + 1/(1 + 1/(n-1 + 1/(n^2-1 + 1/(1 + 1/(n^2-1 + 1/(n^3-1 + 1/(1 + ...))))))))). Examples are given below. Cf. A111317 and A143951.
(End)
a(n) = c * x^(-n) + O((5/3)^n), where c = 0.312363324596741... and x = A347901 = 0.576148769142756... is the lowest root of the equation Q(x) = 0, Q(x) see above (Odlyzko & Wilf 1988). - Vaclav Kotesovec, Jul 18 2013, updated Sep 24 2020
G.f.: G(0), where G(k)= 1 - x^(k+1)/(x^(k+1) - 1/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Aug 06 2013
G.f.: 1 - 1/x + 1/(x*W(0)), where W(k)= 1 - x^(2*k+2)/(1 - x^(2*k+1)/W(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Aug 16 2013

More terms from David W. Wilson, Apr 30 2001

A014551 Jacobsthal-Lucas numbers.

Original entry on oeis.org

2, 1, 5, 7, 17, 31, 65, 127, 257, 511, 1025, 2047, 4097, 8191, 16385, 32767, 65537, 131071, 262145, 524287, 1048577, 2097151, 4194305, 8388607, 16777217, 33554431, 67108865, 134217727, 268435457, 536870911, 1073741825, 2147483647, 4294967297, 8589934591

Offset: 0

Eric W. Weisstein

Also gives the number of points of period n in the subshift of finite type corresponding to the square matrix A=[1,2;1,0] (this is then given by trace(A^n)). - Thomas Ward, Mar 07 2001
Sequence is identical to its signed inverse binomial transform (autosequence of the second kind). - Paul Curtz, Jul 11 2008
a(n) can be expressed in terms of values of the Fibonacci polynomials F_n(x), computed at x=1/sqrt(2). - Tewodros Amdeberhan (tewodros(AT)math.mit.edu), Dec 15 2008
Pisano period lengths: 1, 1, 2, 2, 4, 2, 6, 2, 6, 4, 10, 2, 12, 6, 4, 2, 8, 6, 18, 4, ... - R. J. Mathar, Aug 10 2012
Let F(x) = Product_{n >= 0} (1 - x^(3*n+1))/(1 - x^(3*n+2)). This sequence is the simple continued fraction expansion of the real number 1 + F(-1/2) = 2.83717 78068 73232 99799 ... = 2 + 1/(1 + 1/(5 + 1/(7 + 1/(17 + ...)))). See A111317. - Peter Bala, Dec 26 2012
With different signs,  2, -1, 5, -7, 17, -31, 65, -127, 257, -511, 1025, -2047, ... is the Lucas V(-1,-2) sequence. - R. J. Mathar, Jan 08 2013
The identity 2 = 2/2 + 2^2/(2*1) - 2^3/(2*1*5) - 2^4/(2*1*5*7) + 2^5/(2*1*5*7*17) + 2^6/(2*1*5*7*17*31) - - + + can be viewed as a generalized Engel-type expansion of the number 2 to the base 2. Compare with A062510. - Peter Bala, Nov 13 2013
For n >= 2, a(n) is the number of ways to tile a 2 X n strip, where the first two columns have an extra cell at the top, with 1 X 2 dominoes and 2 X 2 squares. Shown here is one of the a(7)=127 ways for the n=7 case:
  ._.
  |_|_________.
  | |   | |_| |
  ||__|_|_|_|. - Greg Dresden, Sep 26 2021
Named by Horadam (1988) after the German mathematician Ernst Jacobsthal (1882-1965) and the French mathematician Édouard Lucas (1842-1891). - Amiram Eldar, Oct 02 2023

G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. pp. 180, 255.
Lind and Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, 1995. (General material on subshifts of finite type)
Kritkhajohn Onphaeng and Prapanpong Pongsriiam. Jacobsthal and Jacobsthal-Lucas Numbers and Sums Introduced by Jacobsthal and Tverberg. Journal of Integer Sequences, Vol. 20 (2017), Article 17.3.6.
Abdelmoumène Zekiri, Farid Bencherif, Rachid Boumahdi, Generalization of an Identity of Apostol, J. Int. Seq., Vol. 21 (2018), Article 18.5.1.

