A000931 -id:A000931 - cp's OEIS frontend

A176971 Expansion of (1+x)/(1+x-x^3) in powers of x.

1, 0, 0, 1, -1, 1, 0, -1, 2, -2, 1, 1, -3, 4, -3, 0, 4, -7, 7, -3, -4, 11, -14, 10, 1, -15, 25, -24, 9, 16, -40, 49, -33, -7, 56, -89, 82, -26, -63, 145, -171, 108, 37, -208, 316, -279, 71, 245, -524, 595

Offset: 0

Roger L. Bagula, Apr 29 2010

Except for signs the sequence is the essentially same as A078013, A050935 and A104769.

Padovan sequence extended to negative indices. - Hugo Pfoertner, Jul 16 2017

			G.f. = 1 + x^3 - x^4 + x^5 - x^7 + 2*x^8 - 2*x^9 + x^10 + x^11 - 3*x^12 + ...

G. C. Greubel, Table of n, a(n) for n = 0..5000
Taras Goy and Mark Shattuck, Toeplitz-Hessenberg determinant formulas for the sequence F_n-1, Online J. Anal. Comb. (2025) Vol. 19, Paper 1, 1-26. See p. 12.
Yüksel Soykan, Summing Formulas For Generalized Tribonacci Numbers, arXiv:1910.03490 [math.GM], 2019.
Wikipedia, Padovan sequence.
Index entries for linear recurrences with constant coefficients, signature (-1,0,1). [From _R. J. Mathar_, Apr 30 2010]

Cf. A000931, A005251, A050935, A078013, A104769, A182097.

m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+x)/(1+x-x^3))); // G. C. Greubel, Sep 25 2018

a[0] := 1; a[1] = 0; a[2] = 0;
a[n_] := a[n] = a[n - 2] + a[n - 3];
b = Table[a[n], {n, 0, 50}];
Table[b[[n]]^2 - b[[n - 1]]*b[[n + 1]], {n, 1, Length[b] - 1}]
a[ n_] := If[ n >= 0, SeriesCoefficient[ (1 + x) / (1 + x - x^3), {x, 0, n}], SeriesCoefficient[ 1 / (1 - x^2 - x^3), {x, 0, Abs@n}]]; (* Michael Somos, Dec 13 2013 *)

{a(n) = if( n>=0, polcoeff( (1 + x) / (1 + x - x^3) + x * O(x^n), n), polcoeff( 1 / (1 - x^2 - x^3) + x * O(x^-n), -n))}; /* Michael Somos, Dec 13 2013 */

a(n) = A000931(n)^2 -A000931(n-1)*A000931(n+1).
a(n) = -a(n-1) +a(n-3). - R. J. Mathar, Apr 30 2010
a(n) = -A104769(n) - A104769(n+1). - Ralf Stephan, Aug 18 2013
G.f.: 1 / (1 - x^3 / (1 + x)). - Michael Somos, Dec 13 2013
a(n) = A182097(-n) for all n in Z. - Michael Somos, Dec 13 2013
A000931(n) = a(n)^2 - a(n-1) * a(n+1). - Michael Somos, Dec 13 2013
Binomial transform is A005251(n+1). - Michael Somos, Dec 13 2013

Deleted certain dangerous or potentially dangerous links. - N. J. A. Sloane, Jan 30 2021

A337548 Number of compositions (ordered partitions) of n into distinct parts congruent to 2 mod 3.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 2, 1, 0, 2, 1, 0, 4, 1, 6, 4, 1, 6, 6, 1, 12, 6, 1, 18, 8, 25, 24, 8, 25, 30, 10, 49, 42, 10, 73, 48, 12, 121, 60, 132, 145, 72, 134, 217, 84, 254, 265, 96, 376, 361, 114, 616, 433, 126, 858, 553, 864, 1218, 649, 882, 1580, 817, 1620, 2180, 937

Offset: 0

Ilya Gutkovskiy, Nov 22 2020

nonn

			a(15) = 6 because we have [8, 5, 2], [8, 2, 5], [5, 8, 2], [5, 2, 8], [2, 8, 5] and [2, 5, 8].

Index entries for sequences related to compositions

Cf. A000931, A016789, A032020, A032021, A035386, A262928, A337547, A339059, A339060.

nmax = 65; CoefficientList[Series[Sum[k! x^(k (3 k + 1)/2)/Product[1 - x^(3 j), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

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

A033217 Primes of form x^2 + 23*y^2.

