A053581 -id:A053581 - cp's OEIS frontend

A291000 p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - S - S^2 - S^3.

1, 3, 9, 26, 74, 210, 596, 1692, 4804, 13640, 38728, 109960, 312208, 886448, 2516880, 7146144, 20289952, 57608992, 163568448, 464417728, 1318615104, 3743926400, 10630080640, 30181847168, 85694918912, 243312448256, 690833811712, 1961475291648, 5569190816256

Offset: 0

Clark Kimberling, Aug 22 2017

Suppose s = (c(0), c(1), c(2),...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)*x + c(1)*x^2 + c(2)*x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x).  Taking p(S) = 1 - S gives the "INVERT" transform of s, so that p-INVERT is a generalization of the "INVERT" transform (e.g., A033453).
In the following guide to p-INVERT sequences using s = (1,1,1,1,1,...) = A000012, in some cases t(1,1,1,1,1,...) is a shifted version of the cited sequence:
p(S)                             t(1,1,1,1,1,...)
1 - S                                A000079
1 - S^2                              A000079
1 - S^3                              A024495
1 - S^4                              A000749
1 - S^5                              A139761
1 - S^6                              A290993
1 - S^7                              A290994
1 - S^8                              A290995
1 - S - S^2                          A001906
1 - S - S^3                          A116703
1 - S - S^4                          A290996
1 - S^3 - S^6                        A290997
1 - S^2 - S^3                        A095263
1 - S^3 - S^4                        A290998
1 - 2 S^2                            A052542
1 - 3 S^2                            A002605
1 - 4 S^2                            A015518
1 - 5 S^2                            A163305
1 - 6 S^2                            A290999
1 - 7 S^2                            A291008
1 - 8 S^2                            A291001
(1 - S)^2                            A045623
(1 - S)^3                            A058396
(1 - S)^4                            A062109
(1 - S)^5                            A169792
(1 - S)^6                            A169793
(1 - S^2)^2                          A024007
1 - 2 S - 2 S^2                      A052530
1 - 3 S - 2 S^2                      A060801
(1 - S)(1 - 2 S)                     A053581
(1 - 2 S)(1 - 3 S)                   A291002
(1 - S)(1 - 2 S)(1 - 3 S)(1 - 4 S)   A291003
(1 - 2 S)^2                          A120926
(1 - 3 S)^2                          A291004
1 + S - S^2                          A000045  (Fibonacci numbers starting with -1)
1 - S - S^2 - S^3                    A291000
1 - S - S^2 - S^3 - S^4              A291006
1 - S - S^2 - S^3 - S^4 - S^5        A291007
1 - S^2 - S^4                        A290990
(1 - S)(1 - 3 S)                     A291009
(1 - S)(1 - 2 S)(1 - 3 S)            A291010
(1 - S)^2 (1 - 2 S)                  A291011
(1 - S^2)(1 - 2 S)                   A291012
(1 - S^2)^3                          A291013
(1 - S^3)^2                          A291014
1 - S - S^2 + S^3                    A045891
1 - 2 S - S^2 + S^3                  A291015
1 - 3 S + S^2                        A136775
1 - 4 S + S^2                        A291016
1 - 5 S + S^2                        A291017
1 - 6 S + S^2                        A291018
1 - S - S^2 - S^3 + S^4              A291019
1 - S - S^2 - S^3 - S^4 + S^5        A291020
1 - S - S^2 - S^3 + S^4 + S^5        A291021
1 - S - 2 S^2 + 2 S^3                A175658
1 - 3 S^2 + 2 S^3                    A291023
(1 - 2 S^2)^2                        A291024
(1 - S^3)^3                          A291143
(1 - S - S^2)^2                      A209917

Clark Kimberling, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4, -4, 2)

Cf. A000012, A289780.

z = 60; s = x/(1 - x); p = 1 - s - s^2 - s^3;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000012 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291000 *)

G.f.: (-1 + x - x^2)/(-1 + 4 x - 4 x^2 + 2 x^3).

a(n) = 4*a(n-1) - 4*a(n-2) + 2*a(n-3) for n >= 4.

A027649 a(n) = 2*(3^n) - 2^n.