T. D. Noe, Table of n, a(n) for n = 0..200
Kunle Adegoke, Robert Frontczak, and Taras Goy, Partial sum of the products of the Horadam numbers with subscripts in arithmetic progression, Notes on Num. Theor. and Disc. Math. (2021) Vol. 27, No. 2, 54-63.
Tewodros Amdeberhan, A note on Fibonacci-type polynomials, arXiv:0811.4652 [math.NT], 2008.
Hacène Belbachir, Amine Belkhir, and Ihab-Eddin Djellas, Permanent of Toeplitz-Hessenberg Matrices with Generalized Fibonacci and Lucas entries, Applications and Applied Mathematics: An International Journal (AAM 2022), Vol. 17, Iss. 2, Art. 15, 558-570.
Paula Catarino, Helena Campos, and Paulo Vasco. On the Mersenne sequence. Annales Mathematicae et Informaticae, 46 (2016), pp. 37-53.
Charles K. Cook and Michael R. Bacon, Some identities for Jacobsthal and Jacobsthal-Lucas numbers satisfying higher order recurrence relations, Annales Mathematicae et Informaticae, 41 (2013), pp. 27-39.
Fatih Erduvan and Refik Keskin, Fibonacci And Lucas Numbers Which Are Product Of Two Jacobsal-Lucas Numbers [sic], Appl. Math. E-Notes (2023) Vol. 23, 60-70.
M. C. Firengiz and A. Dil, Generalized Euler-Seidel method for second order recurrence relations, Notes on Number Theory and Discrete Mathematics, Vol. 20, 2014, No. 4, 21-32.
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 p. 24.
A. F. Horadam, Jacobsthal and Pell Curves, Fib. Quart. 26, 79-83, 1988.
A. F. Horadam, Jacobsthal Representation Numbers, Fib Quart. 34, 40-54, 1996.
D. Jhala, G. P. S. Rathore, and K. Sisodiya, Some Properties of k-Jacobsthal Numbers with Arithmetic Indexes, Turkish Journal of Analysis and Number Theory, 2014, Vol. 2, No. 4, 119-124.
Thomas Koshy and Ralph P. Grimaldi, Ternary words and Jacobsthal numbers, Fib. Quart., 55 (No. 2, 2017), 129-136.
Kritkhajohn Onphaeng, Tammatada Khemaratchatakumthorn, and Prapanpong Pongsriiam, Inequalities for Inclusion-Exclusion-Like Sums Involving the Ceiling and the Nearest Integer Functions, Integers (2025) Vol. 25, Art. No. A45. See p. 3.
Mihai Prunescu and Lorenzo Sauras-Altuzarra, On the representation of C-recursive integer sequences by arithmetic terms, arXiv:2405.04083 [math.LO], 2024. See p. 16.
Yash Puri and Thomas Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), Article 01.2.1.
M. Rahmani, The Akiyama-Tanigawa matrix and related combinatorial identities, Linear Algebra and its Applications 438 (2013) 219-230. - From _N. J. A. Sloane_, Dec 26 2012
Mario Raso, Integer Sequences in Cryptography: A New Generalized Family and its Application, Ph. D. Thesis, Sapienza University of Rome (Italy 2025). See p. 41.
Yüksel Soykan, On Summing Formulas For Generalized Fibonacci and Gaussian Generalized Fibonacci Numbers, Advances in Research (2019) Vol. 20, No. 2, 1-15, Article AIR.51824.
Yüksal Soykan, On Summing Formulas for Horadam Numbers, Asian Journal of Advanced Research and Reports (2020) Vol. 8, Issue 1, 45-61.
Yüksel Soykan, Generalized Fibonacci Numbers: Sum Formulas, Journal of Advances in Mathematics and Computer Science (2020) Vol. 35, No. 1, 89-104.
Yüksel Soykan, Closed Formulas for the Sums of Squares of Generalized Fibonacci Numbers, Asian Journal of Advanced Research and Reports (2020) Vol. 9, No. 1, 23-39, Article no. AJARR.55441.
Yüksel Soykan, Closed Formulas for the Sums of Cubes of Generalized Fibonacci Numbers: Closed Formulas of Sum_{k=0..n} W_k^3 and Sum_{k=1..n} W_(-k)^3, Archives of Current Research International (2020) Vol. 20, Issue 2, 58-69.
Yüksel Soykan, A Study on Generalized Fibonacci Numbers: Sum Formulas Sum_{k=0..n} k * x^k * W_k^3 and Sum_{k=1..n} k * x^k W_-k^3 for the Cubes of Terms, Earthline Journal of Mathematical Sciences (2020) Vol. 4, No. 2, 297-331.
Yüksel Soykan, On Generalized (r, s)-numbers, Int. J. Adv. Appl. Math. and Mech. (2020) Vol. 8, No. 1, 1-14.
Yüksel Soykan, Erkan Taşdemir, and İnci Okumuş, On Dual Hyperbolic Numbers With Generalized Jacobsthal Numbers Components, Zonguldak Bülent Ecevit University, (Zonguldak, Turkey, 2019).
Anetta Szynal-Liana, Iwona Włoch, and Mirosław Liana, Generalized commutative quaternion polynomials of the Fibonacci type, Annales Math. Sect. A, Univ. Mariae Curie-Skłodowska (Poland 2022) Vol. 76, No. 2, 33-44.
Elif Tan, Luka Podrug, and Vesna Iršič Chenoweth, Horadam-Lucas Cubes, Axioms (2024) Vol. 13, No. 12, 837.
Kai Wang, On Horadam Sequences and Related Infinite Series, (2020).
Kai Wang, General Identities for Horadam Sequences, (2020).
Eric Weisstein's World of Mathematics, Jacobsthal Number.
Wikipedia, Lucas sequence.
OEIS Wiki, Autosequence.
Volkan Yildiz, Some divisibility properties of Jacobsthal numbers, arXiv:2212.08814 [math.CO], 2022.
Index entries for linear recurrences with constant coefficients, signature (1,2).
Index entries for Lucas sequences.
Index entries for sequences related to Chebyshev polynomials.