Original entry on oeis.org

23, 59, 101, 167, 173, 211, 223, 271, 307, 317, 347, 449, 463, 593, 599, 607, 691, 719, 809, 821, 829, 853, 877, 883, 991, 997, 1097, 1117, 1151, 1163, 1181, 1231, 1319, 1451, 1453, 1481, 1553, 1613, 1669, 1697, 1787, 1789, 1867, 1871, 1879, 1889, 1913, 2027, 2053, 2143, 2309, 2339, 2347, 2381, 2393, 2423, 2539, 2647, 2677, 2693, 2707, 2741, 2819

Offset: 1

N. J. A. Sloane

Discriminant -23.
Also primes of the form x^2 + x*y + 6*y^2. - N. J. A. Sloane, Jun 02 2014
Also primes of the form x^2 - x*y + 6*y^2 with x and y nonnegative. - T. D. Noe, May 07 2005
Primes p such that X^3-X+1 is split modulo p. E.g., X^3-X+1 = (X-33)*(X-40)*(X-94) modulo 167. - Julien Freslon (julien.freslon(AT)wanadoo.fr), Feb 24 2007
It appears that, if x > 0, then tau(p) = A000594(p) == 2 (mod 23). - Comment from Jud McCranie
In fact, this sequence appears to be the same as primes p such that RamanujanTau(p) == {1,2} (mod 23). - Ray Chandler, Dec 01 2016
Excluding the first term, this sequence is the intersection of A191021 and A256567. - Arkadiusz Wesolowski, Oct 03 2021
From Amiram Eldar, Jan 10 2025: (Start)
a(2)..a(10000) are the first terms of the sequence of primes p such that tau(p) == 2 (mod 23), where tau is Ramanujan's tau function (A000594).
Moree and Noubissie (2024) proved that the following 3 conditions for a prime p are equivalent:
  1. tau(p) == 2 (mod 23).
  2. p divides A000931(p+3) where A000931 is the Padovan sequence.
  3. The number of distinct roots modulo p of the polynomial x^3 - x - 1 is 3. (End)

David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989.
Joe Roberts, Lure of the Integers, Math. Assoc. America, 1992. See pp. 158-160, "Integer 23 - the Tau function".

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Pieter Moree and Armand Noubissie, Higher reciprocity laws and ternary linear recurrence sequences, International Journal of Number Theory, 2024; arXiv preprint, arXiv:2205.06685 [math.NT], 2022.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
John Raymond Wilton, Congruence properties of Ramanujan's function τ(n), Proceedings of the London Mathematical Society 2.1 (1930): 1-10. The primes are listed in Table II.
John Raymond Wilton, Congruence properties of Ramanujan's function τ(n), annotated copy of page 8 only.

Cf. A000594, A191021, A256567. Primes in A028958.

QuadPrimes2[1, 0, 23, 10000] (* see A106856 *)
Join[{23}, nn=23; pMax=5000; Union[Reap[Do[p=x^2 + nn y^2; If[p<=pMax&&PrimeQ[p], Sow[p]], {x, Sqrt[pMax]}, {y, Sqrt[pMax/nn]}]][[2, 1]]]] (* Vincenzo Librandi, Sep 05 2016 *)

isok(p) = isprime(p) && !(kronecker(-23, p)==-1) && !polisirreducible(Mod(1, p)*(x^3-x-1)); \\ Arkadiusz Wesolowski, Oct 03 2021

isok(p) = p==23 || (isprime(p) && #polrootsmod(x^3-x-1, p)==3); \\ Arkadiusz Wesolowski, Oct 09 2021

A077954 Expansion of 1/(1-x+2*x^2-x^3) in powers of x.

Original entry on oeis.org

1, 1, -1, -2, 1, 4, 0, -7, -3, 11, 10, -15, -24, 16, 49, -7, -89, -26, 145, 108, -208, -279, 245, 595, -174, -1119, -176, 1888, 1121, -2831, -3185, 3598, 7137, -3244, -13920, -295, 24301, 10971, -37926, -35567, 51256, 84464, -53615, -171287, 20407, 309366, 97265, -501060, -386224, 713161

Offset: 0

N. J. A. Sloane, Nov 17 2002

			G.f. = 1 + x - x^2 - 2*x^3 + x^4 + 4*x^5 - 7*x^7 - 3*x^8 + 11*x^9 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
Index entries for linear recurrences with constant coefficients, signature (1,-2,1).