Original entry on oeis.org

1, 4, 14, 46, 146, 454, 1394, 4246, 12866, 38854, 117074, 352246, 1058786, 3180454, 9549554, 28665046, 86027906, 258149254, 774578834, 2323998646, 6972520226, 20918609254, 62757924914, 188277969046, 564842295746, 1694543664454, 5083664547794, 15251060752246

Offset: 0

N. J. A. Sloane

Poly-Bernoulli numbers B_n^(k) with k=-2.
Binomial transform of A007051, if both sequences start at 0. Binomial transform of A000225(n+1). - Paul Barry, Mar 24 2003
Euler expands (1-z)/(1-5z+6z^2) and finds the general term. Section 226 of the Introductio indicates that he could have written down the recursion relation: a(n) = 5 a(n-1)-6 a(n-2). - V. Frederick Rickey (fred-rickey(AT)usma.edu), Feb 10 2006
Let R be a binary relation on the power set P(A) of a set A having n = |A| elements such that for every element x, y of P(A), xRy if x is a subset of y or y is a subset of x. Then a(n) = |R|. - Ross La Haye, Dec 22 2006
With regard to the comment by Ross La Haye: For proper subsets see A056182. - For nonempty subsets see A091344. - For nonempty proper subsets see a(n+1) in A260217. - Manfred Boergens, Aug 02 2023
If x, y are two n-bit binary strings then a(n) gives the number of pairs (x,y) such that XOR(x, y) = ABS(x - y). - Ramasamy Chandramouli, Feb 15 2009
Equals row sums of the triangular version of A038573. - Gary W. Adamson, Jun 04 2009
Inverse binomial transform of A085350. - Paul Curtz, Nov 14 2009
Related to the number of even a's in a nontrivial cycle (should one exist) in the 3x+1 Problem, where a <= floor(log_2(2*(3^n) - 2^n)). The value n correlates to the number of odds in such a nontrivial cycle. See page 1288 of Crandall's paper. Also, this relation gives another proof that the number of odds divided by the number of evens in a nontrivial cycle is bounded by log 2 / log 3 (this observation does not resolve the finite cycles conjecture as the value could be arbitrarily close to this bound). However, the same argument gives that log 2 / log 3 is less than or equal to the number of odds divided by the number of evens in a divergent sequence (should one exist), as log 2 / log 3 is the limit value for a cycle of an arbitrarily large length, where the length is given by the value n. - Jeffrey R. Goodwin, Aug 04 2011
Row sums of Riordan triangle A106516. - Wolfdieter Lang, Jan 09 2015
Number of restricted barred preferential arrangements having 3 bars in which the sections are all restricted sections such that (for fixed sections i and j) section i or section j is empty. - Sithembele Nkonkobe, Oct 12 2015
This is also row 2 of A281891: for n >= 1, when consecutive positive integers are written as a product of primes in nondecreasing order, a factor of 2 or 3 occurs in n-th position a(n) times out of every 6^n. - Peter Munn, May 18 2017
Also row sums of A124929. - Omar E. Pol, Jun 15 2017
This is the sum of A318921(n) for n in the range 2^(k+1) to 2^(k+2)-1. See A318921 for proof. - N. J. A. Sloane, Sep 25 2018
a(n) is also the number of acyclic orientations of the complete bipartite graph K_{2,n}. - Vincent Pilaud, Sep 15 2020
a(n-1) is also the number of n-digit numbers whose largest decimal digit is 2. - Stefano Spezia, Nov 15 2023

Leonhard Euler, Introductio in analysin infinitorum (1748), section 216.

T. D. Noe, Table of n, a(n) for n = 0..200
Taylor Brysiewicz, Holger Eble, and Lukas Kühne, Enumerating chambers of hyperplane arrangements with symmetry, arXiv:2105.14542 [math.CO], 2021.
R. E. Crandall, On the 3x+1 problem, Math. Comp., 32 (1978) 1281-1292.
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata, arXiv:1503.01796 [math.CO], 2015; see also the Accompanying Maple Package.
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, Odd-Rule Cellular Automata on the Square Grid, arXiv:1503.04249 [math.CO], 2015.
John Elias, Illustration of initial terms: Reflected Sierpinski triangle
K. Imatomi, M. Kaneko, and E. Takeda, Multi-Poly-Bernoulli Numbers and Finite Multiple Zeta Values, J. Int. Seq. 17 (2014) # 14.4.5
Ken Kamano, Sums of Products of Poly-Bernoulli Numbers of Negative Index, Journal of Integer Sequences, Vol. 15 (2012), #12.1.3.
K. Kamano, Sums of Products of Bernoulli Numbers, Including Poly-Bernoulli Numbers, J. Int. Seq. 13 (2010), 10.5.2.
Masanobu Kaneko, Poly-Bernoulli numbers, Journal de théorie des nombres de Bordeaux, 9 no. 1 (1997), Pages 221-228.
Takao Komatsu, Some recurrence relations of poly-Cauchy numbers, J. Nonlinear Sci. Appl., (2019) Vol. 12, Issue 12, 829-845.
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
S. Nkonkobe and V. Murali, On some properties and relations between restricted barred preferential arrangements, multi-poly-bernoulli numbers and related numbers, arXiv:1509.07352 [math.CO], 2015.
Eric Weisstein's World of Mathematics, Sierpinski Sieve
Wikipedia, Sierpinski triangle
Index entries for sequences related to Bernoulli numbers.
Index entries for linear recurrences with constant coefficients, signature (5,-6).