Cf. A001045 (companion "autosequence"), A019322, A066845, A111317.
Cf. A135440 (first differences), A166920 (partial sums).
Cf. A000079, A033999, A102345, A105723.
Cf. A006995.

a014551 n = a000079 n + a033999 n
a014551_list = map fst $ iterate (\(x,s) -> (2 * x - 3 * s, -s)) (2, 1)
-- Reinhard Zumkeller, Jan 02 2013

[2^n + (-1)^n: n in [0..30]]; // G. C. Greubel, Dec 17 2017

f[n_]:=2/(n+1);x=4;Table[x=f[x];Denominator[x],{n,0,5!}] (* Vladimir Joseph Stephan Orlovsky, Mar 12 2010 *)
nxt[{n_,a_}]:={n+1,2a-3(-1)^(n+1)}; Transpose[NestList[nxt,{1,2},40]] [[2]] (* Harvey P. Dale, May 27 2013 *)
LinearRecurrence[{1, 2}, {2, 1}, 40] (* Jean-François Alcover, Jan 07 2019 *)

a(n)=2^n+(-1)^n \\ Charles R Greathouse IV, Nov 20 2012

[lucas_number2(n,1,-2) for n in range(0, 32)] # Zerinvary Lajos, Apr 30 2009

a(n+1) = 2 * a(n) - (-1)^n * 3.
From Len Smiley, Dec 07 2001: (Start)
a(n) = 2^n + (-1)^n.
G.f.: (2-x)/(1-x-2*x^2). (End)
E.g.f.: exp(x) + exp(-2*x) produces a signed version. - Paul Barry, Apr 27 2003
a(n+1) = Sum_{k=0..floor(n/2)} binomial(n-1, 2*k)*3^(2*k)/2^(n-2). - Paul Barry, Feb 21 2003
0, 1, 5, 7 ... is 2^n - 2*0^n + (-1)^n, the 2nd inverse binomial transform of (2^n-1)^2 (A060867). - Paul Barry, Sep 05 2003
a(n) = 2*T(n, i/(2*sqrt(2))) * (-i*sqrt(2))^n with i^2=-1. - Paul Barry, Nov 17 2003
a(n) = A078008(n) + A001045(n+1). - Paul Barry, Feb 12 2004
a(n) = 2*A001045(n+1) - A001045(n). - Paul Barry, Mar 22 2004
a(0)=2, a(1)=1, a(n) = a(n-1) + 2*a(n-2) for n > 1. - Philippe Deléham, Nov 07 2006
a(2*n+1) = Product_{d|(2*n+1)} cyclotomic(d,2). a(2^k*(2*n+1)) = Product_{d|(2*n+1)} cyclotomic(2*d,2^(2^k)). - Miklos Kristof, Mar 12 2007
a(n) = 2^{(n-1)/2}F_{n-1}(1/sqrt(2)) + 2^{(n+2)/2}F_{n-2}(1/sqrt(2)). - Tewodros Amdeberhan (tewodros(AT)math.mit.edu), Dec 15 2008
E.g.f.: U(0) where U(k) = 1 + (-1)^k/(2^k - 4^k*x*2/(2*x*2^k + (-1)^k*(k+1)/U(k+1))) ; (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Nov 02 2012
G.f.: U(0) where U(k) = 1 + (-1)^k/(2^k - 4^k*x*2/(2*x*2^k + (-1)^k/U(k+1))) ; (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Nov 02 2012
