A133140 -id:A133140 - cp's OEIS frontend

A133138 Triangle T(n,k) of the coefficients of the polynomials Q(n,x)=(1+x)[(1+x)^(n-1)+x^(n-1)], Q(0,x)=2.

2, 2, 2, 1, 3, 2, 1, 3, 4, 2, 1, 4, 6, 5, 2, 1, 5, 10, 10, 6, 2, 1, 6, 15, 20, 15, 7, 2, 1, 7, 21, 35, 35, 21, 8, 2, 1, 8, 28, 56, 70, 56, 28, 9, 2, 1, 9, 36, 84, 126, 126, 84, 36, 10, 2, 1, 10, 45, 120, 210, 252, 210, 120, 45, 11, 2

Offset: 0

Paul Curtz, Sep 21 2007

			Triangle T(n,k) begins:
n/k 0   1   2    3    4    5    6    7    8    9  10  11  12
0:  2
1:  2   2
2:  1   3   2
3:  1   3   4    2
4:  1   4   6    5    2
5:  1   5  10   10    6    2
6:  1   6  15   20   15    7    2
7:  1   7  21   35   35   21    8    2
8:  1   8  28   56   70   56   28    9    2
9:  1   9  36   84  126  126   84   36   10    2
10: 1  10  45  120  210  252  210  120   45   11   2
11: 1  11  55  165  330  462  462  330  165   55  12   2
12: 1  12  66  220  495  792  924  792  495  220  66  13   2
... - _Franck Maminirina Ramaharo_, May 18 2018

Cf. A133135.

q[n_] := (1+x)*((1+x)^(n-1) + x^(n-1)); t[n_, k_] := Coefficient[q[n], x, k]; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 16 2013 *)

Q(n, x) := (1 + x)*((1 + x)^(n - 1) + x^(n - 1))$
t(n,k) := ratcoef(expand(Q(n, x)), x, k)$
for n:0 thru 20 do print(makelist(t(n, k), k, 0, n)); /* Franck Maminirina Ramaharo, May 18 2018 */

From R. J. Mathar, Jun 12 2008: (Start)
T(n,k) = A007318(n,k), 0 <= k < n-1.
T(n,k) = A007318(n,k)+1, n-1 <= k <= n.
Sum_{k=0..n} T(n,k) = A133140(n). (End)
T(n,k) = A007318(n,k) + A097806(n,k). - Franck Maminirina Ramaharo, May 18 2018

Edited by R. J. Mathar, Jun 12 2008

A283070 Sierpinski tetrahedron or tetrix numbers: a(n) = 2*4^n + 2.

Original entry on oeis.org

4, 10, 34, 130, 514, 2050, 8194, 32770, 131074, 524290, 2097154, 8388610, 33554434, 134217730, 536870914, 2147483650, 8589934594, 34359738370, 137438953474, 549755813890, 2199023255554, 8796093022210, 35184372088834, 140737488355330, 562949953421314

Offset: 0

Peter M. Chema, Feb 28 2017

Number of vertices required to make a Sierpinski tetrahedron or tetrix of side length 2^n. The sum of the vertices (balls) plus line segments (rods) of one tetrix equals the vertices of its larger, adjacent iteration. See formula.
Equivalently, the number of vertices in the (n+1)-Sierpinski tetrahedron graph. - Eric W. Weisstein, Aug 17 2017
Also the independence number of the (n+2)-Sierpinski tetrahedron graph. - Eric W. Weisstein, Aug 29 2021
Final digit alternates 4 and 0.

Colin Barker, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Independence Number
Eric Weisstein's World of Mathematics, Sierpinski Tetrahedron Graph
Eric Weisstein's World of Mathematics, Tetrix
Eric Weisstein's World of Mathematics, Vertex Count
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Subsequence of A016957.
Cf. A052539, A279511, A279512.
First bisection of A052548, A087288; second bisection of A049332, A133140, A135440.
Cf. A002023 (edge count).

Table[2 4^n + 2, {n, 0, 30}] (* Bruno Berselli, Feb 28 2017 *)
2 (4^Range[0, 20] + 1) (* Eric W. Weisstein, Aug 17 2017 *)
LinearRecurrence[{5, -4}, {4, 10}, 20] (* Eric W. Weisstein, Aug 17 2017 *)
CoefficientList[Series[-((2 (-2 + 5 x))/(1 - 5 x + 4 x^2)), {x, 0, 20}], x] (* Eric W. Weisstein, Aug 17 2017 *)

a(n)=2*4^n+2 \\ Charles R Greathouse IV, Feb 28 2017

Vec(2*(2 - 5*x) / ((1 - x)*(1 - 4*x)) + O(x^30)) \\ Colin Barker, Mar 02 2017

def a(n): return 2*4**n + 2
print([a(n) for n in range(25)]) # Michael S. Branicky, Aug 29 2021

G.f.: 2*(2 - 5*x)/((1 - x)*(1 - 4*x)).
a(n) = 5*a(n-1) - 4*a(n-2) for n > 1.
a(n+1) = a(n) + A002023(n).
a(n) = 2*A052539(n) = A188161(n) - 1 = A087289(n) + 1 = A056469(2*n+2) = A261723(4*n+1).
E.g.f.: 2*(exp(4*x) + exp(x)). - G. C. Greubel, Aug 17 2017

Entry revised by Editors of OEIS, Mar 01 2017

A134351 Binomial transform of [1, 5, -1, 5, -1, 5, ...]. Inverse binomial transform of A134350.

Original entry on oeis.org

1, 6, 10, 18, 34, 66, 130, 258, 514, 1026, 2050, 4098, 8194, 16386, 32770, 65538, 131074, 262146, 524290, 1048578, 2097154, 4194306, 8388610, 16777218, 33554434, 67108866, 134217730, 268435458, 536870914, 1073741826, 2147483650

Offset: 1

Gary W. Adamson, Oct 21 2007

nonn

			a(4) = 18 = (1, 3, 3, 1) dot (1, 5, -1, 5) = (1 + 15 - 3 + 5).

Index entries for linear recurrences with constant coefficients, signature (3, -2).

Cf. A134350.

Essentially the same as A133140, A089985, A052548.

1,seq(2^(n+1)+2,n=1..25); # Emeric Deutsch, Oct 24 2007

a(n) = 2 + 2^(n+1) for n >= 2; a(1)=1. - Emeric Deutsch, Oct 24 2007

O.g.f.: (-1-3*x+6*x^2)/((1-x)*(-1+2*x)). - R. J. Mathar, Apr 02 2008

More terms from Emeric Deutsch, Oct 24 2007

More terms from R. J. Mathar, Apr 02 2008

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A133138 Triangle T(n,k) of the coefficients of the polynomials Q(n,x)=(1+x)[(1+x)^(n-1)+x^(n-1)], Q(0,x)=2.

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A283070 Sierpinski tetrahedron or tetrix numbers: a(n) = 2*4^n + 2.

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A134351 Binomial transform of [1, 5, -1, 5, -1, 5, ...]. Inverse binomial transform of A134350.

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