A006720 -id:A006720 - cp's OEIS frontend

A038754 a(2n) = 3^n, a(2n+1) = 2*3^n.

1, 2, 3, 6, 9, 18, 27, 54, 81, 162, 243, 486, 729, 1458, 2187, 4374, 6561, 13122, 19683, 39366, 59049, 118098, 177147, 354294, 531441, 1062882, 1594323, 3188646, 4782969, 9565938, 14348907, 28697814, 43046721, 86093442, 129140163, 258280326, 387420489

Offset: 0

Henry Bottomley, May 03 2000

In general, for the recurrence a(n) = a(n-1)*a(n-2)/a(n-3), all terms are integers iff a(0) divides a(2) and first three terms are positive integers, since a(2n+k) = a(k)*(a(2)/a(0))^n for all nonnegative integers n and k.
Equals eigensequence of triangle A070909; (1, 1, 2, 3, 6, 9, 18, ...) shifts to the left with multiplication by triangle A070909. - Gary W. Adamson, May 15 2010
The a(n) represent all paths of length (n+1), n >= 0, starting at the initial node on the path graph P_5, see the second Maple program. - Johannes W. Meijer, May 29 2010
a(n) is the difference between numbers of multiple of 3 evil (A001969) and odious (A000069) numbers in interval [0, 2^(n+1)). - Vladimir Shevelev, May 16 2012
A "half-geometric progression": to obtain a term (beginning with the third one) we multiply the before previous one by 3. - Vladimir Shevelev, May 21 2012
Pisano periods: 1, 2, 1, 4, 8, 2, 12, 4, 1, 8, 10, 4, 6, 12, 8, 8, 32, 2, 36, 8, ... . - R. J. Mathar, Aug 10 2012
Numbers k such that the k-th cyclotomic polynomial has a root mod 3. - Eric M. Schmidt, Jul 31 2013
Range of row n of the circular Pascal array of order 6. - Shaun V. Ault, Jun 05 2014
Also, the number of walks of length n on the graph 0--1--2--3--4 starting at vertex 1. - Sean A. Irvine, Jun 03 2025

			In the interval [0,2^5) we have 11 multiples of 3 numbers, from which 10 are evil and only one (21) is odious. Thus a(4) = 10 - 1 = 9. - _Vladimir Shevelev_, May 16 2012

Indranil Ghosh, Table of n, a(n) for n = 0..1500 (first 401 terms from T. D. Noe)
S. V. Ault and C. Kicey, Counting paths in corridors using circular Pascal arrays, Discrete Mathematics, Vol. 332 (2014), pp. 45-54.
Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
Nachum Dershowitz, Between Broadway and the Hudson, arXiv:2006.06516 [math.CO], 2020.
Sean A. Irvine, Walks on Graphs.
Richard L. Ollerton and Anthony G. Shannon, Some properties of generalized Pascal squares and triangles, Fib. Q., 36 (1998), 98-109. See pages 106-7.
Vladimir Shevelev, On monotonic strengthening of Newman-like phenomenon on (2m+1)-multiples in base 2m, arXiv:0710.3177 [math.NT], 2007.
M. B. Wells, Elements of Combinatorial Computing, Pergamon, Oxford, 1971. [Annotated scanned copy of pages 237-240]
Index entries for linear recurrences with constant coefficients, signature (0,3).

Cf. A000045, A000069, A000079, A000120, A000244, A001969, A006720, A006721.
Cf. A006722, A006723, A008776, A028495, A030436, A037124, A048328, A061551.
Cf. A068911, A070909, A087503, A124302, A128588, A133464, A140730, A140740.
Cf. A178381, A182751, A182752, A182753, A182754, A182755, A182756, A182757.
Fifth row of the array A094718.

import Data.List (transpose)
a038754 n = a038754_list !! n
a038754_list = concat $ transpose [a000244_list, a008776_list]
-- Reinhard Zumkeller, Oct 19 2015

[n le 2 select n else 3*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Aug 18 2016