a(n) = sqrt(9*(A001045)^2 + (-1)^n*2^(n+2)). - Vladimir Shevelev, Mar 13 2013
G.f.: 2 + G(0)*x*(1+4*x)/(2-x), where G(k) = 1 + 1/(1 - x*(9*k-1)/( x*(9*k+8) - 2/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 13 2013
a(n) = [x^n] ( (1 + x + sqrt(1 + 2*x + 9*x^2))/2 )^n for n >= 1. - Peter Bala, Jun 23 2015
For n >= 1: a(n) = A006995(2^((n+2)/2)) when n is even, a(n) = A006995(3*2^((n-1)/2) - 1) when n is odd. - Bob Selcoe, Sep 04 2017
a(n) = J(n) + 4*J(n-1), a(0)=2, where J is A001045. - Yuchun Ji, Apr 23 2019
For n >= 0, 1/(2*a(n+1)) = Sum_{m>=n} a(m)/(a(m+1)*a(m+2)). - Kai Wang, Mar 03 2020
For 4 > h >= 0, k >= 0, a(4*k+h) mod 5 = a(h) mod 5. - Kai Wang, May 06 2020
From Kai Wang, May 30 2020: (Start)
(2 - a(n+1)/a(n))/9 = Sum_{m>=n} (-2)^m/(a(m)*a(m+1)).
a(n) = 2*A001045(n+1) - A001045(n).
a(n)^2 = a(2*n) + 2*(-2)^n.
a(n)^2 = 9*A001045(n)^2 + 4*(-2)^n.
a(2*n) = 9*A001045(n)^2 + 2*(-2)^n.
2*A001045(m+n) = A001045(m)*a(n) + a(m)*A001045(n).
2*(-2)^n*A001045(m-n) = A001045(m)*a(n) - a(m)*A001045(n).
A001045(m+n) + (-2)^n*A001045(m-n) = A001045(m)*a(n).
A001045(m+n) - (-2)^n*A001045(m-n) = a(m)*A001045(n).
2*a(m+n) = 9*A001045(m)*A001045(n) + a(m)*a(n).
2*(-2)^n*a(m-n) = a(m)*a(n) - 9*A001045(m)*A001045(n).
a(m+n) - (-2)^n*a(m-n) = 9*A001045(m)*A001045(n).
a(m+n) + (-2)^n*a(m-n) = a(m)*a(n).
a(m+n)*a(m-n) - a(m)*a(m) = 9*(-2)^(m-n)*A001045(n)^2.
a(m+1)*a(n) - a(m)*a(n+1) = 9*(-2)^n*A001045(m-n). (End)
a(n) = F(n+1) + F(n-1) + Sum_{k=0..(n-2)} a(k)*F(n-1-k) for F(n) the Fibonacci numbers and for n > 1. - Greg Dresden, Jun 03 2020

A143951 Number of Dyck paths such that the area between the x-axis and the path is n.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 6, 9, 14, 21, 31, 47, 71, 107, 161, 243, 367, 553, 834, 1258, 1898, 2863, 4318, 6514, 9827, 14824, 22361, 33732, 50886, 76762, 115796, 174680, 263509, 397508, 599647, 904579, 1364576, 2058489, 3105269, 4684359, 7066449, 10659877, 16080632, 24257950, 36593598, 55202165, 83273553, 125619799, 189499952

