A025192 -id:A025192 - cp's OEIS frontend

A135293 Differences between successive numbers whose sum of digits in base 3 is 2.

2, 2, 2, 4, 2, 6, 10, 2, 6, 18, 28, 2, 6, 18, 54, 82, 2, 6, 18, 54, 162, 244, 2, 6, 18, 54, 162, 486, 730, 2, 6, 18, 54, 162, 486, 1458, 2188, 2, 6, 18, 54, 162, 486, 1458, 4374, 6562, 2, 6, 18, 54, 162, 486, 1458, 4374, 13122

Offset: 0

Adam Shelly (adam.shelly(AT)gmail.com), Dec 04 2007, Dec 05 2007

First differences of A052216 when the entries in that sequence are interpreted as base 3 numbers.

Can be regarded as a triangle, where T(0,0)=2, T(n+1,0) = T(n,0)+T(n,n), T(n+1,m) = T(n,m) for 0 < m <= n and T(n+1,n+1) = sum of T(n+1,0..n)

			triangle begins:
2
2 2
4 2 6
10 2 6 18
28 2 6 18 54
82 2 6 18 54 162
244 2 6 18 54 162 486.

G. C. Greubel, Table of n, a(n) for the first 50 rows

Cf. A052216.

T[0, 0] := 2; T[n_, 0] := 3^(n - 1) + 1; T[n_, m_] := 2*3^(m - 1); Table[T[n, m], {n, 0, 5}, {m, 0, n}] (* G. C. Greubel, Oct 09 2016 *)
Join[{2},Differences[Select[Range[50000],Total[IntegerDigits[#,3]]==2&]]] (* Harvey P. Dale, Jul 04 2019 *)

T(n,m) = 2*3^(m-1) = A025192(m) for m>0. T(n,0) = 2*A124302(n). - Franklin T. Adams-Watters, Sep 29 2011

Edited by Franklin T. Adams-Watters, Sep 29 2011

A160760 Triangle read by rows, binomial transform of an infinite lower triangular Toeplitz matrix with A078008 in every column.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 9, 5, 3, 1, 27, 14, 8, 4, 1, 81, 41, 22, 12, 5, 1, 243, 122, 63, 34, 17, 6, 1, 729, 365, 185, 97, 51, 23, 7, 1, 2187, 1094, 550, 282, 148, 74, 30, 8, 1, 6561, 3281, 1644, 832, 430, 222, 104, 38, 9, 1

Offset: 0

Gary W. Adamson, May 25 2009

Row sums = A025192: (1, 2, 6, 18, 54, 162, 486, 1458,...).

A triangle formed like Pascal's triangle, but with 3^n for n>=0 on the left border instead of 1. - Boris Putievskiy, Aug 19 2013

			First few rows of the triangle =
     1;
     1,    1;
     3,    2,    1;
     9,    5,    3,   1;
    27,   14,    8,   4,   1;
    81,   41,   22,  12,   5,   1;
   243,  122,   63,  34,  17,   6,   1;
   729,  365,  185,  97,  51,  23,   7,  1;
  2187, 1094,  550, 282, 148,  74,  30,  8, 1;
  6561, 3281, 1644, 832, 430, 222, 104, 38, 9, 1;
...

Cf. A078008, A025192.

A007318 * an infinite lower triangular Toeplitz matrix with A078008 in every column: (1, 0, 2, 2, 6, 10, 22, 42, 86,...).

Closed-form formula for arbitrary left and right borders of Pascal like triangle see A228196. - Boris Putievskiy, Aug 19 2013

T(7,4) corrected by Georg Fischer, Oct 08 2021

A168570 Exponent of 3 in 2^n - 1.

Original entry on oeis.org

0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 4, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0

Offset: 1

Martins Opmanis, Nov 30 2009

nonn

Records: a(A025192(n)) = n and a(k) < n for k < A025192(n). [Joerg Arndt, Apr 07 2014]

			For n=6, 2^6 - 1 = 63. Greatest divisor of 63 which is a power of 3 is 9 (3^2).