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=3*a[n-2]+2 od: seq(a[n]+1, n=0..34); # Zerinvary Lajos, Mar 20 2008
with(GraphTheory): P:=5: G:=PathGraph(P): A:= AdjacencyMatrix(G): nmax:=35; for n from 1 to nmax do B(n):=A^n; a(n):=add(B(n)[1,k],k=1..P) od: seq(a(n),n=1..nmax); # Johannes W. Meijer, May 29 2010

LinearRecurrence[{0,3},{1,2},40] (* Harvey P. Dale, Jan 26 2014 *)
CoefficientList[Series[(1+2x)/(1-3x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Aug 18 2016 *)
Module[{nn=20,c},c=3^Range[0,nn];Riffle[c,2c]] (* Harvey P. Dale, Aug 21 2021 *)

PARI
```
a(n)=(1/6)*(5-(-1)^n)*3^floor(n/2)
```
PARI
```
a(n)=3^(n>>1)<
				
```

[2^(n%2)*3^((n-(n%2))/2) for n in range(61)] # G. C. Greubel, Oct 10 2022

a(n) = a(n-1)*a(n-2)/a(n-3) with a(0)=1, a(1)=2, a(2)=3.
a(2*n) = (3/2)*a(2*n-1) = 3^n, a(2*n+1) = 2*a(2*n) = 2*3^n.
From Benoit Cloitre, Apr 27 2003: (Start)
a(1)=1, a(n)= 2*a(n-1) if a(n-1) is odd, or a(n)= (3/2)*a(n-1) if a(n-1) is even.
a(n) = (1/6)*(5-(-1)^n)*3^floor(n/2).
a(2*n) = a(2*n-1) + a(2*n-2) + a(2*n-3).
a(2*n+1) = a(2*n) + a(2*n-1). (End)
G.f.: (1+2*x)/(1-3*x^2). - Paul Barry, Aug 25 2003
From Reinhard Zumkeller, Sep 11 2003: (Start)
a(n) = (1 + n mod 2) * 3^floor(n/2).
a(n) = A087503(n) - A087503(n-1). (End)
a(n) = sqrt(3)*(2+sqrt(3))*(sqrt(3))^n/6 - sqrt(3)*(2-sqrt(3))*(-sqrt(3))^n/6. - Paul Barry, Sep 16 2003
From Reinhard Zumkeller, May 26 2008: (Start)
a(n) = A140740(n+2,2).
a(n+1) = a(n) + a(n - n mod 2). (End)
If p(i) = Fibonacci(i-3) and if A is the Hessenberg matrix of order n defined by A(i,j) = p(j-i+1), (i<=j), A(i,j)=-1, (i=j+1), and A(i,j)=0 otherwise. Then, for n>=1, a(n-1) = (-1)^n det A. - Milan Janjic, May 08 2010
a(n) = A182751(n) for n >= 2. - Jaroslav Krizek, Nov 27 2010
a(n) = Sum_{i=0..2^(n+1), i==0 (mod 3)} (-1)^A000120(i). - Vladimir Shevelev, May 16 2012
a(0)=1, a(1)=2, for n>=3, a(n)=3*a(n-2). - Vladimir Shevelev, May 21 2012
Sum_(n>=0) 1/a(n) = 9/4. - Alexander R. Povolotsky, Aug 24 2012
a(n) = sqrt(3*a(n-1)^2 + (-3)^(n-1)). - Richard R. Forberg, Sep 04 2013
a(n) = 2^((1-(-1)^n)/2)*3^((2*n-1+(-1)^n)/4). - Luce ETIENNE, Aug 11 2014
From Reinhard Zumkeller, Oct 19 2015: (Start)
a(2*n) = A000244(n), a(2*n+1) = A008776(n).
For n > 0: a(n+1) = a(n) + if a(n) odd then min{a(n), a(n-1)} else max{a(n), a(n-1)}, see also A128588. (End)
E.g.f.: (7*cosh(sqrt(3)*x) + 4*sqrt(3)*sinh(sqrt(3)*x) - 4)/3. - Stefano Spezia, Feb 17 2022
Sum_{n>=0} (-1)^n/a(n) = 3/4. - Amiram Eldar, Dec 02 2022