Row n = 2 of array A099594.
Also occurs as a row, column, diagonal or as row sums in A038573, A085870, A090888, A106516, A217764, A281891.
Cf. A000079, A000225, A001047, A007051, A053581, A056182, A085350, A091344, A124929, A166060, A260217, A318921.

a027649 n = a027649_list !! n
a027649_list = map fst $ iterate (\(u, v) -> (3 * u + v, 2 * v)) (1, 1)
-- Reinhard Zumkeller, Jun 09 2013

[2*(3^n)-2^n: n in [0..30]]; // Vincenzo Librandi, Jul 17 2011

a(n, k):= (-1)^n*sum( (-1)^'m'*'m'!*Stirling2(n,'m')/('m'+1)^k,'m'=0..n);
seq(a(n, -2), n=0..30);

Table[2(3^n)-2^n,{n,0,30}] (* or *) LinearRecurrence[ {5,-6},{1,4},31]  (* Harvey P. Dale, Apr 22 2011 *)

a(n)=2*(3^n)-2^n \\ Charles R Greathouse IV, Jul 16 2011

Vec((1-x)/((1-2*x)*(1-3*x)) + O(x^50)) \\ Altug Alkan, Oct 12 2015

[2*(3^n - 2^(n-1)) for n in (0..30)] # G. C. Greubel, Aug 01 2022

G.f.: (1-x)/((1-2*x)*(1-3*x)).
a(n) = 3*a(n-1) + 2^(n-1), with a(0) = 1.
a(n) = Sum_{k=0..n} binomial(n, k)*(2^(k+1) - 1). - Paul Barry, Mar 24 2003
Partial sums of A053581. - Paul Barry, Jun 26 2003
Main diagonal of array (A085870) defined by T(i, 1) = 2^i - 1, T(1, j) = 2^j - 1, T(i, j) = T(i-1, j) + T(i-1, j-1). - Benoit Cloitre, Aug 05 2003
a(n) = A090888(n, 3). - Ross La Haye, Sep 21 2004
a(n) = Sum_{k=0..n} binomial(n+2, k+1)*Sum_{j=0..floor(k/2)} A001045(k-2j). - Paul Barry, Apr 17 2005
a(n) = Sum_{k=0..n} Sum_{j=0..n} binomial(n,j)*binomial(j+1,k+1). - Paul Barry, Sep 18 2006
a(n) = A166060(n+1)/6. - Philippe Deléham, Oct 21 2009
a(n) = 5*a(n-1) - 6*a(n-2), a(0)=1, a(1)=4. - Harvey P. Dale, Apr 22 2011
a(n) = A217764(n,2). - Ross La Haye, Mar 27 2013
For n>0, a(n) = 3 * a(n-1) + 2^(n-1) = 2 * (a(n-1) + 3^(n-1)). - J. Conrad, Oct 29 2015
for n>0, a(n) = 2 * (1 + 2^(n-2) + Sum_{x=1..n-2} Sum_{k=0..x-1} (binomial(x-1,k)*(2^(k+1) + 2^(n-x+k)))). - J. Conrad, Dec 10 2015
E.g.f.: exp(2*x)*(2*exp(x) - 1). - Stefano Spezia, May 18 2024

Better formulas from David W. Wilson and Michael Somos
Incorrect formula removed by Charles R Greathouse IV, Mar 18 2010
Duplications (due to corrections to A numbers) removed by Peter Munn, Jun 15 2017

A038573 a(n) = 2^A000120(n) - 1.