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A051064 (without the zeros).

a:= n-> padic[ordp](2^n-1, 3):
seq(a(n), n=1..120);  # Alois P. Heinz, Mar 27 2017

Table[IntegerExponent[2^n - 1, 3], {n, 100}] (* T. D. Noe, Apr 13 2014 *)

vector(100,n,valuation(2^n-1,3)) /* Joerg Arndt, Jun 13 2011 */

A196731 Expansion of g.f. (1-x)/(1-12*x).

Original entry on oeis.org

1, 11, 132, 1584, 19008, 228096, 2737152, 32845824, 394149888, 4729798656, 56757583872, 681091006464, 8173092077568, 98077104930816, 1176925259169792, 14123103110037504, 169477237320450048, 2033726847845400576, 24404722174144806912, 292856666089737682944, 3514279993076852195328

Offset: 0

Philippe Deléham, Oct 05 2011

Index entries for linear recurrences with constant coefficients, signature (12).

Cf. A002001, A005054, A025192, A052934, A055272, A055274, A055276, A193722.

a:= n-> ceil(11*12^(n-1)):
seq(a(n), n=0..20);  # Alois P. Heinz, Mar 25 2025

Join[{1},NestList[12#&,11,20]] (* Harvey P. Dale, Sep 19 2018 *)

a(n)=if(n,11*12^n--,1) \\ Charles R Greathouse IV, Oct 05 2011

a(n)=(11+!n)*12^(n-1)  \\ M. F. Hasler, Oct 05 2011

a(n) = Sum_{k=0..n} A193722(n,k)*9^(n-k).
a(n+1) = 12*a(n) for n > 0. - M. F. Hasler, Oct 05 2011
From Elmo R. Oliveira, Mar 18 2025: (Start)
a(n) = 11*12^(n-1) with a(0)=1.
E.g.f.: (11*exp(12*x) + 1)/12. (End)

More terms from Elmo R. Oliveira, Mar 25 2025

A199570 Table, each row contains the previous sequence in odd columns and the row number in even columns.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 3, 2, 3, 1, 4, 1, 4, 2, 4, 1, 4, 3, 4, 1, 4, 3, 4, 2, 4, 3, 4, 1, 5, 1, 5, 2, 5, 1, 5, 3, 5, 1, 5, 3, 5, 2, 5, 3, 5, 1, 5, 4, 5, 1, 5, 4, 5, 2, 5, 4, 5, 1, 5, 4, 5, 3, 5, 4, 5, 1, 5, 4, 5, 3, 5, 4, 5, 2, 5, 4, 5, 3, 5, 4, 5

Offset: 1

Franklin T. Adams-Watters, Nov 08 2011

			The table starts:
  1
  1 2
  1 3 1 3 2 3
  1 4 1 4 2 4 1 4 3 4 1 4 3 4 2 4 3 4
  ...

John Tyler Rascoe, Rows n = 1..9 of table, flattened

Cf. A025192 (row lengths), A070940.

n=4;v=vector(3^n);v[1]=1;for(k=1,n,for(i=(s=3^(k-1))+1,3^k,v[i]=if((i-s)%2,v[(i-s+1)\2],k+1)));v

def A199570_list(row):
    A = [1]
    for i in range(2,row+1):
        z = 2*(3**(i-2))
        for j in range(1,z+1):
            if j%2 != 0: A.append(A[int((j-1)/2)])
            else: A.append(i)
    return(A) # John Tyler Rascoe, Feb 19 2023

A208736 Number of nonisomorphic graded posets with 0 and 1 and non-uniform Hasse graph of rank n, with exactly 2 elements of each rank level between 0 and 1.