A006721 Somos-5 sequence: a(n) = (a(n-1) * a(n-4) + a(n-2) * a(n-3)) / a(n-5), with a(0) = a(1) = a(2) = a(3) = a(4) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 5, 11, 37, 83, 274, 1217, 6161, 22833, 165713, 1249441, 9434290, 68570323, 1013908933, 11548470571, 142844426789, 2279343327171, 57760865728994, 979023970244321, 23510036246274433, 771025645214210753

Offset: 0

N. J. A. Sloane

Using the addition formula for the Weierstrass sigma function it is simple to prove that the subsequence of even terms of a Somos-5 type sequence satisfy a 4th-order recurrence of Somos-4 type and similarly the odd subsequence satisfies the same 4th-order recurrence. - Andrew Hone, Aug 24 2004
log(a(n)) ~ 0.071626946 * n^2. (Hone)
The Brown link article gives interesting information about related sequences including recurrences and numerical approximations.
The n-th term is a divisor of the (n+k*(2*n-4))-th term for all integers n and k. - Peter H van der Kamp, May 18 2015
The elliptic curve y^2 + xy = x^3 + x^2 - 2x (LMFDB label 102.a1) has infinite order point P = (2, 2) and 2-torsion point T = (0, 0). Define d(n) = a(n+2). The x and y coordinates of nP + T have denominators d(n)^2 and d(n)^3. - Michael Somos, Oct 29 2022

Paul C. Kainen, Fibonacci in Somos-5 ..., Fib. Q., 60:4 (2022), 362-364.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=0..100
K. S. Brown, A Quasi-Periodic Sequence
R. H. Buchholz and R. L. Rathbun, An infinite set of Heron triangles with two rational medians, Amer. Math. Monthly, 104 (1997), 107-115.
Xiang-Ke Chang and Xing-Biao Hu, A conjecture based on Somos-4 sequence and its extension, Linear Algebra Appl. 436, No. 11, 4285-4295 (2012).
Bryant Davis, Rebecca Kotsonis, and Jeremy Rouse, The density of primes dividing a term in the Somos-5 sequence, arXiv:1507.05896 [math.NT], 2015.
Harini Desiraju and Brady Haran, The Troublemaker Number, Numberphile video (2022).
S. Fomin and A. Zelevinsky, The Laurent phenomenon, arXiv:math/0104241 [math.CO], 2001.
David Gale, The strange and surprising saga of the Somos sequences, in Mathematical Entertainments, Math. Intelligencer 13(1) (1991), pp. 40-42.
R. W. Gosper and Richard C. Schroeppel, Somos Sequence Near-Addition Formulas and Modular Theta Functions, arXiv:math/0703470 [math.NT]
J. W. E. Harrow and A. N. W. Hone, Casting more light in the shadows: dual Somos-5 sequences, arXiv:2409.00406 [nlin.SI], 2024. See p. 2.
Andrew N. W. Hone, Elliptic curves and quadratic recurrence sequences, Bull. Lond. Math. Soc. 37 (2005) 161-171.
Andrew N. W. Hone, Sigma function solution of the initial value problem for Somos 5 sequences, arXiv:math/0501554 [math.NT], 2005-2006.
Andrew N. W. Hone, Heron triangles with two rational median and Somos-5 sequences, arXiv:2107.03197 [math.NT], 2022.
Andrew N. W. Hone, Heron triangles and the hunt for unicorns, arXiv:2401.05581 [math.NT], 2024.
LMFDB, Elliptic Curve 102.a1 (Cremona label 102a1)
Xinrong Ma, Magic determinants of Somos sequences and theta functions, Discrete Mathematics 310.1 (2010): 1-5.
J. L. Malouf, An integer sequence from a rational recursion, Discr. Math. 110 (1992), 257-261.
J. Propp, The Somos Sequence Site
J. Propp, The 2002 REACH tee-shirt
R. M. Robinson, Periodicity of Somos sequences, Proc. Amer. Math. Soc., 116 (1992), 613-619.
Matthew Christopher Russell, Using experimental mathematics to conjecture and prove theorems in the theory of partitions and commutative and non-commutative recurrences, PhD Dissertation, Mathematics Department, Rutgers University, May 2016; see also.
Vladimir Shevelev and Peter J. C. Moses, On a sequence of polynomials with hypothetically integer coefficients, arXiv preprint arXiv:1112.5715 [math.NT], 2011.
Michael Somos, Somos 6 Sequence
Michael Somos, Brief history of the Somos sequence problem
D. E. Speyer, Perfect matchings and the octahedral recurrence, arXiv:math/0402452 [math.CO], 2004.
Alex Stone, The Astonishing Behavior of Recursive Sequences, Quanta Magazine, Nov 16 2023, 13 pages.
Andrei K. Svinin, Volterra map and related recurrences, arXiv:2502.06908 [nlin.SI], 2025. See p. 27.
Peter H. van der Kamp, Somos-4 and Somos-5 are arithmetic divisibility sequences, arXiv:1505.00194 [math.NT], 2015.
A. J. van der Poorten, Elliptic curves and continued fractions, arXiv:math/0403225 [math.NT], 2004.
A. J. van der Poorten, Elliptic curves and continued fractions, J. Int. Sequences, Volume 8, no. 2 (2005), article 05.2.5.
A. J. van der Poorten, Recurrence relations for elliptic sequences: : every Somos 4 is a Somos k, arXiv:math/0412293 [math.NT], 2004.
A. J. van der Poorten, Hyperelliptic curves, continued fractions and Somos sequences, arXiv:math/0608247 [math.NT], 2006.
Eric Weisstein's World of Mathematics, Somos Sequence.
D. Zagier, Problems posed at the St Andrews Colloquium, 1996
Index entries for two-way infinite sequences