Original entry on oeis.org

0, 1, 1, 3, 1, 3, 3, 7, 1, 3, 3, 7, 3, 7, 7, 15, 1, 3, 3, 7, 3, 7, 7, 15, 3, 7, 7, 15, 7, 15, 15, 31, 1, 3, 3, 7, 3, 7, 7, 15, 3, 7, 7, 15, 7, 15, 15, 31, 3, 7, 7, 15, 7, 15, 15, 31, 7, 15, 15, 31, 15, 31, 31, 63, 1, 3, 3, 7, 3, 7, 7, 15, 3, 7, 7, 15, 7, 15, 15, 31, 3, 7, 7, 15, 7, 15, 15, 31

Offset: 0

Marc LeBrun

Essentially the same sequence as A001316, which has much more information, and also A159913. - N. J. A. Sloane, Jun 05 2009
Smallest number with same number of 1's in its binary expansion as n.
Fixed point of the morphism 0 -> 01, 1 -> 13, 3 -> 37, ... = k -> k, 2k+1, ... starting from a(0) = 0; 1 -> 01 -> 0113 -> 01131337 -> 011313371337377(15) -> ..., . - Robert G. Wilson v, Jan 24 2006
From Gary W. Adamson, Jun 04 2009: (Start)
As an infinite string, 2^n terms per row starting with "1": (1; 1,3; 1,3,3,7; 1,3,3,7,3,7,7,15; 1,3,3,7,3,7,7,15,3,7,7,15,7,15,15,31;...)
Row sums of that triangle = A027649: (1, 4, 14, 46, 454, ...); where the next row sum = current term of A027649 + next term in finite difference row of A027649, i.e., (1, 3, 10, 32, 100, 308, ...) = A053581. (End)
From Omar E. Pol, Jan 24 2016: (Start)
Partial sums give A267700.
a(n) is also the number of cells turned ON at n-th generation of the cellular automaton of A267700 in a 90-degree sector on the square grid.
a(n) is also the number of Y-toothpicks added at n-th generation of the structure of A267700 in a 120-degree sector on the triangular grid. (End)
Row sums of A090971. - Nikolaos Pantelidis, Nov 23 2022

			9 = 1001 -> 0011 -> 3, so a(9)=3.
From _Gary W. Adamson_, Jun 04 2009: (Start)
Triangle read by rows:
  1;
  1, 3;
  1, 3, 3, 7;
  1, 3, 3, 7, 3, 7, 7, 15;
  1, 3, 3, 7, 3, 7, 7, 15, 3, 7, 7, 15, 7, 15, 15, 31;
  ...
Row sums: (1, 4, 14, 46, ...) = A027649 = last row terms + new set of terms such that row 3 = (1, 3, 3, 7,) + (3, 7, 7, 15) = 14 + 32 = A027649(2) + A053581(3). (End)
The rows of this triangle converge to A159913. - _N. J. A. Sloane_, Jun 05 2009
G.f. = x + x^2 + 3*x^3 + x^4 + 3*x^5 + 3*x^6 + 7*x^7 + x^8 + 3*x^9 + 3*x^10 + 7*x^11 + ... - _Michael Somos_, Jul 24 2023

Seiichi Manyama, Table of n, a(n) for n = 0..8191 (terms 0..1023 from T. D. Noe)
David Applegate, The movie version
Michael Gilleland, Some Self-Similar Integer Sequences
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata
Index entries for sequences related to toothpick sequences
Index entries for sequences that are fixed points of mappings

Cf. A007313, A115378, A073138.
This is Guy Steele's sequence GS(3, 6) (see A135416).
Cf. also A000079, A001316, A027649, A053581, A159913, A267700.
Write n in b-ary, sort digits into increasing order: this sequence (b=2), A038574 (b=3), A319652 (b=4), A319653 (b=5), A319654 (b=6), A319655 (b=7), A319656 (b=8), A319657 (b=9), A004185 (b=10).
Column k=0 of A340666.

a038573 0 = 0
a038573 n = (m + 1) * (a038573 n') + m where (n', m) = divMod n 2
-- Reinhard Zumkeller, Oct 10 2012, Feb 07 2011
(Python 3.10+)
def A038573(n): return (1<Chai Wah Wu, Nov 15 2022

seq(2^convert(convert(n,base,2),`+`)-1, n=0..100); # Robert Israel, Jan 24 2016

Array[ 2^Count[ IntegerDigits[ #, 2 ], 1 ]-1&, 100 ]
Nest[ Flatten[ # /. a_Integer -> {a, 2a + 1}] &, {0}, 7] (* Robert G. Wilson v, Jan 24 2006 *)