Offset: 0

Emeric Deutsch, Oct 09 2008

nonn

Column sums of A129182.

			a(5)=3 because we have UDUUDD, UUDDUD and UDUDUDUDUD, where U=(1,1) and D=(1,-1).
From _Peter Bala_, Dec 26 2012: (Start)
F(1/10) = sum {n >= 0} a(n)/10^n has the simple continued fraction expansion 1 + 1/(8 + 1/(1 + 1/(98 + 1/(1 + 1/(998 + 1/(1 + ...)))))).
F(-1/10) = sum {n >= 0} (-1)^n*a(n)/10^n has the simple continued fraction expansion 1/(1 + 1/(10 + 1/(100 + 1/(1000 + ...)))).
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..2000 (first 1001 terms from Vincenzo Librandi)
Paul Barry, On sequences with {-1, 0, 1} Hankel transforms, arXiv preprint arXiv:1205.2565 [math.CO], 2012. - From _N. J. A. Sloane_, Oct 18 2012

Cf. A129182, A291874 (convolution inverse).

Cf. A005169, A111317, A224704.

g:=1/(1-x/(1-x^3/(1-x^5/(1-x^7/(1-x^9/(1-x^11/(1-x^13/(1-x^15)))))))): gser:= series(g,x=0,45): seq(coeff(gser,x,n),n=0..44);
# second Maple program:
b:= proc(x, y, k) option remember;
      `if`(y<0 or y>x or k<0 or k>x^2/2-(y-x)^2/4, 0,
      `if`(x=0, 1, b(x-1, y-1, k-y+1/2) +b(x-1, y+1, k-y-1/2)))
    end:
a:= n-> add(b(2*n-4*t, 0, n), t=0..n/2):
seq(a(n), n=0..50);  # Alois P. Heinz, Aug 24 2018

terms = 50; CoefficientList[1/(1+ContinuedFractionK[-x^(2i-1), 1, {i, 1, Sqrt[terms]//Ceiling}]) + O[x]^terms, x] (* Jean-François Alcover, Jul 11 2018 *)

N=66; q = 'q +O('q^N);
G(k) = if(k>N, 1, 1 - q^(k+1) / G(k+2) );
gf = 1 / G(0);
Vec(gf) \\ Joerg Arndt, Jul 06 2013

G.f.: 1/(1 - x/(1 - x^3/(1 - x^5/(1 - x^7/(1 - x^9/(1 - ...
Derivation: the g.f. G(x,z) of Dyck paths, where x marks area and z marks semilength, satisfies G(x,z)=1+x*z*G(x,z)*G(x,x^2*z). Set z=1.
From Peter Bala, Dec 26 2012: (Start)
Let F(x) denote the o.g.f. of this sequence. For positive integer n >= 3, the real number F(1/n) has the simple continued fraction expansion 1 + 1/(n-2 + 1/(1 + 1/(n^2-2 + 1/(1 + 1/(n^3-2 + 1/(1 + ...)))))).
For n >= 1, F(-1/n) has the simple continued fraction expansion
1/(1 + 1/(n + 1/(n^2 + 1/(n^3 + ...)))). Examples are given below. Cf. A005169 and A111317.
(End)
G.f.: A(x) = 1/(1 - x/(1-x + x/(1+x^2 + x^4/(1-x^3 - x^2/(1+x^4 - x^7/(1-x^5 + x^3/(1+x^6 + x^10/(1-x^7 - x^4/(1+x^8 - x^13/(1-x^9 + x^5/(1+x^10 + x^16/(1 + ...)))))))))))), a continued fraction. - Paul D. Hanna, Aug 08 2016
a(n) ~ c / r^n, where r = 0.66290148514884371255690407749133031115536799774051... and c = 0.337761150388539773466092171229604432776662930886727976914... . - Vaclav Kotesovec, Feb 17 2017, corrected Nov 04 2021
From Peter Bala, Jul 04 2019: (Start)
O.g.f. as a ratio of q-series: N(q)/D(q), where N(q) = Sum_{n >= 0} (-1)^n*q^(2*n^2+n)/( (1-q^2)*(1-q^4)*...*(1-q^(2*n)) ) and D(q) = Sum_{n >= 0} (-1)^n*q^(2*n^2-n)/( (1-q^2)*(1-q^4)*...*(1-q^(2*n)) ). Cf. A224704.
D(q) has its least  positive (and simple) real zero at x = 0.66290 14851 48843 71255 69040 ....
a(n) ~ c*d^n, where  d = 1/x = 1.5085197761707628638804960 ...  and c = - N(x)/(x*D'(x)) = 0.3377611503885397734660921 ... (the prime indicates differentiation w.r.t. q). (End)

b-file corrected and extended by Alois P. Heinz, Aug 24 2018

A080924 Jacobsthal gap sequence.