Cf. A006720, A006722, A006723, A048736.

a006721 n = a006721_list !! n
a006721_list = [1,1,1,1,1] ++
  zipWith div (foldr1 (zipWith (+)) (map b [1..2])) a006721_list
  where b i = zipWith (*) (drop i a006721_list) (drop (5-i) a006721_list)
-- Reinhard Zumkeller, Jan 22 2012

I:=[1,1,1,1,1]; [n le 5 select I[n] else (Self(n-1) * Self(n-4) + Self(n-2) * Self(n-3)) div Self(n-5): n in [1..30]]; // Vincenzo Librandi, May 18 2015

for n from 0 to 4 do a[n]:= 1 od:
for n from 5 to 50 do a[n]:=(a[n-1] * a[n-4] + a[n-2] * a[n-3]) / a[n-5] od:
seq(a[i],i=0..50); # Robert Israel, May 19 2015

a[0] = a[1] = a[2] = a[3] = a[4] = 1; a[n_] := a[n] = (a[n - 1] a[n - 4] + a[n - 2] a[n - 3])/a[n - 5]; Array[a, 27, 0] (* Robert G. Wilson v, Aug 15 2010 *)
a[ n_] := If[ Abs [n - 2] < 3, 1, If[ n < 0, a[4 - n], a[n] = (a[n - 1] a[n - 4] + a[n - 2] a[n - 3]) / a[n - 5]]]; (* Michael Somos, Jul 15 2011 *)
RecurrenceTable[{a[0]==a[1]==a[2]==a[3]==a[4]==1,a[n]==(a[n-1]a[n-4]+ a[n-2]a[n-3])/a[n-5]},a,{n,30}] (* Harvey P. Dale, Dec 25 2011 *)

{a(n) = if( abs(n-2) < 3, 1, if( n<0, a(4-n), (a(n-1) * a(n-4) + a(n-2) * a(n-3)) / a(n-5)))}; /* Michael Somos, Jul 15 2011 */

