A132309 -id:A132309 - cp's OEIS frontend

A077552 Consider the following triangle in which the n-th row contains n distinct numbers whose product is the smallest and has the least possible number of divisors. 1 is a member of only the first row. Sequence contains the final term of the rows (the leading diagonal).

1, 3, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648

Offset: 0

Amarnath Murthy, Nov 10 2002

nonn

N points are chosen on a circle to get n arcs of different lengths. Then a(n-1) counts - Anton Zakharov, Dec 14 2016

			Triangle begins
1
2 3
2 4 8
2 4 8 16
2 4 8 16 32

Cf. A132309.

Table[2^(n + 1) - Boole[n < 2], {n, 0, 30}] (* Michael De Vlieger, Mar 15 2017 *)

For n>2, a(n)=2^n. - Ray Chandler, Aug 21 2003

Row sums of triangle A132309. - Gary W. Adamson, Aug 18 2007

Corrected and extended by Ray Chandler, Aug 21 2003

A132307 2*A007318^(2) - A000012.

Original entry on oeis.org

1, 3, 1, 7, 7, 1, 15, 23, 11, 1, 31, 63, 47, 15, 1, 63, 159, 159, 79, 19, 1, 127, 383, 479, 319, 119, 23, 1, 255, 895, 1343, 1119, 559, 167, 27, 1, 511, 2047, 3583, 3583, 2239, 895, 223, 31, 1, 1023, 4607, 9215, 10751, 8063, 4031, 1343, 287, 35, 1

Offset: 0

Gary W. Adamson, Aug 18 2007

Row sums = A132308: (1, 4, 15, 50, 157, 480, 1451, ...). Inverse binomial transform of A132307 = triangle A132309 (having row sums A077552).

			First few rows of the triangle:
   1;
   3,   1;
   7,   7,   1;
  15,  23,  11,  1;
  31,  63,  47, 15,  1;
  63, 159, 159, 79, 19, 1;
  ...

Cf. A132308, A132309, A077552.

2*A007318^(2) - A000012 as infinite lower triangular matrices.

A132308 a(n) = 2*3^n - n - 1.

Original entry on oeis.org

1, 4, 15, 50, 157, 480, 1451, 4366, 13113, 39356, 118087, 354282, 1062869, 3188632, 9565923, 28697798, 86093425, 258280308, 774840959, 2324522914, 6973568781, 20920706384, 62762119195, 188286357630, 564859072937, 1694577218860

Offset: 0

Gary W. Adamson, Aug 18 2007

nonn

Row sums of triangle A132307.

			a(2) = 15 = sum of row 2 terms of triangle A132307: (7 + 7 + 1).
a(2) = 15 = (1, 2, 1) dot (1, 3, 8) = (1 + 6 + 8); where A077552 = (1, 3, 8, 16, 32, 64, ...).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-7,3).

Cf. A132307, A132309, A077552.

A132308:=n->2*3^n-n-1; seq(A132308(n), n=0..30); # Wesley Ivan Hurt, Mar 26 2014

s=1;lst={s};Do[s+=(n+=s++)+n;AppendTo[lst, s], {n, 0, 5!, 1}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 15 2008 *)
Table[2*3^n-n-1,{n,0,30}] (* or *) LinearRecurrence[{5,-7,3},{1,4,15},30] (* Harvey P. Dale, Aug 06 2013 *)
CoefficientList[Series[(1 - x + 2 x^2)/((1 - 3 x) (1 - x)^2), {x, 0, 50}], x] (* Vincenzo Librandi, Mar 27 2014 *)

Binomial transform of A077552.
a(n) = 2*3^n - n - 1. - Rolf Pleisch, Sep 26 2010
G.f.: (1-x+2*x^2)/((1-3*x)*(1-x)^2). - Bruno Berselli, Mar 31 2011

More terms from Vladimir Joseph Stephan Orlovsky, Nov 15 2008
Wrong formula 2*n^3-n-1 removed by Rolf Pleisch, Oct 19 2010
Simpler definition from Wesley Ivan Hurt, Mar 26 2014

cp's OEIS Frontend

A077552 Consider the following triangle in which the n-th row contains n distinct numbers whose product is the smallest and has the least possible number of divisors. 1 is a member of only the first row. Sequence contains the final term of the rows (the leading diagonal).

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A132307 2*A007318^(2) - A000012.

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A132308 a(n) = 2*3^n - n - 1.

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