Original entry on oeis.org

0, 1, 3, 1, 15, 1, 63, 1, 255, 1, 1023, 1, 4095, 1, 16383, 1, 65535, 1, 262143, 1, 1048575, 1, 4194303, 1, 16777215, 1, 67108863, 1, 268435455, 1, 1073741823, 1, 4294967295, 1, 17179869183, 1, 68719476735, 1, 274877906943, 1, 1099511627775, 1

Offset: 0

Paul Barry, Feb 26 2003

Inverse binomial transform of A080925
From Peter Bala, Dec 26 2012: (Start)
Let F(x) = product {n >= 0} (1 - x^(3*n+1))/(1 - x^(3*n+2)). This sequence is the simple continued fraction expansion of the real number F(1/4) = 0.79761 68651 30459 16010 ... = 1/(1 + 1/(3 + 1/(1 + 1/(15 + 1/(1 + 1/(63 + 1/(1 + 1/(255 + ...)))))))). See A111317. (End)
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 3", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. - Robert Price, Apr 19 2017

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

Vincenzo Librandi, Table of n, a(n) for n = 0..300
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 (-1,4,4).

Cf. A001045, A002450, A080926, A080927, A111317.

CoefficientList[Series[x (1 + 4 x) / ((1 + x) (1 + 2 x) (1 - 2 x)), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 05 2013 *)
LinearRecurrence[{-1, 4, 4}, {0, 1, 3}, 42] (* Jean-François Alcover, Sep 21 2017 *)

a(2n) = 3*A001045(2n) = 3*A002450(n) = 4^n-1, a(2n+1)=1.
a(n) = (2^n-2*(-1)^n+(-2)^n)/2.
G.f.: x*(1+4*x)/((1+x)*(1+2*x)*(1-2*x)).
E.g.f.: (exp(2*x)-2*exp(-x)+exp(-2*x))/2.

A094024 Alternating 1 with one less than the powers of 2.

Original entry on oeis.org

1, 1, 1, 3, 1, 7, 1, 15, 1, 31, 1, 63, 1, 127, 1, 255, 1, 511, 1, 1023, 1, 2047, 1, 4095, 1, 8191, 1, 16383, 1, 32767, 1, 65535, 1, 131071, 1, 262143, 1, 524287, 1, 1048575, 1, 2097151, 1, 4194303, 1, 8388607, 1, 16777215, 1, 33554431, 1, 67108863, 1

Offset: 0

Paul Barry, Apr 22 2004

Inverse binomial transform of A052542. Partial sums are A075427.

Let F(x) = product {n >= 0} (1 - x^(3*n+1))/(1 - x^(3*n+2)). This sequence is the simple continued fraction expansion of the real number F(1/2) = 0.64227 25013 85234 96714 ... = 1/(1 + 1/(1 + 1/(1 + 1/(3 + 1/(1 + 1/(7 + 1/(1 + 1/(15 + ...)))))))). See A111317. - Peter Bala, Dec 26 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (-1,2,2).

Cf. A000129, A016116.

Cf. A111317.

[Ceiling((-1)^n+((Sqrt(2))^n-(-Sqrt(2))^n)/Sqrt(2)): n in [0..50]]; // Vincenzo Librandi, Aug 17 2011

LinearRecurrence[{-1, 2, 2}, {1, 1, 1}, 60] (* Jean-François Alcover, Jul 02 2018 *)

PARI
```
a(n)=(1-(-1)^n)*2^floor(n/2)+(-1)^n
```

G.f.: (1+2*x) / ((1+x) * (1-2*x^2)).
E.g.f.: exp(-x) + 2*sinh(sqrt(2)*x) / sqrt(2).
a(n) = (-1)^n + ((sqrt(2))^n - (-sqrt(2))^n) / sqrt(2).
a(n) = (1-(-1)^n) * 2^floor(n/2) + (-1)^n. - Ralf Stephan, Aug 19 2013
a(n) = -a(n-1) + 2*a(n-2) + 2*a(n-3). - Andrew Howroyd, Feb 21 2018

Better name from Ralf Stephan, Aug 19 2013

Even terms for n >= 60 corrected in b-file by Andrew Howroyd, Feb 21 2018

A358369 Euler transform of 2^floor(n/2), (A016116).