{a(n) = my(E = ellinit([1, 1, 0, -2, 0]), P = [2, 2], T = [0, 0]); if(n == 2, 1, n = abs(n-2); sqrtint(denominator(elladd(E, T, ellmul(E, P, n))[1])))}; /* Michael Somos, Oct 29 2022 */

from gmpy2 import divexact
A006721 = [1,1,1,1,1]
for n in range(5,1001):
    A006721.append(int(divexact(A006721[n-1]*A006721[n-4]+A006721[n-2]*A006721[n-3], A006721[n-5]))) # Chai Wah Wu, Aug 15 2014

Comments from Andrew Hone, Aug 24 2004: "Both the even terms b(n)=a(2n) and odd terms b(n)=a(2n+1) satisfy the fourth-order recurrence b(n)=(b(n-1)*b(n-3)+8*b(n-2)^2)/b(n-4).
"Hence the general formula is a(2n)=A*B^n*sigma(c+n*k)/sigma(k)^(n^2), a(2n+1)=D*E^n*sigma(f+n*k)/sigma(k)^(n^2) where sigma is the Weierstrass sigma function associated to the elliptic curve y^2=4*x^3-(121/12)*x+845/216 (this is birationally equivalent to the minimal model V^2+U*V+6*V=U^3+7*U^2+12*U given by van der Poorten).
"The real/imaginary half-periods of the curve are w1=1.181965956, w3=0.973928783*I and the constants are A=0.142427718-1.037985022*I, B=0.341936209+0.389300717*I, c=0.163392411+w3, k=1.018573545, D=-0.363554228-0.803200610*I, E=0.644801269+0.734118205*I, f=c+k/2-w1 all to 9 decimal places."
a(4 - n) = a(n). a(n+2) * a(n-2) = 2 * a(n+1) * a(n-1) - a(n)^2 if n is even. a(n+2) * a(n-2) = 3 * a(n+1) * a(n-1) - a(n)^2 if n is odd.

a(26)-a(27) from Robert G. Wilson v, Aug 15 2010

Definition corrected by Chai Wah Wu, Aug 15 2014

A011848 a(n) = floor(binomial(n, 2)/2).

Original entry on oeis.org

0, 0, 0, 1, 3, 5, 7, 10, 14, 18, 22, 27, 33, 39, 45, 52, 60, 68, 76, 85, 95, 105, 115, 126, 138, 150, 162, 175, 189, 203, 217, 232, 248, 264, 280, 297, 315, 333, 351, 370, 390, 410, 430, 451, 473, 495, 517, 540, 564, 588, 612, 637, 663, 689, 715, 742, 770, 798

Offset: 0

N. J. A. Sloane, Dec 11 1996

Column sums of an array of the odd numbers repeatedly shifted 4 places to the right:
1 3 5 7  9 11 13 15 17...
         1  3  5  7  9...
                     1...
.........................
-------------------------
1 3 5 7 10 14 18 22 27...
Floor of the area under the polygon connecting the lattice points (n, floor(n/2)) from 0..n. - Wesley Ivan Hurt, Jun 09 2014
Beginning with a(4)=3, the sequence might be called the "off-axis" Ulam-Spiral numbers because they are the numbers in ascending order on the horizontal and vertical spokes (heading outward) starting with the first turning points on the spiral (i.e., 3, 5, 7 and 10). That is, starting with: 3 (upward); 5 (leftward); 7 (downward) and 10 (rightward). These are A033991 (starting at a(1)), A007742 (starting at a(1)), A033954 (starting at a(1)) and A001107 (starting at a(2)), respectively. These quadri-sections are summarized in the formulas of Sep 26 2015. - Bob Selcoe, Oct 05 2015
Conjecture: For n = 2, a(n) is the greatest k such that A123663(k) < A000217(n - 2). - Peter Kagey, Nov 18 2016
a(n) is also the matching number of the n-triangular graph, (n-1)-triangular honeycomb queen graph, (n-1)-triangular honeycomb bishop graphs, and (for n > 7) (n-1)-triangular honeycomb obtuse knight graphs. - Eric W. Weisstein, Jun 02 2017 and Apr 03 2018
After 0, 0, 0, add 1, then add 2 three times, then add 3, then add 4 three times, then add 5, etc.; i.e., first differences are A004524 = (0, 0, 0, 1, 2, 2, 2, 3, 4, 4, 4, 5, ...). - M. F. Hasler, May 09 2018
Let s(0) = s(1) = 1, s(-1) = s(2) = x, and s(n+2)*s(n-2) = s(n+1)*s(n-1) + s(n)^2 for all n in Z. Then s(n) = p(n) / x^e(n) is a Laurent polynomial in x with p(n) a polynomial with nonnegative integer coefficients of degree a(n) for all n in Z. If x = 1, then s(n) = p(n) = A006720(n+1). - Michael Somos, Mar 22 2023

