A134347 -id:A134347 - cp's OEIS frontend

A048473 a(0)=1, a(n) = 3a(n-1) + 2; a(n) = 23^n - 1.

1, 5, 17, 53, 161, 485, 1457, 4373, 13121, 39365, 118097, 354293, 1062881, 3188645, 9565937, 28697813, 86093441, 258280325, 774840977, 2324522933, 6973568801, 20920706405, 62762119217, 188286357653, 564859072961, 1694577218885, 5083731656657, 15251194969973, 45753584909921

Offset: 0

Clark Kimberling

The number of triangles (of all sizes, including holes) in Sierpiński's triangle after n inscriptions. - Lee Reeves, May 10 2004
The sequence is not only related to Sierpiński's triangle, but also to "Floret's cube" and the quaternion factor space Q X Q / {(1,1), (-1,-1)}. It can be written as a_n = ves((A+1)x)^n) as described at the Math Forum Discussions link. - Creighton Dement, Jul 28 2004
Relation to C(n) = Collatz function iteration using only odd steps: If we look for record subsequences where C(n) > n, this subsequence starts at 2^n - 1 and stops at the local maximum of 2*3^n - 1. Examples: [3,5], [7,11,17], [15,23,35,53], ..., [127,191,287,431,647,971,1457]. - Lambert Klasen, Mar 11 2005
Group the natural numbers so that the (2n-1)-th group sum is a multiple of the (2n)-th group containing one term. (1,2),(3),(4,5,6,7,8,9,10,11),(12),(13,14,15,16,17,18,19,...,38),(39),(40,41,...,118,119),(120), (121,122,123,...) ... a(n) = {the sum of the terms of (2n-1)-th group}/{the term of (2n)th group}. The first term of the odd numbered group is given by A003462. The only term of even numbered group is given by A029858. - Amarnath Murthy, Aug 01 2005
a(n)+1 = A008776(n); it appears that this gives the number of terms in the (n+1)-th "gap" of numbers missing in A171884. - M. F. Hasler, May 09 2013
Sum of n-th row of triangle of powers of 3: 1; 1 3 1; 1 3 9 3 1; 1 3 9 27 9 3 1; ... - Philippe Deléham, Feb 23 2014
For n >= 3, also the number of dominating sets in the n-helm graph. - Eric W. Weisstein, May 28 2017
The number of elements of length <= n in the free group on two generators. - Anton Mellit, Aug 10 2017
In general, a first order inhomogeneous recurrence of the form s(0) = a, s(n) = m*s(n-1) + k, n>0, will have a closed form of a*m^n + ((m^n-1)/(m-1))*k. - Gary Detlefs, Jun 07 2024

			a(0) = 1;
a(1) = 1 + 3 + 1 = 5;
a(2) = 1 + 3 + 9 + 3 + 1 = 17;
a(3) = 1 + 3 + 9 + 27 + 9 + 3 + 1 = 53; etc. - _Philippe Deléham_, Feb 23 2014

Theoni Pappas, Math Stuff, Wide World Publ/Tetra, San Carlos CA, page 15, 2002.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Creighton Dement, A paper on floretions and quaternions, some questions, The Math Forum.
Eric Weisstein's World of Mathematics, Dominating Set.
Eric Weisstein's World of Mathematics, Helm Graph.
Index entries for linear recurrences with constant coefficients, signature (4,-3).

a(n)=T(2,n), array T given by A048471.
Cf. A003462, A029858. A column of A119725.
Cf. A008776, A010684, A046055, A112468, A112739, A134347, A171884.