Original entry on oeis.org

1, 1, 3, 5, 12, 20, 43, 73, 146, 250, 475, 813, 1499, 2555, 4592, 7800, 13761, 23253, 40421, 67963, 116723, 195291, 332026, 552882, 932023, 1544943, 2585243, 4267081, 7094593, 11662769, 19281018, 31575874, 51937608, 84753396, 138772038, 225693778, 368017636

Offset: 0

Peter Luschny, Nov 17 2022

nonn

Cf. A002513, A016116.

Sequences that can be represented as a EulerTransform(BinaryRecurrenceSequence()) include A000009, A000041, A000712, A001970, A002513, A010054, A015128, A022567, A034691, A111317, A111335, A117410, A156224, A166861, A200544, A261031, A261329, A358449.

BinaryRecurrenceSequence := proc(b, c, u0:=0, u1:=1) local u;
u := proc(n) option remember; if n < 2 then return [u0, u1][n + 1] fi;
b*u(n - 1) + c*u(n - 2) end; u end:
EulerTransform := proc(a) local b;
b := proc(n) option remember; if n = 0 then return 1 fi; add(add(d * a(d),
d = NumberTheory:-Divisors(j)) * b(n-j), j = 1..n) / n end; b end:
a := EulerTransform(BinaryRecurrenceSequence(0, 2, 1)): seq(a(n), n=0..36);

from typing import Callable
from functools import cache
from sympy import divisors
def BinaryRecurrenceSequence(b:int, c:int, u0:int=0, u1:int=1) -> Callable:
    @cache
    def u(n: int) -> int:
        if n < 2:
            return [u0, u1][n]
        return b * u(n - 1) + c * u(n - 2)
    return u
def EulerTransform(a: Callable) -> Callable:
    @cache
    def b(n: int) -> int:
        if n == 0:
            return 1
        s = sum(sum(d * a(d) for d in divisors(j)) * b(n - j)
            for j in range(1, n + 1))
        return s // n
    return b
b = BinaryRecurrenceSequence(0, 2, 1)
a = EulerTransform(b)
print([a(n) for n in range(37)])

# uses[EulerTransform from A166861]
b = BinaryRecurrenceSequence(0, 2, 1)
a = EulerTransform(b)
print([a(n) for n in range(37)])

A087404 a(n) = 4a(n-1) + 5a(n-2) for n > 1, with a(0) = 2 and a(1) = 4.

Original entry on oeis.org

2, 4, 26, 124, 626, 3124, 15626, 78124, 390626, 1953124, 9765626, 48828124, 244140626, 1220703124, 6103515626, 30517578124, 152587890626, 762939453124, 3814697265626, 19073486328124, 95367431640626, 476837158203124, 2384185791015626, 11920928955078124, 59604644775390626

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Sep 01 2003

Let F(x) = Product_{n>=0} (1 - x^(3*n+1))/(1 - x^(3*n+2)). This sequence is the simple continued fraction expansion of the real number 1 + F(-1/5) = 2.24761 97788 60361 46849 ... = 2 + 1/(4 + 1/(26 + 1/(124 + 1/(626 + ...)))). See A111317. - Peter Bala, Dec 26 2012

Michael De Vlieger, Table of n, a(n) for n = 0..1430
Weerayuth Nilsrakoo and Achariya Nilsrakoo, On One-Parameter Generalization of Jacobsthal Numbers, WSEAS Trans. Math. (2025) Vol. 24, 51-61. See p. 3.
Index entries for linear recurrences with constant coefficients, signature (4,5).

Cf. A081340, A111317.

