A000244 -id:A000244 - cp's OEIS frontend

A027465 Cube of lower triangular normalized binomial matrix.

1, 3, 1, 9, 6, 1, 27, 27, 9, 1, 81, 108, 54, 12, 1, 243, 405, 270, 90, 15, 1, 729, 1458, 1215, 540, 135, 18, 1, 2187, 5103, 5103, 2835, 945, 189, 21, 1, 6561, 17496, 20412, 13608, 5670, 1512, 252, 24, 1, 19683, 59049, 78732, 61236, 30618, 10206, 2268

Offset: 0

Olivier Gérard, N. J. A. Sloane

Rows of A013610 reversed. - Michael Somos, Feb 14 2002
Row sums are powers of 4 (A000302), antidiagonal sums are A006190 (a(n) = 3*a(n-1) + a(n-2)). - Gerald McGarvey, May 17 2005
Triangle of coefficients in expansion of (3+x)^n.
Also: Pure Galton board of scheme (3,1). Also: Multiplicity (number) of pairs of n-dimensional binary vectors with dot product (overlap) k. There are 2^n = A000079(n) binary vectors of length n and 2^(2n) = 4^n = A000302(n) different pairs to form dot products k = Sum_{i=1..n} v[i]*u[i] between these, 0 <= k <= n. (Since dot products are symmetric, there are only 2^n*(2^n-1)/2 different non-ordered pairs, actually.) - R. J. Mathar, Mar 17 2006
Mirror image of A013610. - Zerinvary Lajos, Nov 25 2007
T(i,j) is the number of i-permutations of 4 objects a,b,c,d, with repetition allowed, containing j a's. - Zerinvary Lajos, Dec 21 2007
The antidiagonals of the sequence formatted as a square array (see Examples section) and summed with alternating signs gives a bisection of Fibonacci sequence, A001906. Example: 81-(27-1)=55. Similar rule applied to rows gives A000079. - Mark Dols, Sep 01 2009
Triangle T(n,k), read by rows, given by (3,0,0,0,0,0,0,0,...)DELTA (1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 09 2011
T(n,k) = binomial(n,k)*3^(n-k), the number of subsets of [2n] with exactly k symmetric pairs, where elements i and j of [2n] form a symmetric pair if i+j=2n+1. Equivalently, if n couples attend a (ticketed) event that offers door prizes, then the number of possible prize distributions that have exactly k couples as dual winners is T(n,k). - Dennis P. Walsh, Feb 02 2012
T(n,k) is the number of ordered pairs (A,B) of subsets of {1,2,...,n} such that the intersection of A and B contains exactly k elements. For example, T(2,1) = 6 because we have ({1},{1}); ({1},{1,2}); ({2},{2}); ({2},{1,2}); ({1,2},{1}); ({1,2},{2}). Sum_{k=0..n} T(n,k)*k = A002697(n) (see comment there by Ross La Haye). - Geoffrey Critzer, Sep 04 2013
Also the convolution triangle of A000244. - Peter Luschny, Oct 09 2022

			Example: n = 3 offers 2^3 = 8 different binary vectors (0,0,0), (0,0,1), ..., (1,1,0), (1,1,1). a(3,2) = 9 of the 2^4 = 64 pairs have overlap k = 2: (0,1,1)*(0,1,1) = (1,0,1)*(1,0,1) = (1,1,0)*(1,1,0) = (1,1,1)*(1,1,0) = (1,1,1)*(1,0,1) = (1,1,1)*(0,1,1) = (0,1,1)*(1,1,1) = (1,0,1)*(1,1,1) = (1,1,0)*(1,1,1) = 2.
For example, T(2,1)=6 since there are 6 subsets of {1,2,3,4} that have exactly 1 symmetric pair, namely, {1,4}, {2,3}, {1,2,3}, {1,2,4}, {1,3,4}, and {2,3,4}.
The present sequence formatted as a triangular array:
     1
     3     1
     9     6     1
    27    27     9     1
    81   108    54    12    1
   243   405   270    90   15    1
   729  1458  1215   540  135   18   1
  2187  5103  5103  2835  945  189  21  1
  6561 17496 20412 13608 5670 1512 252 24 1
  ...
A013610 formatted as a triangular array:
  1
  1  3
  1  6   9
  1  9  27   27
  1 12  54  108   81
  1 15  90  270  405   243
  1 18 135  540 1215  1458   729
  1 21 189  945 2835  5103  5103  2187
  1 24 252 1512 5670 13608 20412 17496 6561
   ...
A099097 formatted as a square array:
      1     0     0    0   0 0 0 0 0 0 0 ...
      3     1     0    0   0 0 0 0 0 0 ...
      9     6     1    0   0 0 0 0 0 ...
     27    27     9    1   0 0 0 0 ...
     81   108    54   12   1 0 0 ...
    243   405   270   90  15 1 ...
    729  1458  1215  540 135 ...
   2187  5103  5103 2835 ...
   6561 17496 20412 ...
  19683 59049 ...
  59049 ...

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
B. N. Cyvin et al., Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, Match, No. 34 (Oct 1996), pp. 109-121.
Erich Neuwirth, Recursively defined combinatorial functions: Extending Galton's board, Discrete Math. 239 No. 1-3, 33-51 (2001).

Cf. A000244, A007318, A013610, A013610, A099097, A027471, A027472, A036216, A036217, A036219, A036220, A036221, A036222, A036223.

a027465 n k = a027465_tabl !! n !! k
a027465_row n = a027465_tabl !! n
a027465_tabl = iterate (\row ->
   zipWith (+) (map (* 3) (row ++ [0])) (map (* 1) ([0] ++ row))) [1]
-- Reinhard Zumkeller, May 26 2013

for i from 0 to 12 do seq(binomial(i, j)*3^(i-j), j = 0 .. i) od; # Zerinvary Lajos, Nov 25 2007
# Uses function PMatrix from A357368. Adds column 1, 0, 0, ... to the left.
PMatrix(10, n -> 3^(n-1)); # Peter Luschny, Oct 09 2022

t[n_, k_] := Binomial[n, k]*3^(n-k); Table[t[n, n-k], {n, 0, 9}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Sep 19 2012 *)

