A122553 -id:A122553 - cp's OEIS frontend

A215712 Numerator of sum(i=1..n, 3*i/4^i ).

3, 9, 81, 21, 1359, 2727, 21837, 21843, 349515, 699045, 5592393, 2796201, 89478471, 178956963, 1431655749, 1431655761, 22906492227, 45812984481, 366503875905, 22906492245, 5864062014783, 11728124029599, 93824992236861, 93824992236879, 1501199875790139

Offset: 1

Alonso del Arte, Aug 21 2012

The limit as n goes to infinity is 4/3.

			a(4) = 21 because 3/4 + 6/16 + 9/64 + 12/256 = 3/4 + 3/8 + 9/64 + 3/64 = 48/64 + 24/64 + 9/64 + 3/64 = 84/64 = 21/16.

Calvin C. Clawson, The Beauty and Magic of Numbers. New York: Plenum Press (1996): 96.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A215713 for the denominators.
A036295/A036296 is the same with i/2^i instead of 3i/4^i.
Cf. A122553.

[Numerator(&+[3*i/4^i: i in [1..n]]): n in [1..25]]; // Bruno Berselli, Sep 03 2012

Table[Numerator[Sum[3i/4^i, {i, n}]], {n, 40}]

a(17) corrected by Vincenzo Librandi, Sep 04 2012

A215713 Denominator of sum(i=1..n, 3*i/4^i).

Original entry on oeis.org

4, 8, 64, 16, 1024, 2048, 16384, 16384, 262144, 524288, 4194304, 2097152, 67108864, 134217728, 1073741824, 1073741824, 17179869184, 34359738368, 274877906944, 17179869184, 4398046511104, 8796093022208, 70368744177664, 70368744177664, 1125899906842624

Offset: 1

Alonso del Arte, Aug 21 2012

The odd-indexed terms are the even-indexed powers of 4 (A013709).

			a(4) = 16 because 3/4 + 6/16 + 9/64 + 12/256 = 3/4 + 3/8 + 9/64 + 3/64 = 48/64 + 24/64 + 9/64 + 3/64 = 84/64 = 21/16.

Calvin C. Clawson, The Beauty and Magic of Numbers. New York: Plenum Press (1996): 96.

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A215712 for the numerators. A036295/A036296 is very similar but with i/2^i instead of 3i/4^i. Cf. also A122553.

[Denominator(&+[3*i/4^i: i in [1..n]]): n in [1..25]]; // Bruno Berselli, Sep 03 2012

Table[Denominator[Sum[3i/4^i, {i, n}]], {n, 40}]

vector(100, n, denominator(sum(i=1, n, 3*i/4^i))) \\ Colin Barker, Nov 09 2014

A251635 Riordan array (1-2x,x), inverse of Riordan array (1/(1-2x), x) = A130321.

Original entry on oeis.org

1, -2, 1, 0, -2, 1, 0, 0, -2, 1, 0, 0, 0, -2, 1, 0, 0, 0, 0, -2, 1, 0, 0, 0, 0, 0, -2, 1, 0, 0, 0, 0, 0, 0, -2, 1, 0, 0, 0, 0, 0, 0, 0, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 1, 0, 0, 0, 0, 0

Offset: 0

Wolfdieter Lang, Jan 10 2015

This is a simple Riordan array, an infinite lower triangular matrix. It is the inverse matrix of A130321 (with zeros above the diagonal).
Row sums have o.g.f. (1-2*x)/(1-x) and give 1, repeat(-1), i.e., A153881(n+1), n >= 0.
Alternate row sums have o.g.f. (1-2*x)/(1+x) and give 1, repeat(-3,3), i.e., (-1)^n*A122553(n).

			The triangle T(n, k) begins:
n\k  0  1  2  3  4  5  6  7  8  9 10 ...
0:   1
1:  -2  1
2:   0 -2  1
3:   0  0 -2  1
4:   0  0  0 -2  1
5:   0  0  0  0 -2  1
6:   0  0  0  0  0 -2 1
7:   0  0  0  0  0  0 -2  1
8:   0  0  0  0  0  0  0 -2  1
9:   0  0  0  0  0  0  0  0 -2  1
10:  0  0  0  0  0  0  0  0  0 -2  1
...