Original entry on oeis.org

0, 0, 0, 1, 5, 22, 91, 361, 1392, 5265, 19653, 72694, 267179, 977593, 3565600, 12975457, 47142021, 171075606, 620303547, 2247803785, 8141857808, 29481675889, 106728951109, 386314552438, 1398132674955, 5059626441177, 18308871648576, 66249898660801

Offset: 0

David Nacin, Mar 01 2012

Uniform used in the sense of Retakh, Serconek and Wilson. We use Stanley's definition of graded poset: all maximal chains have the same length n (which also implies all maximal elements have maximal rank.)

R. Stanley, Enumerative combinatorics. Vol. 1, Cambridge University Press, Cambridge, 1997, pp. 96-100.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
V. Retakh, S. Serconek, and R. Wilson, Hilbert Series of Algebras Associated to Directed Graphs and Order Homology, arXiv:1010.6295 [math.RA], 2010-2011.
Wikipedia, Graded poset
Index entries for linear recurrences with constant coefficients, signature (8,-21,20,-5).

Cf. A208737, A206901, A206902, A206947-A206950, A001906, A025192, A081567, A124302, A124292, A088305, A086405, A012781.

Join[{0, 0}, LinearRecurrence[{8, -21, 20, -5}, {0, 1, 5, 22}, 40]]

def a(n, d={0:0,1:0,2:0,3:1,4:5,5:22}):
    if n in d:
        return d[n]
    d[n]=8*a(n-1) - 21*a(n-2) + 20*a(n-3) - 5*a(n-4)
    return d[n]

a(n) = 8*a(n-1) - 21*a(n-2) + 20*a(n-3) - 5*a(n-4), a(2) = 0, a(3) = 1, a(4) = 5, a(5) = 22.
G.f.: (x^3 - 3*x^4 + 3*x^5)/(1 - 8*x + 21*x^2 - 20*x^3 + 5*x^4); (x^3 * (1 - 3*x + 3*x^2))/((1 - 3*x + x^2)*(1 - 5*x + 5*x^2)) .
a(n) = A081567(n-2) - A001519(n-1).

A208737 Number of nonisomorphic graded posets with 0 and 1 and non-uniform Hasse graph of rank n, with no 3-element antichain.

Original entry on oeis.org

0, 0, 0, 1, 7, 37, 175, 778, 3325, 13837, 56524, 227866, 909832, 3607294, 14227447, 55894252, 218937532, 855650749, 3338323915, 13007422705, 50631143323, 196928737582, 765495534433, 2974251390529, 11552064922624, 44856304154086

Offset: 0

David Nacin, Mar 01 2012

Uniform used in the sense of Retakh, Serconek and Wilson. We use Stanley's definition of graded poset: all maximal chains have the same length n (which also implies all maximal elements have maximal rank.)

R. Stanley, Enumerative combinatorics. Vol. 1, Cambridge University Press, Cambridge, 1997, pp. 96-100.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
V. Retakh, S. Serconek, and R. Wilson, Hilbert Series of Algebras Associated to Directed Graphs and Order Homology, arXiv:1010.6295 [math.RA], 2010-2011.
Wikipedia, Graded poset
Index entries for linear recurrences with constant coefficients, signature (10,-36,57,-39,9).

Cf. A208736, A206901, A206902, A206947-A206950, A001906, A025192, A081567, A124302, A124292, A088305, A086405, A012781.

Join[{0}, LinearRecurrence[{10, -36, 57, -39, 9}, {0, 0, 1, 7, 37}, 40]]

def a(n, d={0:0,1:0,2:0,3:1,4:7,5:37}):
    if n in d:
        return d[n]
    d[n]=10*a(n-1) - 36*a(n-2) + 57*a(n-3) - 39*a(n-4) + 9*a(n-5)
    return d[n]

a(n) = 10*a(n-1) - 36*a(n-2) + 57*a(n-3) - 39*a(n-4) + 9*a(n-5), a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 7, a(5) = 37.
G.f: (x^3 - 3*x^4 + 3*x^5)/(1 - 10*x + 36*x^2 - 57*x^3 + 39*x^4 - 9*x^5); (x^3*(1 - 3*x + 3*x^2)) / ((1 - x) (1 - 3*x) (1 - 6*x + 9*x^2 - 3*x^3)).
a(n) = A124292(n) - A124302(n).