{T(n, k) = polcoeff( (3 + x)^n, k)}; /* Michael Somos, Feb 14 2002 */

Numerators of lower triangle of (b^2)[ i, j ] where b[ i, j ] = binomial(i-1, j-1)/2^(i-1) if j <= i, 0 if j > i.
Triangle whose (i, j)-th entry is binomial(i, j)*3^(i-j).
a(n, m) = 4^(n-1)*Sum_{j=m..n} b(n, j)*b(j, m) = 3^(n-m)*binomial(n-1, m-1), n >= m >= 1; a(n, m) := 0, n < m. G.f. for m-th column: (x/(1-3*x))^m (m-fold convolution of A000244, powers of 3). - Wolfdieter Lang, Feb 2006
G.f.: 1 / (1 - x(3+y)).
a(n,k) = 3*a(n-1,k) + a(n-1,k-1) - R. J. Mathar, Mar 17 2006
From the formalism of A133314, the e.g.f. for the row polynomials of A027465 is exp(x*t)*exp(3x). The e.g.f. for the row polynomials of the inverse matrix is exp(x*t)*exp(-3x). p iterates of the matrix give the matrix with e.g.f. exp(x*t)*exp(p*3x). The results generalize for 3 replaced by any number. - Tom Copeland, Aug 18 2008
T(n,k) = A164942(n,k)*(-1)^k. - Philippe Deléham, Oct 09 2011
Let P and P^T be the Pascal matrix and its transpose and H = P^3 = A027465. Then from the formalism of A132440 and A218272,
  exp[x*z/(1-3z)]/(1-3z) = exp(3z D_z z) e^(x*z)= exp(3D_x x D_x) e^(z*x)
  = (1 z z^2 z^3 ...) H (1 x x^2/2! x^3/3! ...)^T
  = (1 x x^2/2! x^3/3! ...) H^T (1 z z^2 z^3 ...)^T = Sum_{n>=0} (3z)^n L_n(-x/3), where D is the derivative operator and L_n(x) are the regular (not normalized) Laguerre polynomials. - Tom Copeland, Oct 26 2012
E.g.f. for column k: x^k/k! * exp(3x). - Geoffrey Critzer, Sep 04 2013

A108411 a(n) = 3^floor(n/2). Powers of 3 repeated.

Original entry on oeis.org

1, 1, 3, 3, 9, 9, 27, 27, 81, 81, 243, 243, 729, 729, 2187, 2187, 6561, 6561, 19683, 19683, 59049, 59049, 177147, 177147, 531441, 531441, 1594323, 1594323, 4782969, 4782969, 14348907, 14348907, 43046721, 43046721, 129140163, 129140163, 387420489, 387420489, 1162261467

Offset: 0

Ralf Stephan, Jun 05 2005

a(n) is the Parker sequence for the automorphism group of the limit of the class of oriented graphs; a(n) counts the finite circulant structures in that class. - N-E. Fahssi, Feb 18 2008
Complete sequence: every positive integer is the sum of members of this sequence. - Charles R Greathouse IV, Jul 19 2012
Conjecture: a(n+1) is the number of distinct subsets S of {0,1,2,...,n} such that the sumset S+S does not contain n. - Michael Chu, Oct 05 2021. Andrew Howroyd, Nov 20 2021: The conjecture is true: If there are m pairs of numbers that add to n then inclusion/exclusion gives sum(k=0, m, binomial(m,k)*(-1)^k*2^(2*m-2*k)) as the number of sets that don't contain any of those pairs which equals 3^m. For even n , n/2 cannot be included in any set.
Also, number of walks of length n in the graph K_{1,3} (the graph with edges {1,2}, {1,3}, {1,4}) starting at one of the degree 1 vertices. - Sean A. Irvine, May 30 2025

			a(6) = 27; 3^floor(6/2) = 3^floor(3) = 3^3 = 27.

Vincenzo Librandi, Table of n, a(n) for n = 0..3000
D. A. Gewurz and F. Merola, Sequences realized as Parker vectors of oligomorphic permutation groups, J. Integer Seq., 6 (2003), 03.1.6.
Index entries for linear recurrences with constant coefficients, signature (0,3).

Essentially the same as A056449 and A162436.

Cf. A000244, A004526, A016116, A152815, A158020.

a108411 = (3 ^) . flip div 2  -- Reinhard Zumkeller, May 01 2014

[3^Floor(n/2): n in [0..50]]; // Vincenzo Librandi, Aug 17 2011

A108411:=n->3^floor(n/2); seq(A108411(k), k=0..100); # Wesley Ivan Hurt, Nov 01 2013

Table[3^Floor[n/2], {n,0,100}] (* Wesley Ivan Hurt, Nov 01 2013 *)

PARI
```
a(n)=3^floor(n/2);
```

def A108411(n): return 3**(n>>1) # Chai Wah Wu, Oct 28 2024

O.g.f.: (1+x)/(1-3*x^2). - R. J. Mathar, Apr 01 2008
a(n) = 3^(n/2)*((1+(-1)^n)/2+(1-(-1)^n)/(2*sqrt(3))). - Paul Barry, Nov 12 2009
a(n+3) = a(n+2)*a(n+1)/a(n). - Reinhard Zumkeller, Mar 04 2011
a(n) = (-1)^n*sum(A158020(n,k)*2^k, 0<=k<=n). - Philippe Deléham, Dec 01 2011
a(n) = sum(A152815(n,k)*2^k, 0<=k<=n). - Philippe Deléham, Apr 22 2013
a(n) = 3^A004526(n). - Michel Marcus, Aug 30 2014
E.g.f.: cosh(sqrt(3)*x) + sinh(sqrt(3)*x)/sqrt(3). - Stefano Spezia, Dec 31 2022

Incorrect formula removed by Michel Marcus, Oct 06 2021

A036561 Nicomachus triangle read by rows, T(n, k) = 2^(n - k)*3^k, for 0 <= k <= n.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 8, 12, 18, 27, 16, 24, 36, 54, 81, 32, 48, 72, 108, 162, 243, 64, 96, 144, 216, 324, 486, 729, 128, 192, 288, 432, 648, 972, 1458, 2187, 256, 384, 576, 864, 1296, 1944, 2916, 4374, 6561, 512, 768, 1152, 1728, 2592, 3888, 5832, 8748, 13122, 19683

