A083855 -id:A083855 - cp's OEIS frontend

A083584 a(n) = (8*4^n - 5)/3.

1, 9, 41, 169, 681, 2729, 10921, 43689, 174761, 699049, 2796201, 11184809, 44739241, 178956969, 715827881, 2863311529, 11453246121, 45812984489, 183251937961, 733007751849, 2932031007401, 11728124029609, 46912496118441

Offset: 0

Paul Barry, May 01 2003

a(n) = A007583(n+1) - 2 = A020988(n) - 1 = A039301(n+2) - 3. - Ralf Stephan, Jun 14 2003

Sum of n-th row of triangle of powers of 4: 1; 4 1 4; 16 4 1 4 16; 64 16 4 1 4 16 64; .... - Philippe Deléham, Feb 24 2014

			a(0) = 1;
a(1) = 4 + 1 + 4 = 9;
a(2) = 16 + 4 + 1 + 4 + 16 = 41;
a(3) = 64 + 16 + 4 + 1 + 4 + 16 + 64 = 169; etc. - _Philippe Deléham_, Feb 24 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..170
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A083855.

[(8*4^n-5)/3: n in [0..40] ]; // Vincenzo Librandi, Apr 28 2011

(8 4^Range[0,30]-5)/3 (* or *) LinearRecurrence[{5,-4},{1,9},30] (* Harvey P. Dale, Oct 23 2011 *)

a(n)=(8*4^n-5)/3 \\ Charles R Greathouse IV, Oct 07 2015

print([8*4**n//3 - 1 for n in range(50)]) # Karl V. Keller, Jr., May 21 2022

a(n) = (8*4^n - 5)/3.
G.f.: (1+4*x)/((1-x)*(1-4*x)).
E.g.f.: (8*exp(4*x) - exp(x))/3.
a(0)=1, a(1)=9, a(n) = 5*a(n-1) - 4*a(n-2). - Harvey P. Dale, Oct 23 2011
a(n) = 4*a(n-1) + 5, a(0) = 1. - Philippe Deléham, Feb 24 2014
a(n+1) = 2^(2^n+1) + a(n), a(1)=1. - Ben Paul Thurston, Dec 27 2015

A083854 Numbers that are squares, twice squares, three times squares, or six times squares, i.e., numbers whose squarefree part divides 6.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 25, 27, 32, 36, 48, 49, 50, 54, 64, 72, 75, 81, 96, 98, 100, 108, 121, 128, 144, 147, 150, 162, 169, 192, 196, 200, 216, 225, 242, 243, 256, 288, 289, 294, 300, 324, 338, 361, 363, 384, 392, 400, 432, 441, 450, 484, 486, 507

Offset: 0

Henry Bottomley, May 06 2003

nonn

It is simple to divide equilateral triangles into these numbers of congruent parts: squares by making smaller equilateral triangles; 6*squares by dividing each small equilateral triangle by its medians into small right triangles; and 2*squares or 3*squares by recombining three or two of these small right triangles.

Ivan Neretin, Table of n, a(n) for n = 0..10000

Cf. A000290, A007913, A001105, A028982, A033428, A033581, A083855, A195055.

mx = 23; Sort@Select[Flatten@Table[{1, 2, 3, 6} n^2, {n, mx}], # <= mx^2 &] (* Ivan Neretin, Nov 08 2016 *)

a(n) is bounded below by 0.137918...*n^2 where 0.137918... = 3*(3-2*sqrt(2))*(2-sqrt(3)); the error appears to be O(n).

Sum_{n>=1} 1/a(n) = Pi^2/3 (A195055). - Amiram Eldar, Dec 19 2020

cp's OEIS Frontend

A083584 a(n) = (8*4^n - 5)/3.

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