{a(n) = 2^subst(Pol(binary(n)), x, 1) - 1};

a(n) = 2^hammingweight(n)-1; \\ Michel Marcus, Jan 24 2016

a(2n) = a(n), a(2n+1) = 2*a(n)+1, a(0) = 0. a(n) = A001316(n)-1 = 2^A000120(n) - 1. - Daniele Parisse
a(n) = number of positive integers k < n such that n XOR k = n-k (cf. A115378). - Paul D. Hanna, Jan 21 2006
a(n) = f(n, 1) with f(x, y) = if x = 0 then y - 1 else f(floor(x/2), y*(1 + x mod 2)). - Reinhard Zumkeller, Nov 21 2009
a(n) = (n mod 2 + 1) * a(floor(n/2)) + n mod 2. - Reinhard Zumkeller, Oct 10 2012
a(n) = Sum_{i=1..n} C(n,i) mod 2. - Wesley Ivan Hurt, Nov 17 2017
G.f.: -1/(1 - x) + Product_{k>=0} (1 + 2*x^(2^k)). - Ilya Gutkovskiy, Aug 20 2019
G.f. A(x) = x + x^2*A(x) + (1 + 2*x)*(1 - x^2)*A(x^2). - Michael Somos, Jul 24 2023

More terms from Erich Friedman

New definition from N. J. A. Sloane, Mar 01 2008

A093833 3^n-Jacobsthal(n).

Original entry on oeis.org

1, 2, 8, 24, 76, 232, 708, 2144, 6476, 19512, 58708, 176464, 530076, 1591592, 4777508, 14337984, 43024876, 129096472, 387333108, 1162086704, 3486434876, 10459654152, 31379661508, 94140382624, 282423944076, 847277424632

Offset: 0

Paul Barry, Apr 17 2004

Binomial transform of A052992. Binomial transform is A093834. Partial sums are A004054. Sums of consecutive pairs yield A053581.
Contribution from Johannes W. Meijer, Aug 15 2010: (Start)
An elephant sequence, see A175654. For the corner squares four A[5] vectors, with decimal values 343, 349, 373 and 469, lead to this sequence. For the central square these vectors lead to the companion sequence A175659.
(End)

Index entries for linear recurrences with constant coefficients, signature (4,-1,-6)

G.f.: (1-x)^2/((1+x)(1-2x)(1-3x)); a(n)=3^n-2^n/3+(-1)^n/3; a(n)=3^n-A001045(n).

A385178 Triangle T(n,k) read by rows in which the n-th diagonal lists the n-th differences of A001047, 0 <= k <= n.

Original entry on oeis.org

0, 1, 1, 3, 4, 5, 7, 10, 14, 19, 15, 22, 32, 46, 65, 31, 46, 68, 100, 146, 211, 63, 94, 140, 208, 308, 454, 665, 127, 190, 284, 424, 632, 940, 1394, 2059, 255, 382, 572, 856, 1280, 1912, 2852, 4246, 6305, 511, 766, 1148, 1720, 2576, 3856, 5768, 8620, 12866, 19171

Offset: 0

Paul Curtz, Jun 20 2025

			Triangle begins:
    0;
    1,   1;
    3,   4,    5;
    7,  10,   14,   19;
   15,  22,   32,   46,   65;
   31,  46,   68,  100,  146,  211;
   63,  94,  140,  208,  308,  454,  665;
  127, 190,  284,  424,  632,  940, 1394, 2059;
  255, 382,  572,  856, 1280, 1912, 2852, 4246,  6305;
  511, 766, 1148, 1720, 2576, 3856, 5768, 8620, 12866, 19171;
  ...

Columns k=0..2: A000225, A033484, A053209 (sans 1).
Diagonals: A001047, A027649, A053581 (sans 1), A291012 (sans 2).
Cf. A028243 (row sums), A036561, A059268, A081656, A248216, A321003.

/* As triangle */ [[2^(n-k)*3^k - 2^k : k in [0..n]]: n in [0..9]]; // Vincenzo Librandi, Jun 27 2025

T:= proc(n,k) option remember;
     `if`(n=k, 3^n-2^n, T(n, k+1)-T(n-1, k))
    end:
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Jun 24 2025

t[n_, 0] := 3^n - 2^n; t[n_, k_] := t[n, k] = t[n + 1, k - 1] - t[n, k - 1]; Table[t[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Jun 20 2025 *)

T(n,n) = 3^n - 2^n = A001047(n).
T(n,k) = T(n,k+1) - T(n-1,k) for 0 <= k < n.
T(n,k) = 2^(n-k)*3^k - 2^k = A036561(n,k) - A059268(n,k).
T(2n,n) = A248216(n+1).

cp's OEIS Frontend

A291000 p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - S - S^2 - S^3.

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A027649 a(n) = 2*(3^n) - 2^n.

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A038573 a(n) = 2^A000120(n) - 1.

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A093833 3^n-Jacobsthal(n).

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A385178 Triangle T(n,k) read by rows in which the n-th diagonal lists the n-th differences of A001047, 0 <= k <= n.

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