Offset: 0

N. J. A. Sloane

The triangle pertaining to this sequence has the property that every row, every column and every diagonal contains a nontrivial geometric progression. More interestingly every line joining any two elements contains a nontrivial geometric progression. - Amarnath Murthy, Jan 02 2002
Kappraff states (pp. 148-149): "I shall refer to this as Nicomachus' table since an identical table of numbers appeared in the Arithmetic of Nicomachus of Gerasa (circa 150 A.D.)" The table was rediscovered during the Italian Renaissance by Leon Battista Alberti, who incorporated the numbers in dimensions of his buildings and in a system of musical proportions. Kappraff states "Therefore a room could exhibit a 4:6 or 6:9 ratio but not 4:9. This ensured that ratios of these lengths would embody musical ratios". - Gary W. Adamson, Aug 18 2003
After Nichomachus and Alberti several Renaissance authors described this table. See for instance Pierre de la Ramée in 1569 (facsimile of a page of his Arithmetic Treatise in Latin in the links section). - Olivier Gérard, Jul 04 2013
The triangle sums, see A180662 for their definitions, link Nicomachus's table with eleven different sequences, see the crossrefs. It is remarkable that these eleven sequences can be described with simple elegant formulas. The mirror of this triangle is A175840. - Johannes W. Meijer, Sep 22 2010
The diagonal sums Sum_{k} T(n - k, k) give A167762(n + 2). - Michael Somos, May 28 2012
Where d(n) is the divisor count function, then d(T(i,j)) = A003991, the rows of which sum to the tetrahedral numbers A000292(n+1). For example, the sum of the divisors of row 4 of this triangle (i = 4), gives d(16) + d(24) + d(36) + d(54) + d(81) = 5 + 8 + 9 + 8 + 5 = 35 = A000292(5). In fact, where p and q are distinct primes, the aforementioned relationship to the divisor function and tetrahedral numbers can be extended to any triangle of numbers in which the i-th row is of form {p^(i-j)*q^j, 0<=j<=i}; i >= 0 (e.g., A003593, A003595). - Raphie Frank, Nov 18 2012, corrected Dec 07 2012
Sequence (or tree) generated by these rules: 1 is in S, and if x is in S, then 2*x and 3*x are in S, and duplicates are deleted as they occur; see A232559. - Clark Kimberling, Nov 28 2013
Partial sums of rows produce Stirling numbers of the 2nd kind: A000392(n+2) = Sum_{m=1..(n^2+n)/2} a(m). - Fred Daniel Kline, Sep 22 2014
A permutation of A003586. - L. Edson Jeffery, Sep 22 2014
Form a word of length i by choosing a (possibly empty) word on alphabet {0,1} then concatenating a word of length j on alphabet {2,3,4}. T(i,j) is the number of such words. - Geoffrey Critzer, Jun 23 2016
Form of Zorach additive triangle (see A035312) where each number is sum of west and northwest numbers, with the additional condition that each number is GCD of the two numbers immediately below it. - Michel Lagneau, Dec 27 2018

			The start of the sequence as a triangular array read by rows:
   1
   2   3
   4   6   9
   8  12  18  27
  16  24  36  54  81
  32  48  72 108 162 243
  ...
The start of the sequence as a table T(n,k) n, k > 0:
    1    2    4    8   16   32 ...
    3    6   12   24   48   96 ...
    9   18   36   72  144  288 ...
   27   54  108  216  432  864 ...
   81  162  324  648 1296 2592 ...
  243  486  972 1944 3888 7776 ...
  ...
- _Boris Putievskiy_, Jan 08 2013

Jay Kappraff, Beyond Measure, World Scientific, 2002, p. 148.
Flora R. Levin, The Manual of Harmonics of Nicomachus the Pythagorean, Phanes Press, 1994, p. 114.

Reinhard Zumkeller and Matthew House, Rows n = 0..300 of triangle, flattened [Rows 0 through 120 were computed by Reinhard Zumkeller; rows 121 through 300 by Matthew House, Jul 09 2015]
Fred Daniel Kline, How do I convert this Nicomachus' Triangle to one with edges?
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Pierre de la Ramée (Petrus Ramus), P. Rami Arithmeticae (anno 1569) Liber 2, Cap. XVI "De inventione continue proportionalium" p.46 (leaf 0055) describes this integer triangle in a layout close to the current OEIS 'tabl' layout.
Marko Riedel, Proof of identity by Egorychev method.
Thomas Scheuerle, Version of this triangle from Boethius (480-524), Anicius Manlius Severinus Boethius, De institutione arithmetica, Medeltidshandskrift 1 (Mh 1), Lund University Library, early 10th century, page 4r.
Robert Sedgewick, Analysis of shellsort and related algorithms, Fourth European Symposium on Algorithms, Barcelona, September, 1996.

Cf. A001047 (row sums), A000400 (central terms), A013620, A007318.
Cf. A003586, A000079, A000244, A007283, A025197, A005010, A003946, A005051.
Triangle sums (see the comments): A001047 (Row1); A015441 (Row2); A005061 (Kn1, Kn4); A016133 (Kn2, Kn3); A016153 (Fi1, Fi2); A016140 (Ca1, Ca4); A180844 (Ca2, Ca3); A180845 (Gi1, Gi4); A180846 (Gi2, Gi3); A180847 (Ze1, Ze4); A016185 (Ze2, Ze3). - Johannes W. Meijer, Sep 22 2010, Sep 10 2011
Antidiagonal cumulative sum: A000392; square arrays cumulative sum: A160869. Antidiagonal products: 6^A000217; antidiagonal cumulative products: 6^A000292; square arrays products: 6^A005449; square array cumulative products: 6^A006002.

a036561 n k = a036561_tabf !! n !! k
a036561_row n = a036561_tabf !! n
a036561_tabf = iterate (\xs@(x:_) -> x * 2 : map (* 3) xs) [1]
-- Reinhard Zumkeller, Jun 08 2013

/* As triangle: */ [[(2^(i-j)*3^j)/3: j in [1..i]]: i in [1..10]]; // Vincenzo Librandi, Oct 17 2014

A036561 := proc(n,k): 2^(n-k)*3^k end:
seq(seq(A036561(n,k),k=0..n),n=0..9);
T := proc(n,k) option remember: if k=0 then 2^n elif k>=1 then procname(n,k-1) + procname(n-1,k-1) fi: end: seq(seq(T(n,k),k=0..n),n=0..9);
# Johannes W. Meijer, Sep 22 2010, Sep 10 2011

Flatten[Table[ 2^(i-j) 3^j, {i, 0, 12}, {j, 0, i} ]] (* Flatten added by Harvey P. Dale, Jun 07 2011 *)

for(i=0,9,for(j=0,i,print1(3^j<<(i-j)", "))) \\ Charles R Greathouse IV, Dec 22 2011