Cf. A000931, A005314.

a:=[1,1,-1];; for n in [4..50] do a[n]:=a[n-1]-2*a[n-2]+a[n-3]; od; a; # G. C. Greubel, Aug 07 2019

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( 1/(1-x+2*x^2-x^3) )); // G. C. Greubel, Aug 07 2019

seq(coeff(series(1/(1-x+2*x^2-x^3), x, n+1), x, n), n = 0..50); # G. C. Greubel, Aug 07 2019

a[ n_] := If[ n < 0, SeriesCoefficient[ x^3 / (1 - 2 x + x^2 - x^3), {x, 0, -n}], SeriesCoefficient[ 1 / (1 - x + 2 x^2 - x^3), {x, 0, n}]]
LinearRecurrence[{1,-2,1}, {1,1,-1}, 50] (* G. C. Greubel, Aug 07 2019 *)

{a(n) = if( n<0,  polcoeff( x^3 / (1 - 2*x + x^2 - x^3) + x * O(x^-n), -n), polcoeff( 1 / (1 - x + 2*x^2 - x^3) + x * O(x^n), n))} /* Michael Somos, Sep 18 2012 */

(1/(1-x+2*x^2-x^3)).series(x, 50).coefficients(x, sparse=False) # G. C. Greubel, Aug 07 2019

a(0)=1, a(1)=1, a(2)=-1, a(n) = a(n-1) -2*a(n-2) +a(n-3) for n>=3. - Philippe Deléham, Sep 15 2006
a(n) = A000931(-2*n). - Michael Somos, Sep 18 2012
a(n) = A005314(-n-2). - Michael Somos, Dec 13 2013
a(n) = a(n-1) - 2*a(n-2) + a(n-3) for all n in Z. - Michael Somos, Dec 13 2013

A100891 Prime Padovan numbers.

Original entry on oeis.org

2, 3, 5, 7, 37, 151, 3329, 23833, 13091204281, 3093215881333057, 1363005552434666078217421284621279933627102780881053358473, 1558877695141608507751098941899265975115403618621811951868598809164180630185566719

Offset: 1

John Lien, Jan 10 2005

nonn

Next term corresponds to Padovan(1262) and has 154 decimal digits.

Midhat J. Gazale, "Gnomon: From Pharaohs to Fractals", Princeton University Press, 1999.

Hugo Pfoertner, Table of n, a(n) for n = 1..20
Ian Stewart, Tales of a Neglected Number.
Eric Weisstein's World of Mathematics, Padovan Sequence

Cf. A000931, A122498, A291216, A291673.

Indices of prime Padovan numbers are A112882.

Rest[Select[LinearRecurrence[{0,1,1},{1,1,2},1000],PrimeQ]] (* Harvey P. Dale, Mar 31 2012 *)

More terms from Robert G. Wilson v, Jan 14 2005

A127682 Number of non-isomorphic (i.e., defined up to a rotation and a reflection) maximal independent sets of the n-cycle graph having at least one symmetry axis. Also: Number of cyclic and palindromic compositions of n in which each term is either 2 or 3.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 2, 2, 3, 2, 4, 3, 5, 4, 7, 5, 9, 7, 12, 9, 16, 12, 21, 16, 28, 21, 37, 28, 49, 37, 65, 49, 86, 65, 114, 86, 151, 114, 200, 151, 265, 200, 351, 265, 465, 351, 616, 465, 816, 616, 1081, 816, 1432, 1081, 1897, 1432, 2513, 1897, 3329, 2513, 4410, 3329

Offset: 1

Jean-Luc Marichal (jean-luc.marichal(AT)uni.lu), Jan 24 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
R. Bisdorff and J.-L. Marichal, Counting non-isomorphic maximal independent sets of the n-cycle graph, arXiv:math/0701647 [math.CO], 2007-2008 and JIS 11 (2008) 08.5.7.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1,0,1).

Cf. A000931.

Rest[CoefficientList[Series[-x^2*(x^4+x^3+x^2+x+1)/(x^6+x^4-1),{x,0,63}],x]] (* Vaclav Kotesovec, Mar 29 2014 *)
LinearRecurrence[{0,0,0,1,0,1},{0,1,1,1,1,2},70] (* Harvey P. Dale, Jul 17 2014 *)

concat(0, Vec(-x^2*(x^4+x^3+x^2+x+1)/(x^6+x^4-1) + O(x^100))) \\ Colin Barker, Mar 29 2014

a(n) = A000931(k+3) if n=2k-1 and a(n) = A000931(k+5) if n=2k.
a(n) = a(n-4) + a(n-6).
G.f.: -x^2*(x^4+x^3+x^2+x+1) / (x^6+x^4-1). - Colin Barker, Mar 29 2014

A183368 T(n,k)=Number of nXk binary arrays with no element equal to the sum mod 3 of its horizontal and vertical neighbors.