			G.f. = x^3 + 3*x^4 + 5*x^5 + 7*x^6 + 10*x^7 + 14*x^8 + 18*x^9 + 22*x^10 + ...
p(0) = p(1) = 1, p(2) = 1 + x, p(3) = 1 + x + x^3, p(4) = 1 + 2*x + 2*x^2 + x^3 + x^5. - _Michael Somos_, Mar 22 2023

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Kyu-Hwan Lee and Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.
Eric Weisstein's World of Mathematics, Matching Number.
Eric Weisstein's World of Mathematics, Triangular Graph.
Index entries for linear recurrences with constant coefficients, signature (3,-4,4,-3,1).

A column of triangle A011857.
First differences are in A004524.
Cf. A007318, A033991, A007742, A033954, A001107, A006720, A035608 (bisection), A156859 (bisection).

List([0..60],n->Int(Binomial(n,2)/2)); # Muniru A Asiru, Apr 05 2018

a011848 n = if n < 2 then 0 else flip div 2 $ a007318 n 2
-- Reinhard Zumkeller, Mar 04 2015

[ Floor(n*(n-1)/4) : n in [0..50] ]; // Wesley Ivan Hurt, Jun 09 2014

seq(floor(binomial(n,2)/2), n=0..57); # Zerinvary Lajos, Jan 12 2009

Table[Floor[n (n - 1)/4], {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, Jun 28 2011 *)
CoefficientList[Series[x^3/((1 + x^2) (1 - x)^3), {x, 0, 70}], x] (* Vincenzo Librandi, Jun 21 2013 *)
LinearRecurrence[{3, -4, 4, -4, 1}, {0, 0, 1, 3, 5}, {0, 20}] (* Eric W. Weisstein, Jun 02 2017 *)
Table[Floor[Binomial[n, 2]/2], {n, 0, 20}] (* Eric W. Weisstein, Jun 02 2017 *)
Table[1/4 (-1 + (-1 + n) n + Cos[n Pi/2] + Sin[n Pi/2]), {n, 0, 20}] (* Eric W. Weisstein, Jun 02 2017 *)
Floor[Binomial[Range[0, 20], 2]/2] (* Eric W. Weisstein, Apr 03 2018 *)