[2*3^n - 1: n in [0..30]]; // Vincenzo Librandi, Sep 23 2011

g:= ((1+x)/(1-3*x)/(1-x)): gser:=series(g, x=0, 43): seq(coeff(gser, x, n), n=0..30); # Zerinvary Lajos, Jan 11 2009; typo fixed by Marko Mihaily, Mar 07 2009

NestList[3 # + 2 &, 1, 30] (* Harvey P. Dale, Mar 06 2012 *)
LinearRecurrence[{4, -3}, {1, 5}, 30] (* Harvey P. Dale, Mar 06 2012 *)
Table[2 3^n - 1, {n, 20}] (* Eric W. Weisstein, May 28 2017 *)
2 3^Range[20] - 1 (* Eric W. Weisstein, May 28 2017 *)

first(m)=vector(m,n,n--;2*3^n - 1) \\ Anders Hellström, Dec 11 2015

n-th difference of a(n), a(n-1), ..., a(0) is 2^(n+1) for n=1, 2, 3, ...
a(0)=1, a(n) = a(n-1) + 3^n + 3^(n-1). - Lee Reeves, May 10 2004
a(n) = (3^n + 3^(n+1) - 2)/2. - Creighton Dement, Jul 31 2004
(1, 5, 17, 53, 161, ...) = Ternary (1, 12, 122, 1222, 12222, ...). - Gary W. Adamson, May 02 2005
Row sums of triangle A134347. Also, binomial transform of A046055: (1, 4, 8, 16, 32, 64, ...); and double binomial transform of A010684: (1, 3, 1, 3, 1, 3, ...). - Gary W. Adamson, Oct 21 2007
G.f.: (1+x)/((1-3*x)*(1-x)). - Zerinvary Lajos, Jan 11 2009; corrected by R. J. Mathar, Jan 21 2009
a(0)=1, a(1)=5, a(n) = 4*a(n-1) - 3*a(n-2). - Harvey P. Dale, Mar 06 2012
a(n) = Sum_{k=0..n} A112468(n,k)*4^k. - Philippe Deléham, Feb 23 2014
E.g.f.: exp(x)*(2*exp(2*x) - 1). - Elmo R. Oliveira, Mar 08 2025

Better description from Amarnath Murthy, May 27 2001

A134346 Triangle read by rows: T(n,k) = (2^(n+1)-1)*binomial(n,k).

Original entry on oeis.org

1, 3, 3, 7, 14, 7, 15, 45, 45, 15, 31, 124, 186, 124, 31, 63, 315, 630, 630, 315, 63, 127, 762, 1905, 2540, 1905, 762, 127, 255, 1785, 5355, 8925, 8925, 5355, 1785, 255, 511, 4088, 14308, 28616, 35770, 28616, 14308, 4088, 511

Offset: 0

Gary W. Adamson, Oct 21 2007

Inverse binomial transform: A134347.
From Wolfdieter Lang, Jul 27 2022: (Start)
Also the triangle t with offset 1 and elements t(n, m) = T(n-1, m-1) read by rows, giving in row n >= 1 the sums of the entries of A356028 of like m.
Also triangle t with offset 1 read by rows, giving in row n >= 1 the sum of the numbers from 1, 2, ..., 2^n - 1 with binary weight m, for m = 1, 2, ..., n. [Observation by Kevin Ryde.] (End)
T(n,k) is the sum of the entries in the (k+2)-th column of the Christmas tree pattern (A367562) of order n+1. - Paolo Xausa, Dec 20 2023

			First few rows of the triangle:
n\k    0    1     2     3      4      5     6     7    8   9 ...
0:     1
1:     3    3
2:     7   14     7
3:    15   45    45    15
4:    31  124   186   124     31
5:    63  315   630   630    315     63
6:   127  762  1905  2540   1905    762   127
7:   255 1785  5355  8925   8925   5355  1785   255
8:   511 4088 14308 28616  35770  28616 14308  4088  511
9:  1023 9207 36828 85932 128898 128898 85932 36828 9207 1023
... reformatted by _Wolfdieter Lang_, Aug 21 2022
----------------------------------------------------------------------------------
T(3, 1) = 12 + 10 + 9 + 6 + 5 + 3 = 45. (From A356028 row n = 4, m = 2.)
Recurrences: T(4, 1) = 45 + 15 + 4*16 = 2*(45 + 15) +4 = 124. - _Wolfdieter Lang_, Jul 27 2022

Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of the triangle, flattened).