A355387 Number of ways to choose a distinct subsequence of an integer composition of n.

Original entry on oeis.org

1, 2, 5, 14, 37, 98, 259, 682, 1791, 4697, 12303, 32196, 84199, 220087, 575067, 1502176, 3923117, 10244069, 26746171, 69825070, 182276806, 475804961, 1241965456, 3241732629, 8461261457, 22084402087, 57640875725, 150442742575, 392652788250, 1024810764496

Offset: 0

Gus Wiseman, Jul 04 2022

nonn

By "distinct" we mean equal subsequences are counted only once. For example, the pair (1,1)(1) is counted only once even though (1) is a subsequence of (1,1) in two ways. The version with multiplicity is A025192.

			The a(3) = 14 pairings of a composition with a chosen subsequence:
  (3)()     (3)(3)
  (21)()    (21)(1)   (21)(2)    (21)(21)
  (12)()    (12)(1)   (12)(2)    (12)(12)
  (111)()   (111)(1)  (111)(11)  (111)(111)

Christian Sievers, Table of n, a(n) for n = 0..2000

For partitions we have A000712, composable A339006.
The homogeneous version is A011782, without containment A000302.
With multiplicity we have A025192, for partitions A070933.
The strict case is A032005.
The case of strict subsequences is A236002.
The composable case is A355384, homogeneous without containment A355388.
A075900 counts compositions of each part of a partition.
A304961 counts compositions of each part of a strict partition.
A307068 counts strict compositions of each part of a composition.
A336127 counts compositions of each part of a strict composition.
Cf. A011782, A022811, A032020, A063834, A133494, A181591, A323583, A331330, A336128, A336130, A336139, A355382, A355383.

Table[Sum[Length[Union[Subsets[y]]],{y,Join@@Permutations/@IntegerPartitions[n]}],{n,0,6}]

lista(n)=my(f=sum(k=1,n,(x^k+x*O(x^n))/(1-x/(1-x)+x^k)));Vec((1-x)/((1-2*x)*(1-f))) \\ Christian Sievers, May 06 2025

G.f.: (1-x)/((1-2*x)*(1-f)) where f = Sum_{k>=1} x^k/(1-x/(1-x)+x^k) is the generating function for A331330. - Christian Sievers, May 06 2025

a(16) and beyond from Christian Sievers, May 06 2025

A049299 a(n) = Product_{k = 0..n-1} (a(k) + a(n-1-k)), with a(0) = 1.

Original entry on oeis.org

1, 2, 9, 400, 19456921, 1101216948902114953248, 76796373204229717290826972582321984854855228022915711475735049

Offset: 0

Leroy Quet

			a(3)=400 because 400=(1+9)*(2+2)*(9+1).

Andrew Howroyd, Table of n, a(n) for n = 0..8

Cf. A000108 (Catalan numbers) where a(0) = 1, a(n) = Sum_{k=0..n-1} a(k)*a(n-k), A000012 (constant 1) where a(0) = 1, a(n) = Product_{k=0..n-1} a(k)*a(n-k) and A025192 (2*3^(n-1)) where a(0) = 1, a(n) = Sum_{k=0..n-1} a(k)+a(n-k). - Henry Bottomley, May 16 2000

a(n)={my(v=vector(n+1)); for(n=1, #v, v[n]=prod(k=1, n-1, v[k]+v[n-k])); v[#v]} \\ Andrew Howroyd, Jan 02 2020

lim_{m -> oo} log(a(m+1))/log(a(m)) exists and equals 3. - Roland Bacher, Sep 06 2004.

Offset corrected and terms a(6) and beyond from Andrew Howroyd, Jan 02 2020

A136523 Triangle T(n,k) = A053120(n,k) + A053120(n-1,k), read by rows.