{T(n, k) = if( k<0 || k>n, 0, 2^(n - k) * 3^k)} /* Michael Somos, May 28 2012 */

T(n,k) = A013620(n,k)/A007318(n,k). - Reinhard Zumkeller, May 14 2006
T(n,k) = T(n,k-1) + T(n-1,k-1) for n>=1 and 1<=k<=n with T(n,0) = 2^n for n>=0. - Johannes W. Meijer, Sep 22 2010
T(n,k) = 2^(k-1)*3^(n-1), n, k > 0 read by antidiagonals. - Boris Putievskiy, Jan 08 2013
a(n) = 2^(A004736(n)-1)*3^(A002260(n)-1), n > 0, or a(n) = 2^(j-1)*3^(i-1) n > 0, where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor[(-1+sqrt(8*n-7))/2]. - Boris Putievskiy, Jan 08 2013
G.f.: 1/((1-2x)(1-3yx)). - Geoffrey Critzer, Jun 23 2016
T(n,k) = (-1)^n * Sum_{q=0..n} (-1)^q * C(k+3*q, q) * C(n+2*q, n-q). - Marko Riedel, Jul 01 2024

A006234 a(n) = n*3^(n-4).

Original entry on oeis.org

1, 4, 15, 54, 189, 648, 2187, 7290, 24057, 78732, 255879, 826686, 2657205, 8503056, 27103491, 86093442, 272629233, 860934420, 2711943423, 8523250758, 26732013741, 83682825624, 261508830075, 815907549834, 2541865828329

Offset: 3

N. J. A. Sloane

For n >= 1 a(n) is also the determinant of the n-3 X n-3 matrix with 4's on the diagonal and 1's elsewhere. - Ahmed Fares (ahmedfares(AT)my-deja.com), May 06 2001
a(n+3) = det(M(n)) where M(n) is the n X n matrix with m(i,i) = 4, m(i,j) = i/j for i != j. - Benoit Cloitre, Feb 01 2003
Main diagonal of array defined by m(1,j) = j; m(i,1) = i and m(i,j) = m(i-1,j) + 2*m(i-1,j-1). - Benoit Cloitre, Jun 13 2003
a(n+3) is the number of words of length n on {A, B, C, D} with no D appearing anywhere to the right of an A. - Rob Pratt, Aug 04 2004
Number of spanning trees in the book graph of order n-2, i.e., S_{n-2} X P_2 (S_k = the star graph on k nodes) (conjectured). This conjecture is true - see Doslic (2013). - N. J. A. Sloane, Dec 28 2013
Conjecture: a(n+2) is the total number of parts used in the compositions of n if the parts can be runs of any length from 1 to n, and contain any integers from 1 to n. (The number of such compositions is given by A000244(n-1).) - Gregory L. Simay, May 27 2017
a(n+3) is the number of words of length n defined on 4 letters where one of the letters is used at most once. - Enrique Navarrete, Mar 14 2024

			For n=3, the total number of parts is (3+2)3^(3+2-4)=(5)(3)=15 (each part indicated by "[]"): [3]; [2,1]; [1,2]; [2],[1]; [1],[2]; [1,1,1]; [1,1],[1]; [1],[1,1]; [1],[1],[1]. Note that these 15 parts are arranged into 9 = A000244(3-1)compositions. - _Gregory L. Simay_, May 27 2017

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 3..1000
Tomislav Doslic, Planar polycyclic graphs and their Tutte polynomials, Journal of Mathematical Chemistry, Volume 51, Issue 6, 2013, pp. 1599-1607.
Guillermo Esteban, Clemens Huemer, and Rodrigo I. Silveira, New production matrices for geometric graphs, arXiv:2003.00524 [math.CO], 2020.
Germain Kreweras, Complexité et circuits Eulériens dans les sommes tensorielles de graphes, J. Combin. Theory, B 24 (1978), 202-212.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Eric Weisstein's World of Mathematics, Book Graph.
Eric Weisstein's World of Mathematics, Spanning Tree.
Index entries for linear recurrences with constant coefficients, signature (6,-9).

Binomial transform of A001792.

Cf. A000244, A036290, A050914, A112626.

[ n*3^(n-4): n in [3..30] ]; // Vincenzo Librandi, Aug 19 2011

Table[n 3^(n-4), {n, 3, 30}] (* or *)
CoefficientList[Series[(1-2 x)/(1-3 x)^2, {x,0,30}], x] (* Michael De Vlieger, May 28 2017 *)
LinearRecurrence[{6,-9},{1,4},30] (* Harvey P. Dale, Aug 17 2020 *)

a(n)=n*3^(n-4) \\ Charles R Greathouse IV, Sep 24 2015

[n*3^(n-4) for n in range(3,31)] # G. C. Greubel, Dec 27 2023

G.f.: (1-2*x)/(1-3*x)^2. - Simon Plouffe in his 1992 dissertation.
a(n+3) = Sum_{k=0..n} A112626(n, k). - Ross La Haye, Jan 11 2006
G.f.: Hypergeometric2F1([1,4],[3],3*x). - R. J. Mathar, Aug 09 2015
From Amiram Eldar, Jan 18 2021: (Start)
Sum_{n>=1} 1/a(n) = 81*log(3/2).
Sum_{n>=1} (-1)^(n+1)/a(n) = 81*log(4/3). (End)
E.g.f.: x*(exp(3*x) - 3*x - 1)/27. - Stefano Spezia, Mar 04 2023
E.g.f. (with offset 0): exp(3*x)*(1+x). - Enrique Navarrete, Mar 14 2024

A006899 Numbers of the form 2^i or 3^j.

Original entry on oeis.org

1, 2, 3, 4, 8, 9, 16, 27, 32, 64, 81, 128, 243, 256, 512, 729, 1024, 2048, 2187, 4096, 6561, 8192, 16384, 19683, 32768, 59049, 65536, 131072, 177147, 262144, 524288, 531441, 1048576, 1594323, 2097152, 4194304, 4782969, 8388608, 14348907, 16777216, 33554432

Offset: 1

Simon Plouffe and N. J. A. Sloane

Complement of A033845 with respect to A003586. - Reinhard Zumkeller, Sep 25 2008
In the 14th century, Levi Ben Gerson proved that the only pairs of terms which differ by 1 are (1, 2), (2, 3), (3, 4), and (8, 9); see A235365, A235366, A236210. - Jonathan Sondow, Jan 20 2014
Numbers n such that absolute value of the greatest prime factor of n minus the smallest prime not dividing n is 1 (that is, abs(A006530(n)-A053669(n)) = 1). - Anthony Browne, Jun 26 2016
Deficient 3-smooth numbers, i.e., intersection of A005100 and A003586. - Amiram Eldar, Jun 03 2022

G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, p. 78.
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 = 1..500
Boris Alexeev, Minimal DFAs for testing divisibility, arXiv:cs/0309052 [cs.CC], 2003.
Jung-Chao Ban, Wen-Guei Hu, and Song-Sun Lin, Pattern generation problems arising in multiplicative integer systems, arXiv preprint arXiv:1207.7154 [math.DS], 2012.
Lukas Spiegelhofer, Collisions of the binary and ternary sum-of-digits functions, arXiv:2105.11173 [math.NT], 2021.
Eric Weisstein's World of Mathematics, Pillai's Theorem.