Cf. A000225, A006516(n+1) (row sums), A124929, A134347, A356028, A356117.

Cf. A367508, A367562.

A134346 := proc(n,k)
    (2^(n+1)-1)*binomial(n,k) ;
end proc:
seq(seq( A134346(n,k),k=0..n),n=0..10) ; # R. J. Mathar, Aug 15 2022
ser := series((1/2 - x)^(k - n - 1) - (1 - x)^(k - n - 1), x, 10):
seq(seq(coeff(ser, x, k), k = 0..n), n = 0..9); # Peter Luschny, Aug 22 2022

A134346[n_,k_]:=(2^(n+1)-1)Binomial[n,k];
Table[A134346[n,k],{n,0,10},{k,0,n}] (* Paolo Xausa, Dec 20 2023 *)

T(n,k) = my(b=binomial(n,k)); b<<(n+1) - b; \\ Kevin Ryde, Aug 15 2022

T(n, m) = A000225(n+1)*A007318(n, m).
From Wolfdieter Lang, Aug 21 2022: (Start)
T(n, k) = 0 for n < k, T(n, 0) = 2^(n+1) - 1, and
T(n, k) = T(n-1, k) + T(n-1, k-1) + binomial(n, k)*2^n, or
T(n, k) = 2*(T(n-1, k) + T(n-1, k-1)) + binomial(n-1, k-1).
(Proof for T(n-1, m-1) = t(n, m), offset 1, by separating in the list of the binary code of the numbers 1, 2, ..., 2^n-1 of length n and weight m the sublists with first entry 1 and 0. The total number of elements of the list for n and m is binomial(n, m).) (End)
T(n, k) = [x^k] ((1/2 - x)^(k - n - 1) - (1 - x)^(k - n - 1)). - Peter Luschny, Aug 22 2022

Name simplified by R. J. Mathar, Aug 15 2022

A210196 Triangle of coefficients of polynomials v(n,x) jointly generated with A210195; see the Formula section.

Original entry on oeis.org

1, 1, 4, 1, 8, 8, 1, 12, 24, 16, 1, 16, 48, 64, 32, 1, 20, 80, 160, 160, 64, 1, 24, 120, 320, 480, 384, 128, 1, 28, 168, 560, 1120, 1344, 896, 256, 1, 32, 224, 896, 2240, 3584, 3584, 2048, 512, 1, 36, 288, 1344, 4032, 8064, 10752, 9216, 4608, 1024, 1, 40

Offset: 1

Clark Kimberling, Mar 18 2012

Periodic alternating row sums: 1, -3, 1, -3, 1, -3, ...

For a discussion and guide to related arrays, see A208510.

			First five rows:
  1;
  1,  4;
  1,  8,  8;
  1, 12, 24, 16;
  1, 16, 48, 64, 32;
First three polynomials v(n,x): 1, 1 + 4x, 1 + 8x + 8x^2.

Cf. A134347, A210195, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + v[n - 1, x] + 1;
v[n_, x_] := 2 x*u[n - 1, x] + 2 x*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]   (* A210195 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A210196 *)

u(n,x) = u(n-1,x) + v(n-1,x) + 1, v(n,x) = 2*x*u(n-1,x) + 2*x*v(n-1,x) + 1, where u(1,x)=1, v(1,x)=1.

Conjecture: T(n,0) = 1 and T(n,k) = 2^(k+1)*binomial(n-1,k) if k>0. - Knud Werner, Jan 10 2022

cp's OEIS Frontend

A134348 A007318^(-1) * A134347.

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A048473 a(0)=1, a(n) = 3*a(n-1) + 2; a(n) = 2*3^n - 1.

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A134346 Triangle read by rows: T(n,k) = (2^(n+1)-1)*binomial(n,k).

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A210196 Triangle of coefficients of polynomials v(n,x) jointly generated with A210195; see the Formula section.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A048473 a(0)=1, a(n) = 3a(n-1) + 2; a(n) = 23^n - 1.