Original entry on oeis.org

1, 2, 2, 2, 3, 2, 3, 5, 5, 3, 4, 12, 18, 12, 4, 5, 21, 39, 39, 21, 5, 7, 41, 108, 138, 108, 41, 7, 9, 84, 288, 548, 548, 288, 84, 9, 12, 171, 795, 2129, 3102, 2129, 795, 171, 12, 16, 355, 2278, 8311, 17101, 17101, 8311, 2278, 355, 16, 21, 732, 6438, 32933, 95674, 137688

Offset: 1

R. H. Hardin Jan 04 2011

Table starts
..1...2.....2......3........4.........5...........7............9.............12
..2...3.....5.....12.......21........41..........84..........171............355
..2...5....18.....39......108.......288.........795.........2278...........6438
..3..12....39....138......548......2129........8311........32933.........130750
..4..21...108....548.....3102.....17101.......95674.......543837........3094889
..5..41...288...2129....17101....137688.....1107441......8969225.......72989578
..7..84...795...8311....95674...1107441....12617557....145766281.....1687526938
..9.171..2278..32933...543837...8969225...145766281...2402945556....39693410307
.12.355..6438.130750..3094889..72989578..1687526938..39693410307...936460285557
.16.732.18394.519980.17654882.594531764.19534278264.655471213213.22047188457210

			Some solutions for 6X5
..0..1..0..0..1....1..1..1..0..1....0..0..1..1..1....1..1..1..1..1
..0..0..0..0..0....1..0..1..0..0....1..0..1..0..1....1..0..1..0..1
..1..1..1..1..1....1..1..1..0..1....0..0..1..1..1....1..1..1..1..1
..1..1..0..1..1....1..1..1..0..0....0..1..1..0..1....0..0..0..0..0
..0..0..1..0..0....1..0..1..1..0....1..1..0..1..1....1..1..1..0..0
..1..0..0..0..1....1..1..1..1..0....1..1..1..1..0....1..1..1..0..1

R. H. Hardin, Table of n, a(n) for n = 1..199

Column 1 is A000931(n+6)

A183380 T(n,k)=Number of nXk binary arrays with no element equal to the mod 3 sum of its king-move neighbors.

Original entry on oeis.org

1, 2, 2, 2, 5, 2, 3, 9, 9, 3, 4, 20, 28, 20, 4, 5, 37, 68, 68, 37, 5, 7, 61, 143, 225, 143, 61, 7, 9, 144, 364, 627, 627, 364, 144, 9, 12, 289, 952, 2152, 2767, 2152, 952, 289, 12, 16, 497, 2509, 7036, 11345, 11345, 7036, 2509, 497, 16, 21, 1100, 6412, 22392, 51706, 64530

Offset: 1

R. H. Hardin Jan 04 2011

Table starts
..1....2.....2......3.......4........5..........7...........9...........12
..2....5.....9.....20......37.......61........144.........289..........497
..2....9....28.....68.....143......364........952........2509.........6412
..3...20....68....225.....627.....2152.......7036.......22392........74381
..4...37...143....627....2767....11345......51706......223151.......974387
..5...61...364...2152...11345....64530.....377217.....2204567.....12979097
..7..144...952...7036...51706...377217....2952294....22811993....176950342
..9..289..2509..22392..223151..2204567...22811993...238729120...2477358069
.12..497..6412..74381..974387.12979097..176950342..2477358069..34520238211
.16.1100.16544.241819.4317835.75859054.1369406125.25692566781.477784675604

			Some solutions for 6X5
..0..0..0..0..0....0..0..0..0..0....0..0..0..0..0....1..0..1..0..1
..1..1..0..1..0....1..1..0..0..1....0..1..0..1..0....1..1..1..1..1
..0..1..1..1..0....1..1..0..0..0....0..1..1..1..0....1..0..1..1..0
..0..1..1..0..0....0..0..0..0..0....0..0..1..0..0....0..0..0..1..1
..1..1..1..0..0....1..1..1..1..0....1..0..0..0..1....1..0..1..0..1
..0..0..1..1..0....1..1..1..1..0....1..1..1..1..1....0..0..0..1..0

R. H. Hardin, Table of n, a(n) for n = 1..179

Column 1 is A000931(n+6)

A247917 Expansion of 1 / (1 + x - x^3) in powers of x.