Union of A000079 and A000244. - Reinhard Zumkeller, Sep 25 2008
Cf. A085238, A085239.
Cf. A003586, A005100, A006530, A053669.
Cf. A170803, A235365, A235366, A236210.
A186927 and A186928 are subsequences.
Cf. A108906 (first differences), A006895, A227928.

a006899 n = a006899_list !! (n-1)
a006899_list = 1 : m (tail a000079_list) (tail a000244_list) where
   m us'@(u:us) vs'@(v:vs) = if u < v then u : m us vs' else v : m us' vs
-- Reinhard Zumkeller, Oct 09 2013

A:={seq(2^n,n=0..63)}: B:={seq(3^n,n=0..40)}: C:=sort(convert(A union B,list)): seq(C[j],j=1..39); # Emeric Deutsch, Aug 03 2005

seqMax = 10^20; Union[2^Range[0, Floor[Log[2, seqMax]]], 3^Range[0, Floor[Log[3, seqMax]]]] (* Stefan Steinerberger, Apr 08 2006 *)

is(n)=n>>valuation(n,2)==1 || n==3^valuation(n,3) \\ Charles R Greathouse IV, Aug 29 2016

upto(n) = my(res = vector(logint(n, 2) + logint(n, 3) + 1), t = 1); res[1] = 1; for(i = 2, 3, for(j = 1, logint(n, i), t++; res[t] = i^j)); vecsort(res) \\ David A. Corneth, Oct 26 2017

a(n) = my(i0= logint(3^(n-1),6), i= logint(3^n,6)); if(i > i0, 2^i, my(j=logint(2^n,6)); 3^j) \\ Ruud H.G. van Tol, Nov 10 2022

from sympy import integer_log
def A006899(n): return 1<Chai Wah Wu, Oct 01 2024

a(n) = A085239(n)^A085238(n). - Reinhard Zumkeller, Jun 22 2003
A086411(a(n)) = A086410(a(n)). - Reinhard Zumkeller, Sep 25 2008
A053669(a(n)) - A006530(a(n)) = (-1)^a(n) n > 1. - Anthony Browne, Jun 26 2016
Sum_{n>=1} 1/a(n) = 5/2. - Amiram Eldar, Jun 03 2022
a(n)^(1/n) tends to 3^(log(2)/log(6)) = 2^(log(3)/log(6)) = 1.529592328491883538... - Vaclav Kotesovec, Sep 19 2024

More terms from Reinhard Zumkeller, Jun 22 2003

A028243 a(n) = 3^(n-1) - 2^n + 1 (essentially Stirling numbers of second kind).

Original entry on oeis.org

0, 0, 2, 12, 50, 180, 602, 1932, 6050, 18660, 57002, 173052, 523250, 1577940, 4750202, 14283372, 42915650, 128878020, 386896202, 1161212892, 3484687250, 10456158900, 31372671002, 94126401612, 282395982050, 847221500580, 2541731610602, 7625329049532

Offset: 1

N. J. A. Sloane, Doug McKenzie (mckfam4(AT)aol.com)

For n >= 3, a(n) is equal to the number of functions f: {1,2,...,n-1} -> {1,2,3} such that Im(f) contains 2 fixed elements. - Aleksandar M. Janjic and Milan Janjic, Mar 08 2007
Let P(A) be the power set of an n-element set A. Then a(n+1) = the number of pairs of elements {x,y} of P(A) for which x and y are intersecting and for which either x is a proper subset of y or y is a proper subset of x. - Ross La Haye, Jan 02 2008
Let P(A) be the power set of an n-element set A and R be a relation on P(A) such that for all x, y of P(A), xRy if x is not a subset of y and y is not a subset of x and x and y are disjoint. Then a(n+1) = |R|. - Ross La Haye, Mar 19 2009
Let P(A) be the power set of an n-element set A and R be a relation on P(A) such that for all x, y of P(A), xRy if either 0) x is a proper subset of y or y is a proper subset of x, or 1) x is not a subset of y and y is not a subset of x and x and y are disjoint. Then a(n+2) = |R|. - Ross La Haye, Mar 19 2009
In the terdragon curve, a(n) is the number of triple-visited points in expansion level n.  The first differences of this sequence (A056182) are the number of enclosed unit triangles since on segment expansion each unit triangle forms a new triple-visited point, and existing triple-visited points are unchanged. - Kevin Ryde, Oct 20 2020
a(n+1) is the number of ternary strings of length n that contain at least one 0 and one 1; for example, for n=3, a(4)=12 since the strings are the 3 permutations of 100, the 3 permutations of 110, and the 6 permutations of 210. - Enrique Navarrete, Aug 13 2021
From Sanjay Ramassamy, Dec 23 2021: (Start)
a(n+1) is the number of topological configurations of n points and n lines where the points lie at the vertices of a convex cyclic n-gon and the lines are the perpendicular bisectors of its sides.
a(n+1) is the number of 2n-tuples composed of n 0's and n 1's which have an interlacing signature. The signature of a 2n-tuple (v_1,...,v_{2n}) is the n-tuple (s_1,...,s_n) defined by s_i=v_i+v_{i+n}. The signature is called interlacing if after deleting the 1's, there are letters remaining and the remaining 0's and 2's are alternating. (End)
a(n+1) is the number of pairs (A,B) where B is a nonempty subset of {1,2,...,n} and A is a nonempty proper subset of B. If either "nonempty" or "proper" is omitted then see A001047. If "nonempty" and "proper" are omitted then see A000244. - Manfred Boergens, Mar 28 2023
a(n) is the number of (n-1) X (n-1) nilpotent Boolean relation matrices with rank equal to 1. a(n) = A060867(n-1) - A005061(n-1) (since every rank 1 matrix is either idempotent or nilpotent). - Geoffrey Critzer, Jul 13 2023
For odd n > 3, a(n) is also the number of minimum vertex colorings in the (n-1)-prism graph. - Eric W. Weisstein, Mar 05 2024