Cf. A130321, A153881, A122553.

a251635 n k = a251635_tabl !! n !! k
a251635_row n = a251635_tabl !! n
a251635_tabl = [1] : iterate (0 :) [-2, 1]
-- Reinhard Zumkeller, Jan 11 2015

T(n, k) = 0 if n < k and k = 0..(n-2) for n >= 2, and T(n, n) = 1 and T(n, n-1) = -2.
G.f. for row polynomials P(n, x) = -2^x^(n-1) + x^n is (1-2*z)/(1-x*z).
G.f. for k-th column: (1-2*x)*x^k, k >= 0.

A210622 Decimal expansion of 377/120.

Original entry on oeis.org

3, 1, 4, 1, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6

Offset: 1

N. J. A. Sloane, Mar 24 2012

Claudius Ptolemy's estimate for Pi.

			3.1416666666666666666666666666666666666666666666666666666666...

Petr Beckmann, A History of Pi, 3rd Ed., Boulder, Colorado: The Golem Press (1974): p. 26.
Florian Cajori, A History of Mathematical Notations, Dover edition (2012), par. 44.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §3.6 The Quest for Pi, p. 90.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 49.

Dario Castellanos, The ubiquitous Pi, Math. Mag., 61 (1988), pp. 67-98 and pp. 148-163.
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A021016, A021028, A122553.

RealDigits[377/120,10,120][[1]] (* or *) PadRight[{3,1,4,1},120,{6}] (* Harvey P. Dale, Dec 06 2017 *)

377/120. \\ Charles R Greathouse IV, Mar 25 2014

Equals A021028 plus 3.1. - R. J. Mathar, Mar 27 2012
Equals 3 + A122553 + A021016/10. - Stefano Spezia, Jul 08 2022
From Elmo R. Oliveira, Aug 02 2024: (Start)
G.f.: x*(3 + x + 4*x^2 + x^3) + 6*x^5/(1 - x).
a(n) = 6 for n >= 5. (End)

A255910 Decimal expansion of 16/9.

Original entry on oeis.org

1, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 1

Derek Orr, Mar 10 2015

Cutting the unit square [0,1] x [0,1] into two equal areas with a parabolic curve y = A*x^2 requires A to be 16/9. If you extend this to an arbitrary square [0,s] x [0,s], A = (16/9)*s.

Except for the first terms, identical to A186684, A021040 and A010727.

			1.7777777777777777777777777777...

Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A010727, A021040, A122553, A186684.

RealDigits[16/9, 10, 100][[1]] (* Vincenzo Librandi, Mar 24 2015 *)

x=16/9; for(n=1, 100, d=floor(x); x=(x-d)*10; print1(d,", "))

From Elmo R. Oliveira, Aug 05 2024: (Start)
G.f.: x + 7*x^2/(1 - x).
E.g.f.: 7*(exp(x) - 1) - 6*x.
a(n) = 7 - 6*0^(n-1).
a(n) = 7, n > 1. (End)

A278597 One half of A278481.

Original entry on oeis.org

1, 2, 2, 2, 3, 2, 2, 3, 3, 2, 2, 3, 3, 3, 2, 2, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2

Offset: 1

Omar E. Pol, Nov 23 2016

Apart from the left border and the right border, the rest of the elements are 3's.

			The sequence written as a triangle begins:
                      1;
                    2,  2;
                  2,  3,  2;
                2,  3,  3,  2;
              2,  3,  3,  3,  2;
            2,  3,  3,  3,  3,  2;
          2,  3,  3,  3,  3,  3,  2;
        2,  3,  3,  3,  3,  3,  3,  2;
      2,  3,  3,  3,  3,  3,  3,  3,  2;
    2,  3,  3,  3,  3,  3,  3,  3,  3,  2;
  ...

Row sums give A016777.
Left border gives A040000, the same as the right border.
Middle column gives A122553.
Every diagonal that is parallel to any of the borders gives the elements greater than 1 of A158799.
Cf. A278481.

a(n) = A278481(n)/2.

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A215712 Numerator of sum(i=1..n, 3*i/4^i ).

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A215713 Denominator of sum(i=1..n, 3*i/4^i).

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A251635 Riordan array (1-2*x,x), inverse of Riordan array (1/(1-2*x), x) = A130321.

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A210622 Decimal expansion of 377/120.

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A255910 Decimal expansion of 16/9.

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A278597 One half of A278481.

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A251635 Riordan array (1-2x,x), inverse of Riordan array (1/(1-2x), x) = A130321.