Original entry on oeis.org

1, -1, 1, 0, -1, 2, -2, 1, 1, -3, 4, -3, 0, 4, -7, 7, -3, -4, 11, -14, 10, 1, -15, 25, -24, 9, 16, -40, 49, -33, -7, 56, -89, 82, -26, -63, 145, -171, 108, 37, -208, 316, -279, 71, 245, -524, 595, -350, -174, 769, -1119, 945, -176, -943, 1888, -2064, 1121

Offset: 0

Michael Somos, Sep 26 2014

			G.f. = 1 - x + x^2 - x^4 + 2*x^5 - 2*x^6 + x^7 + x^8 - 3*x^9 + 4*x^10 + ...

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

Cf. A000931, A050935, A078013, A104769, A176971.

m:=60; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1 + x - x^3)));  // G. C. Greubel, Aug 04 2018

CoefficientList[Series[1/(1 + x - x^3), {x, 0, 100}], x] (* Vincenzo Librandi, Sep 27 2014 *)
LinearRecurrence[{-1,0,1},{1,-1,1},60] (* Harvey P. Dale, Apr 10 2025 *)

{a(n) = if( n<0, n = -3-n; polcoeff( 1 / (1 - x^2 - x^3) + x * O(x^n), n), polcoeff( 1 / (1 + x - x^3) + x * O(x^n), n))};

G.f.: 1 / (1 + x - x^3).
0 = a(n) - a(n+2) - a(n+3) for all n in Z.
a(-n) = A000931(n) for all n in Z.
a(n) = A176971(n+3) for all n in Z.
-a(n) = A104769(n+1) for all n in Z.
(-1)^n * a(n) = A050935(n+3) for all n in Z.
-(-1)^n * a(n) = A078013(n+3) for all n in Z.

A269175 a(n) = number of distinct k for which A269174(k) = n; number of finite predecessors for pattern encoded in the binary expansion of n in Wolfram's Rule 124 cellular automaton.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 2, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 2, 1, 0, 2, 2, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 2, 0, 0, 2, 1, 1

Offset: 0

Antti Karttunen, Feb 22 2016

nonn

At positions A000225 seems to occur the record values of this sequence: 1, 0, 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265, 351, 465, ... which seem to match with A000931 (Padovan sequence), or more exactly, with A182097 (Number of compositions (ordered partitions) into parts 2 and 3). Note that these values give also the number of predecessors for each "repunit-pattern" (2^n)-1 in Rule 110 cellular automaton, as rules 110 and 124 are mirror images of each other.

Antti Karttunen, Table of n, a(n) for n = 0..32768
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

Cf. A000225, A000931, A182097, A269174.

Cf. A269176 (indices of zeros), A269177 (of nonzeros), A269178 (of ones).

(definec (A269175 n) (let loop ((p 0) (s 0)) (cond ((> p n) s) (else (loop (+ 1 p) (+ s (if (= n (A269174 p)) 1 0))))))) ;; Very straightforward and very slow.
;; Somewhat optimized version:
(definec (A269175 n) (if (zero? n) 1 (let ((nwid-1 (- (A000523 n) 1))) (let loop ((p (if (< n 2) 0 (A000079 nwid-1))) (s 0)) (cond ((> (A000523 p) nwid-1) s) (else (loop (+ 1 p) (+ s (if (= n (A269174 p)) 1 0)))))))))

cp's OEIS Frontend

A176971 Expansion of (1+x)/(1+x-x^3) in powers of x.

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A337548 Number of compositions (ordered partitions) of n into distinct parts congruent to 2 mod 3.

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A033217 Primes of form x^2 + 23*y^2.

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A077954 Expansion of 1/(1-x+2*x^2-x^3) in powers of x.

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A100891 Prime Padovan numbers.

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A127682 Number of non-isomorphic (i.e., defined up to a rotation and a reflection) maximal independent sets of the n-cycle graph having at least one symmetry axis. Also: Number of cyclic and palindromic compositions of n in which each term is either 2 or 3.

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A183368 T(n,k)=Number of nXk binary arrays with no element equal to the sum mod 3 of its horizontal and vertical neighbors.

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