Seiichi Manyama, Table of n, a(n) for n = 1..2096
Ovidiu Bagdasar, On some functions involving the lcm and gcd of integer tuples, Scientific Publications of the State University of Novi Pazar, Appl. Maths. Inform. and Mech., Vol. 6, 2 (2014), 91-100.
J. Brandts and C. Cihangir, Counting triangles that share their vertices with the unit n-cube, in Conference Applications of Mathematics 2013 in honor of the 70th birthday of Karel Segeth. Jan Brandts, Sergey Korotov, et al., eds., Institute of Mathematics AS CR, Prague 2013.
K. S. Immink, Coding Schemes for Multi-Level Channels that are Intrinsically Resistant Against Unknown Gain and/or Offset Using Reference Symbols, Electronics Letters, Volume: 50, Issue: 1, January 2 2014.
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
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.
P. Melotti, S. Ramassamy and P. Thévenin, Points and lines configurations for perpendicular bisectors of convex cyclic polygons, arXiv:2003.11006 [math.CO], 2020.
Rajesh Kumar Mohapatra and Tzung-Pei Hong, On the Number of Finite Fuzzy Subsets with Analysis of Integer Sequences, Mathematics (2022) Vol. 10, No. 7, 1161.
Kevin Ryde, Iterations of the Terdragon Curve, see index "T triple-visited points".
Eric Weisstein's World of Mathematics, Minimum Vertex Coloring.
Eric Weisstein's World of Mathematics, Prism Graph.
Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Cf. A000392, A008277, A163626, A056182 (first differences), A000244, A001047.

[3^(n-1) - 2*2^(n-1) + 1: n in [1..30]]; // G. C. Greubel, Nov 19 2017

Table[2 StirlingS2[n, 3], {n, 24}] (* or *)
Table[3^(n - 1) - 2*2^(n - 1) + 1, {n, 24}] (* or *)
Rest@ CoefficientList[Series[-2 x^3/(-1 + x)/(-1 + 3 x)/(-1 + 2 x), {x, 0, 24}], x] (* Michael De Vlieger, Sep 24 2016 *)

a(n) = 3^(n-1) - 2*2^(n-1) + 1 \\ G. C. Greubel, Nov 19 2017

[stirling_number2(i,3)*2 for i in range(1,30)] # Zerinvary Lajos, Jun 26 2008

a(n) = 2*S(n, 3) = 2*A000392(n). - Emeric Deutsch, May 02 2004
G.f.: -2*x^3/(-1+x)/(-1+3*x)/(-1+2*x) = -1/3 - (1/3)/(-1+3*x) + 1/(-1+2*x) - 1/(-1+x). - R. J. Mathar, Nov 22 2007
E.g.f.: (exp(3*x) - 3*exp(2*x) + 3*exp(x) - 1)/3. - Wolfdieter Lang, May 03 2017
E.g.f. with offset 0: exp(x)*(exp(x)-1)^2. - Enrique Navarrete, Aug 13 2021
a(n) = Sum_{k = 1..n-2} binomial(n-1, k) * (2^(n-k-1)-1). - Ocean Wong, Jan 03 2025

A039966 a(0) = 1; thereafter a(3n+2) = 0, a(3n) = a(3n+1) = a(n).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane, Dec 11 1999

nonn

Number of partitions of n into distinct powers of 3.
Trajectory of 1 under the morphism: 1 -> 110, 0 -> 000. Thus 1 -> 110 ->110110000 -> 110110000110110000000000000 -> ... - Philippe Deléham, Jul 09 2005
Also, an example of a d-perfect sequence.
This is a composite of two earlier sequences contributed at different times by N. J. A. Sloane and by Reinhard Zumkeller, Mar 05 2005. Christian G. Bower extended them and found that they agreed for at least 512 terms. The proof that they were identical was found by Ralf Stephan, Jun 13 2005, based on the fact that they were both 3-regular sequences.

			The triples of elements (a(3k), a(3k+1), a(3k+2)) are (1,1,0) if a(k) = 1 and (0,0,0) if a(k) = 0.  So since a(2) = 0, a(6) = a(7) = a(8) = 0, and since a(3) = 1, a(9) = a(10) = 1 and a(11) = 0. - _Michael B. Porter_, Jul 11 2016

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
D. Kohel, S. Ling and C. Xing, Explicit Sequence Expansions, in Sequences and their Applications, C. Ding, T. Helleseth, and H. Niederreiter, eds., Proceedings of SETA'98 (Singapore, 1998), 308-317, 1999.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences that are fixed points of mappings
Index entries for characteristic functions

For generating functions Product_{k>=0} (1+a*x^(b^k)) for the following values of (a,b) see: (1,2) A000012 and A000027, (1,3) A039966 and A005836, (1,4) A151666 and A000695, (1,5) A151667 and A033042, (2,2) A001316, (2,3) A151668, (2,4) A151669, (2,5) A151670, (3,2) A048883, (3,3) A117940, (3,4) A151665, (3,5) A151671, (4,2) A102376, (4,3) A151672, (4,4) A151673, (4,5) A151674.
Cf. A062051, A000009, A000244, A004642.
Characteristic function of A005836 (and apart from offset of A003278).

a039966 n = fromEnum (n < 2 || m < 2 && a039966 n' == 1)
   where (n',m) = divMod n 3
-- Reinhard Zumkeller, Sep 29 2011

a := proc(n) option remember; if n <= 1 then RETURN(1) end if; if n = 2 then RETURN(0) end if; if n mod 3 = 2 then RETURN(0) end if; if n mod 3 = 0 then RETURN(a(1/3*n)) end if; if n mod 3 = 1 then RETURN(a(1/3*n - 1/3)) end if end proc; # Ralf Stephan, Jun 13 2005

(* first do *) Needs["DiscreteMath`Combinatorica`"] (* then *) s = Rest[ Sort[ Plus @@@ Table[UnrankSubset[n, Table[3^i, {i, 0, 4}]], {n, 32}]]]; Table[ If[ Position[s, n] == {}, 0, 1], {n, 105}] (* Robert G. Wilson v, Jun 14 2005 *)
CoefficientList[Series[Product[(1 + x^(3^k)), {k, 0, 5}], {x, 0, 111}], x] (* or *)
Nest[ Flatten[ # /. {0 -> {0, 0, 0}, 1 -> {1, 1, 0}}] &, {1}, 5] (* Robert G. Wilson v, Mar 29 2006 *)
Nest[ Join[#, #, 0 #] &, {1}, 5] (* Robert G. Wilson v, Jul 27 2014 *)