Original entry on oeis.org

1, 1, 1, -1, 1, 2, -1, -3, 2, 4, 1, -3, -8, 4, 8, 1, 5, -8, -20, 8, 16, -1, 5, 18, -20, -48, 16, 32, -1, -7, 18, 56, -48, -112, 32, 64, 1, -7, -32, 56, 160, -112, -256, 64, 128, 1, 9, -32, -120, 160, 432, -256, -576, 128, 256, -1, 9, 50, -120, -400, 432, 1120, -576, -1280, 256, 512

Offset: 0

Roger L. Bagula, Mar 23 2008

			Triangle begins as:
   1;
   1,  1;
  -1,  1,   2;
  -1, -3,   2,    4;
   1, -3,  -8,    4,    8;
   1,  5,  -8,  -20,    8,   16;
  -1,  5,  18,  -20,  -48,   16,   32;
  -1, -7,  18,   56,  -48, -112,   32,   64;
   1, -7, -32,   56,  160, -112, -256,   64,   128;
   1,  9, -32, -120,  160,  432, -256, -576,   128, 256;
  -1,  9,  50, -120, -400,  432, 1120, -576, -1280, 256, 512;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000007, A001792, A001793, A001794, A000330, A008794, A011782.
Cf. A025192, A040000 (row sums), A053120, A057077, A081277, A109613.
Cf. A124182.

function A053120(n,k)
  if ((n+k) mod 2) eq 1 then return 0;
  elif n eq 0 then return 1;
  else return (-1)^Floor((n-k)/2)*(n/(n+k))*Binomial(Floor((n+k)/2), k)*2^k;
  end if;
end function;
A136523:= func< n,k | A053120(n,k) + A053120(n-1,k) >;
[A136523(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 26 2023

A053120[n_, k_]:= Coefficient[ChebyshevT[n,x], x, k];
T[n_, k_]:= T[n, k]= A053120[n,k] + A053120[n-1,k];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten

def A053120(n,k):
    if (n+k)%2==1: return 0
    elif n==0: return 1
    else: return floor((-1)^((n-k)//2)*(n/(n+k))*binomial((n+k)//2, k)*2^k)
def A136523(n,k): return A053120(n,k) + A053120(n-1,k)
flatten([[A136523(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 26 2023

T(n, k) = A053120(n,k) + A053120(n-1,k).
Sum_{k=0..n} T(n, k) = A040000(n).
From G. C. Greubel, Jul 26 2023: (Start)
T(n, 0) = A057077(n).
T(n, 1) = (-1)^floor((n-1)/2) * A109613(n-1).
T(n, 2) = (-1)^floor((n-2)/2) * A008794(n-1).
T(n, 3) = (-1)^floor((n+1)/2) * A000330(n-1).
T(n, n) = A011782(n).
T(n, n-1) = A011782(n-1).
T(n, n-2) = -A001792(n-2).
T(n, n-4) = A001793(n-3).
T(n, n-6) = -A001794(n-6).
Sum_{k=0..n} (-1)^k*T(n,k) = A000007(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A000007(n) + [n=1].
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = (-1)^floor(n/2)*A025192(floor(n/2)). (End)

Edited by G. C. Greubel, Jul 26 2023

cp's OEIS Frontend

A135293 Differences between successive numbers whose sum of digits in base 3 is 2.

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A160760 Triangle read by rows, binomial transform of an infinite lower triangular Toeplitz matrix with A078008 in every column.

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A168570 Exponent of 3 in 2^n - 1.

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A196731 Expansion of g.f. (1-x)/(1-12*x).

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A199570 Table, each row contains the previous sequence in odd columns and the row number in even columns.

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A208736 Number of nonisomorphic graded posets with 0 and 1 and non-uniform Hasse graph of rank n, with exactly 2 elements of each rank level between 0 and 1.

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A208737 Number of nonisomorphic graded posets with 0 and 1 and non-uniform Hasse graph of rank n, with no 3-element antichain.

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