CoefficientList[Series[(2 - 4x)/(1 - 4x - 5x^2), {x, 0, 25}], x]
LinearRecurrence[{4,5},{2,4},30] (* Harvey P. Dale, May 13 2022 *)

[lucas_number2(n,4,-5) for n in range(0, 22)] # Zerinvary Lajos, May 14 2009

G.f.: (2 - 4*x)/(1 - 4*x - 5*x^2).
a(n) = 5^n + (-1)^n.
From Elmo R. Oliveira, Aug 23 2024: (Start)
E.g.f.: exp(-x)*(exp(6*x) + 1).
a(n) = 2*A081340(n). (End)

a(22)-a(24) from Elmo R. Oliveira, Aug 23 2024

A084182 a(n) = 3^n + (-1)^n - [1/(n+1)], where [] represents the floor function.

Original entry on oeis.org

1, 2, 10, 26, 82, 242, 730, 2186, 6562, 19682, 59050, 177146, 531442, 1594322, 4782970, 14348906, 43046722, 129140162, 387420490, 1162261466, 3486784402, 10460353202, 31381059610, 94143178826, 282429536482, 847288609442, 2541865828330, 7625597484986

Offset: 0

Paul Barry, May 19 2003

Binomial transform of A084181.
From Peter Bala, Dec 26 2012: (Start)
Let F(x) = product {n >= 0} (1 - x^(3*n+1))/(1 - x^(3*n+2)). This sequence is the simple continued fraction expansion of the real number F(-1/3)  = 1.47627 73316 74531 44215 ... = 1 + 1/(2 + 1/(10 + 1/(26 + 1/(82 + ...)))). See A111317.
(End)

Index entries for linear recurrences with constant coefficients, signature (2,3).

Except for leading term, same as A102345.

Cf. A046717, A111317.

LinearRecurrence[{2,3},{1,2,10},30] (* Harvey P. Dale, Apr 27 2016 *)

a(n) = 3^n + (-1)^n - 0^n.
G.f.: (1+3*x^2)/((1+x)*(1-3*x)).
E.g.f.: exp(3*x)-exp(0)+exp(-x).
a(n) = 2 * A046717(n) for n >= 1.

A111165 Let qf(a,q) = Product(1-a*q^j,j=0..infinity); g.f. is qf(q,q^3)/qf(q^2,q^3).

Original entry on oeis.org

1, -1, 1, -1, 0, 1, -1, 0, 1, -2, 2, 0, -2, 2, -1, -1, 3, -2, -1, 3, -3, 0, 4, -5, 2, 3, -6, 4, 2, -7, 6, 0, -7, 9, -2, -7, 10, -5, -6, 13, -8, -5, 15, -13, -1, 16, -17, 2, 16, -22, 8, 16, -27, 14, 12, -30, 22, 9, -34, 29, 3, -36, 39, -5, -37, 47, -14, -36, 58, -26, -33, 66, -41, -26, 75, -56, -18, 81, -74, -4, 87, -94, 12, 87, -113, 34

Offset: 0

N. J. A. Sloane, Nov 09 2005

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A111375. Convolution inverse of A111317.

a:= proc(n) option remember; `if`(n=0, 1,
      add(add(d*[0, -1, 1][irem(d, 3)+1],
      d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Apr 02 2014

a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*{0, -1, 1}[[Mod[d, 3]+1]], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Apr 28 2014, after Alois P. Heinz *)

{a(n) = if( n<0, 0, polcoeff( prod(k=0, n\3, (1 - x^(3*k+1)) / (1 - x^(3*k+2)), 1 + x * O(x^n)), n))} /* Michael Somos, Dec 23 2007 */

Euler transform of period 3 sequence [ -1, 1, 0, ...]. - Michael Somos, Dec 23 2007

G.f.: Product_{k>=0} (1 - x^(3*k+1)) / (1 - x^(3*k+2)).

cp's OEIS Frontend

A049347 Period 3: repeat [1, -1, 0].

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Maxima

PARI

PARI

PARI

Sage

Formula

Extensions

A005169 Number of fountains of n coins.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

PARI

PARI

Formula

Extensions

A014551 Jacobsthal-Lucas numbers.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

PARI

Sage

Formula

A143951 Number of Dyck paths such that the area between the x-axis and the path is n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A080924 Jacobsthal gap sequence.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

A094024 Alternating 1 with one less than the powers of 2.

A087404 a(n) = 4a(n-1) + 5a(n-2) for n > 1, with a(0) = 2 and a(1) = 4.