{a(n)=local(A,m); if(n<0, 0, m=1; A=1+O(x); while(m<=n, m*=3; A=(1+x)*subst(A,x,x^3)); polcoeff(A,n))} /* Michael Somos, Jul 15 2005 */

A039966(n)=vecmax(digits(n+!n,3))<2;
apply(A039966, [0..99]) \\ M. F. Hasler, Feb 15 2023

def A039966(n):
    while n > 2:
        n,r = divmod(n,3)
        if r==2: return 0
    return int(n!=2) # M. F. Hasler, Feb 15 2023

a(0) = 1, a(1) = 0, a(n) = b(n-2), where b is the sequence defined by b(0) = 1, b(3n+2) = 0, b(3n) = b(3n+1) = b(n). - Ralf Stephan
a(n) = A005043(n-1) mod 3. - Christian G. Bower, Jun 12 2005
a(n) = A002426(n) mod 3. - John M. Campbell, Aug 24 2011
a(n) = A000275(n) mod 3. - John M. Campbell, Jul 08 2016
Properties: 0 <= a(n) <= 1, a(A074940(n)) = 0, a(A005836(n)) = 1; A104406(n) = Sum(a(k), 1 <= k <= n). - Reinhard Zumkeller, Mar 05 2005
Euler transform of sequence b(n) where b(3^k) = 1, b(2*3^k) = -1 and zero otherwise. - Michael Somos, Jul 15 2005
G.f. A(x) satisfies A(x) = (1+x)*A(x^3). - Michael Somos, Jul 15 2005
G.f.: Product{k>=0} 1+x^(3^k). Exponents give A005836.

Entry revised Jun 30 2005

Offset corrected by John M. Campbell, Aug 24 2011

A004094 Powers of 2 written backwards.

Original entry on oeis.org

1, 2, 4, 8, 61, 23, 46, 821, 652, 215, 4201, 8402, 6904, 2918, 48361, 86723, 63556, 270131, 441262, 882425, 6758401, 2517902, 4034914, 8068838, 61277761, 23445533, 46880176, 827712431, 654534862, 219078635, 4281473701, 8463847412, 6927694924, 2954399858, 48196897171

Offset: 0

N. J. A. Sloane

Freeman Dyson believes that A014963(a(n)) <> 5 is true but cannot be proved, see link. - Reinhard Zumkeller, Jan 05 2005

T. D. Noe, Table of n, a(n) for n = 0..1000
Edge Foundation, Annual Question 2005
Richard Lipton, More on testing Dyson's conjecture (2014)
N. J. A. Sloane, Exciting Number Sequences (video of talk), Mar 05 2021.

Cf. A014963, A102382, A102383, A102384, A102385.
The following are parallel families: A000079 (2^n), A004094 (2^n reversed), A028909 (2^n sorted up), A028910 (2^n sorted down), A036447 (double and reverse), A057615 (double and sort up), A263451 (double and sort down); A000244 (3^n), A004167 (3^n reversed), A321540 (3^n sorted up), A321539 (3^n sorted down), A163632 (triple and reverse), A321542 (triple and sort up), A321541 (triple and sort down).
Cf. A004086 (read n backwards).
For indices of primes see A057708.

a004094 = a004086 . a000079  -- Reinhard Zumkeller, Apr 02 2014

[Seqint(Reverse(Intseq(2^n))): n in [0..35]]; // Vincenzo Librandi, Jan 22 2020

a:= n-> (s-> parse(cat(s[-i]$i=1..length(s))))(""||(2^n)):
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 21 2020

Table[FromDigits[Reverse[IntegerDigits[2^n]]], {n, 0, 35}] (* Vincenzo Librandi, Jan 22 2020 *)

rev(n)=subst(Polrev(digits(n)),'x,10)
a(n)=rev(2^n) \\ Charles R Greathouse IV, Oct 20 2014

apply( {A004094(n)=fromdigits(Vecrev(digits(2^n)))}, [0..44]) \\ M. F. Hasler, Feb 18 2021

def A004094(n):
    return int(str(2**n)[::-1]) # Chai Wah Wu, Feb 19 2021

a(n) = A004086(A000079(n)). - Reinhard Zumkeller, Apr 02 2014

More terms from Reinhard Zumkeller, Jan 05 2005

A009971 Powers of 27.

Original entry on oeis.org

1, 27, 729, 19683, 531441, 14348907, 387420489, 10460353203, 282429536481, 7625597484987, 205891132094649, 5559060566555523, 150094635296999121, 4052555153018976267, 109418989131512359209, 2954312706550833698643, 79766443076872509863361, 2153693963075557766310747

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 27), L(1, 27), P(1, 27), T(1, 27). Essentially same as Pisot sequences E(27, 729), L(27, 729), P(27, 729), T(27, 729). See A008776 for definitions of Pisot sequences.

The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n >= 1, a(n) equals the number of 27-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011

T. D. Noe, Table of n, a(n) for n = 0..100
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (27).

Cf. A000244, A001019, A008585, A008776.

[27^n: n in [0..100]]; // Vincenzo Librandi, Nov 21 2010

27^Range[0, 20] (* Paolo Xausa, Jul 19 2024 *)

a(n)=27^n \\ Charles R Greathouse IV, Sep 24 2015

[lucas_number1(n,27,0) for n in range(1, 17)] # Zerinvary Lajos, Apr 29 2009

G.f.: 1/(1-27*x). - Philippe Deléham, Nov 24 2008
a(n) = 27^n; a(n) = 27*a(n-1), n > 0; a(0)=1. - Vincenzo Librandi, Nov 21 2010
From Elmo R. Oliveira, Jul 10 2025: (Start)
E.g.f.: exp(27*x).
a(n) = A000244(n)*A001019(n) = A000244(A008585(n)). (End)

A038622 Triangular array that counts rooted polyominoes.