PARI
```
a(n) = binomial(n, 2)\2;
```

vector(100, n, n--; floor(n*(n-1)/4)) \\ Altug Alkan, Sep 30 2015

def a(n): return n*(n-1)//4 # Christoph B. Kassir, Oct 07 2022

[floor(binomial(n,2)/2) for n in range(0,58)] # Zerinvary Lajos, Dec 01 2009

G.f.: x^3*(1-x^2)/((1-x)^3*(1-x^4)).
G.f.: x^3/((1+x^2)*(1-x)^3). - Jon Perry, Mar 31 2004
a(n) = +3*a(n-1) -4*a(n-2) +4*a(n-3) -3*a(n-4) +a(n-5). - R. J. Mathar, Apr 15 2010
a(n) = floor((n/(1+e^(1/n)))^2). - Richard R. Forberg, Jun 19 2013
a(n) = floor(n*(n-1)/4). - T. D. Noe, Jun 20 2013
a(n) = (1/4) * ( n^2 - n - 1 + (-1)^floor(n/2) ). - Ralf Stephan, Aug 11 2013
a(n) = A054925(n) - A133872(n+2). - Wesley Ivan Hurt, Jun 09 2014
a(4*n) = A033991(n). a(4*n+1) = A007742(n). a(4*n+2) = A033954(n). a(4*n+3) = A001107(n+1). - Bob Selcoe, Sep 26 2015
E.g.f.: (sin(x) + cos(x) + (x^2 - 1)*exp(x))/4. - Ilya Gutkovskiy, Nov 18 2016
A054925(n) = a(-n). A035608(n) = a(2*n+1). Wesley Ivan Hurt, Jun 09 2014
A156859(n) = a(2*n+2). - Michael Somos, Nov 18 2016
Euler transform of length 4 sequence [ 3, -1, 0, 1]. - Michael Somos, Nov 18 2016
From Amiram Eldar, Mar 18 2022: (Start)
Sum_{n>=3} 1/a(n) = 40/9 - 2*Pi/3.
Sum_{n>=3} (-1)^(n+1)/a(n) = 32/9 - 4*log(2). (End)
0 = a(n+2)*(a(n)*(a(n) -6*a(n+1) +4*a(n+2)) +a(n+1)*(8*a(n+1) -10*a(n+2)) + 3*a(n+2)^2) +a(n+3)*(a(n)*(+a(n) -2*a(n+1)) +a(n+2)*(2*a(n+1) -a(n+2))) for all n in Z. - Michael Somos, Mar 22 2023
2*a(n) + 2*a(n-2) = (n-1)*(n-2). - R. J. Mathar, Feb 12 2024

A048736 Dana Scott's sequence: a(n) = (a(n-2) + a(n-1) * a(n-3)) / a(n-4), a(0) = a(1) = a(2) = a(3) = 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 5, 13, 22, 41, 111, 191, 361, 982, 1693, 3205, 8723, 15042, 28481, 77521, 133681, 253121, 688962, 1188083, 2249605, 6123133, 10559062, 19993321, 54419231, 93843471, 177690281, 483649942, 834032173, 1579219205, 4298430243, 7412446082, 14035282561, 38202222241, 65877982561

Offset: 0

David Johnson-Davies

The recursion has the Laurent property. If a(0), a(1), a(2), a(3) are variables, then a(n) is a Laurent polynomial (a rational function with a monic monomial denominator). - Michael Somos, Feb 05 2012

A generalization is if the recursion is modified to a(n) = (a(n-2) + a(n-1) * b*a(n-3)) / a(n-4) where b is a constant, and with arbitrary nonzero initial values, (a(0), a(1), a(2), a(3)), then a(n) = c*(a(n-3) - a(n-6)) + a(n-9) for all n in Z where c is another constant. - Michael Somos, Oct 28 2021

			G.f. = 1 + x + x^2 + x^3 + 2*x^4 + 3*x^5 + 13*x^6 + 22*x^7 + 41*x^8 + 111*x^9 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..3165 (first 501 terms from T. D. Noe)
Joshua Alman, Cesar Cuenca, and Jiaoyang Huang, Laurent phenomenon sequences, Journal of Algebraic Combinatorics 43(3) (2015), 589-633.
Hal Canary, The Dana Scott Recurrence [From _Jaume Oliver Lafont_, Sep 25 2009]
S. Fomin and A. Zelevinsky, The Laurent phenomenon, arXiv:math/0104241 [math.CO], 2001.
S. Fomin and A. Zelevinsky, The Laurent Phenomenon, Advances in Applied Mathematics, 28 (2002), 119-144.
David Gale, The strange and surprising saga of the Somos sequences, Math. Intelligencer 13(1) (1991), pp. 40-42.
D. Gale, Tracking the Automatic Ant And Other Mathematical Explorations, A Collection of Mathematical Entertainments Columns from The Mathematical Intelligencer, Springer, 1998, p. 4.
Matthew Christopher Russell, Using experimental mathematics to conjecture and prove theorems in the theory of partitions and commutative and non-commutative recurrences, PhD Dissertation, Mathematics Department, Rutgers University, May 2016.
Eric Weisstein's World of Mathematics, Laurent Polynomial
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (0,0,10,0,0,-10,0,0,1)

Cf. A006720, A006721, A006722, A006723, A092420, A072881.
Cf. A192241, A192242 (primes and where they occur).
Cf. A276531.

a048736 n = a048736_list !! n
a048736_list = 1 : 1 : 1 : 1 :
   zipWith div
     (zipWith (+)
       (zipWith (*) (drop 3 a048736_list)
                    (drop 1 a048736_list))
       (drop 2 a048736_list))
     a048736_list
-- Reinhard Zumkeller, Jun 26 2011

I:=[1,1,1,1]; [n le 4 select I[n] else (Self(n-2) + Self(n-1)*Self(n-3)) / Self(n-4): n in [1..30]]; // G. C. Greubel, Feb 20 2018

P:=proc(q) local n,v; v:=[1,1,1,1]; for n from 1 to q do
v:=[op(v),(v[-2]+v[-1]*v[-3])/v[-4]] od: op(v); end: P(35); # Paolo P. Lava, Aug 24 2025

RecurrenceTable[{a[0] == a[1] == a[2] == a[3] == 1, a[n] == (a[n - 2] + a[n - 1]a[n - 3])/a[n - 4]}, a[n], {n, 40}] (* or *) LinearRecurrence[{0, 0, 10, 0, 0, -10, 0, 0, 1}, {1, 1, 1, 1, 2, 3, 5, 13, 22}, 41] (* Harvey P. Dale, Oct 22 2011 *)

Vec((1+x+x^2-9*x^3-8*x^4-7*x^5+5*x^6+3*x^7+2*x^8) / (1-10*x^3+10*x^6-x^9)+O(x^99)) \\ Charles R Greathouse IV, Jul 01 2011

a(n) = 9*a(n-3) - a(n-6) - 3 - ( ceiling(n/3) - floor(n/3) ), with a(0) = a(1) = a(2) = a(3) = 1, a(4) = 2, a(5) = 3. - Michael Somos
From Jaume Oliver Lafont, Sep 17 2009: (Start)
a(n) = 10*a(n-3) - 10*a(n-6) + a(n-9).
G.f.: (1 + x + x^2 - 9*x^3 - 8*x^4 - 7*x^5 + 5*x^6 + 3*x^7 + 2*x^8)/(1 - 10*x^3 + 10*x^6 - x^9). (End)
a(n) = a(3-n) for all n in Z. - Michael Somos, Feb 05 2012

More terms from Michael Somos

cp's OEIS Frontend

A161314 Number of partitions of n into Somos-4 sequence numbers A006720 where every part appears at least 9 times.

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A161315 Number of partitions of n into Somos-4 sequence numbers A006720 where every part appears at least 10 times.

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A161316 Number of partitions of n into Somos-4 sequence numbers A006720 where every part appears at least 11 times.

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A161317 Number of partitions of n into Somos-4 sequence numbers A006720 where every part appears at least 12 times.

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A161318 Number of partitions of n into Somos-4 sequence numbers A006720 where every part appears at least 13 times.

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A064185 a(n) = (b(n+3) * b(n-1) * a(n-1)^2 - 1) / b(n+2) where b(n) = A006720(n).

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A038754 a(2n) = 3^n, a(2n+1) = 2*3^n.

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A006721 Somos-5 sequence: a(n) = (a(n-1) * a(n-4) + a(n-2) * a(n-3)) / a(n-5), with a(0) = a(1) = a(2) = a(3) = a(4) = 1.

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A011848 a(n) = floor(binomial(n, 2)/2).

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