Original entry on oeis.org

1, 2, 1, 5, 3, 1, 13, 9, 4, 1, 35, 26, 14, 5, 1, 96, 75, 45, 20, 6, 1, 267, 216, 140, 71, 27, 7, 1, 750, 623, 427, 238, 105, 35, 8, 1, 2123, 1800, 1288, 770, 378, 148, 44, 9, 1, 6046, 5211, 3858, 2436, 1296, 570, 201, 54, 10, 1, 17303, 15115, 11505, 7590, 4302, 2067, 825, 265

Offset: 0

N. J. A. Sloane, torsten.sillke(AT)lhsystems.com

The PARI program gives any row k and any n-th term for this triangular array in square or right triangle array format. - Randall L Rathbun, Jan 20 2002
Triangle T(n,k), 0 <= k <= n, read by rows given by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = 2*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-1,k+1) for k >= 1. - Philippe Deléham, Mar 27 2007
This triangle belongs to the family of triangles defined by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = x*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + y*T(n-1,k) + T(n-1,k+1) for k >= 1. Other triangles arise by choosing different values for (x,y): (0,0) -> A053121; (0,1) -> A089942; (0,2) -> A126093; (0,3) -> A126970; (1,0)-> A061554; (1,1) -> A064189; (1,2) -> A039599; (1,3) -> A110877; (1,4) -> A124576; (2,0) -> A126075; (2,1) -> A038622; (2,2) -> A039598; (2,3) -> A124733; (2,4) -> A124575; (3,0) -> A126953; (3,1) -> A126954; (3,2) -> A111418; (3,3) -> A091965; (3,4) -> A124574; (4,3) -> A126791; (4,4) -> A052179; (4,5) -> A126331; (5,5) -> A125906. - Philippe Deléham, Sep 25 2007
Triangle read by rows = partial sums of A064189 terms starting from the right. - Gary W. Adamson, Oct 25 2008
Column k has e.g.f. exp(x)*(Bessel_I(k,2x)+Bessel_I(k+1,2x)). - Paul Barry, Mar 08 2011

			From _Paul Barry_, Mar 08 2011: (Start)
Triangle begins
     1;
     2,    1;
     5,    3,    1;
    13,    9,    4,   1;
    35,   26,   14,   5,   1;
    96,   75,   45,  20,   6,   1;
   267,  216,  140,  71,  27,   7,  1;
   750,  623,  427, 238, 105,  35,  8, 1;
  2123, 1800, 1288, 770, 378, 148, 44, 9, 1;
Production matrix is
  2, 1,
  1, 1, 1,
  0, 1, 1, 1,
  0, 0, 1, 1, 1,
  0, 0, 0, 1, 1, 1,
  0, 0, 0, 0, 1, 1, 1,
  0, 0, 0, 0, 0, 1, 1, 1,
  0, 0, 0, 0, 0, 0, 1, 1, 1,
  0, 0, 0, 0, 0, 0, 0, 1, 1, 1
(End)

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
D. Gouyou-Beauchamps and G. Viennot, Equivalence of the two-dimensional directed animal problem to a one-dimensional path problem, Adv. in Appl. Math. 9 (1988), no. 3, 334-357. See Table 1 on page 340.

Cf. A005773 (1st column), A005774 (2nd column), A005775, A066822, A000244 (row sums).

Cf. A064189, A007318, A061554, A092392.

import Data.List (transpose)
a038622 n k = a038622_tabl !! n !! k
a038622_row n = a038622_tabl !! n
a038622_tabl = iterate (\row -> map sum $
   transpose [tail row ++ [0,0], row ++ [0], [head row] ++ row]) [1]
-- Reinhard Zumkeller, Feb 26 2013

T := (n,k) -> simplify(GegenbauerC(n-k,-n+1,-1/2)+GegenbauerC(n-k-1,-n+1,-1/2)):
for n from 1 to 9 do seq(T(n,k),k=1..n) od; # Peter Luschny, May 12 2016

nmax = 10; t[n_ /; n > 0, k_ /; k >= 1] := t[n, k] = t[n-1, k-1] + t[n-1, k] + t[n-1, k+1]; t[0, 0] = 1; t[0, ] = 0; t[?Negative, ?Negative] = 0; t[n, 0] := 2 t[n-1, 0] + t[n-1, 1]; Flatten[ Table[ t[n, k], {n, 0, nmax}, {k, 0, n}]](* Jean-François Alcover, Nov 09 2011 *)

s=[0,1]; {A038622(n,k)=if(n==0,1,t=(2*(n+k)*(n+k-1)*s[2]+3*(n+k-1)*(n+k-2)*s[1])/((n+2*k-1)*n); s[1]=s[2]; s[2]=t; t)}

a(n, k) = a(n-1, k-1) + a(n-1, k) + a(n-1, k+1) for k>0, a(n, k) = 2*a(n-1, k) + a(n-1, k+1) for k=0.
Riordan array ((sqrt(1-2x-3x^2)+3x-1)/(2x(1-3x)),(1-x-sqrt(1-2x-3x^2))/(2x)). Inverse of Riordan array ((1-x)/(1+x+x^2),x/(1+x+x^2)). First column is A005773(n+1). Row sums are 3^n (A000244). If L=A038622, then L*L' is the Hankel matrix for A005773(n+1), where L' is the transpose of L. - Paul Barry, Sep 18 2006
T(n,k) = GegenbauerC(n-k,-n+1,-1/2) + GegenbauerC(n-k-1,-n+1,-1/2). In this form also the missing first column of the triangle 1,1,1,3,7,19,... (cf. A002426) can be computed. - Peter Luschny, May 12 2016
From Peter Bala, Jul 12 2021: (Start)
T(n,k) = Sum_{j = k..n} binomial(n,j)*binomial(j,floor((j-k)/2)).
Matrix product of Riordan arrays ( 1/(1 - x), x/(1 - x) ) * ( (1 - x*c(x^2))/(1 - 2*x), x*c(x^2) ) = A007318 * A061554 (triangle version), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108.
Triangle equals A007318^(-1) * A092392 * A007318. (End)
The n-th row polynomial R(n,x) equals the n-th degree Taylor polynomial of the function (1 + x)*(1 + x + x^2)^n expanded about the point x = 0. - Peter Bala, Sep 06 2022

More terms from David W. Wilson

cp's OEIS Frontend

A027465 Cube of lower triangular normalized binomial matrix.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

A108411 a(n) = 3^floor(n/2). Powers of 3 repeated.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

Python

Formula

Extensions

A036561 Nicomachus triangle read by rows, T(n, k) = 2^(n - k)*3^k, for 0 <= k <= n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

PARI

Formula

A006234 a(n) = n*3^(n-4).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

A006899 Numbers of the form 2^i or 3^j.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

PARI

PARI

Python