A017911 -id:A017911 - cp's OEIS frontend

A057048 a(n) = A017911(n+1) = round(sqrt(2)^(n+1)).

1, 2, 3, 4, 6, 8, 11, 16, 23, 32, 45, 64, 91, 128, 181, 256, 362, 512, 724, 1024, 1448, 2048, 2896, 4096, 5793, 8192, 11585, 16384, 23170, 32768, 46341, 65536, 92682, 131072, 185364, 262144, 370728, 524288, 741455, 1048576

Offset: 0

M. F. Hasler, Feb 20 2012

If the natural numbers A000027 are written as a triangle, then a(n) gives the row of the triangle in which the number 2^n can be found. See A017911 for a more elaborate explanation and relation with A000217. [Original definition by Clark Kimberling, Jul 30 2000, clarified by M. F. Hasler, Feb 20 2012, following an observation from T. D. Noe, Apr 27 2003]

			Write the natural numbers A000027 as a triangle:
row 1: 1 . . . <- 2^0 in row 1=a(0)
row 2: 2 3 . . . <- 2^1 in row 2=a(1)
row 3: 4 5 6 . . . <- 2^2 in row 3=a(2)
row 4: 7 8 9 10 . . <- 2^3 in row 4=a(3)
row 5: 11 12 13 14 15
row 6: 16 17 18 19 20 21 <- 2^4 in row 6=a(4).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

[Round(Sqrt(2)^(n+1)): n in [0..50]]; // Vincenzo Librandi, Mar 24 2013

Table[Round[Sqrt[2]^(n+1)], {n, 0, 50}] (* Vincenzo Librandi, Mar 24 2013 *)

A057048(n)=round(sqrt(2^(n+1)))  /* for large values, an implementation using integer arithmetic would be preferable */ \\ M. F. Hasler, Feb 20 2012

a(n)=sqrtint(2^(n+1)) \\ Charles R Greathouse IV, Aug 19 2016

from math import isqrt
def A057048(n): return -isqrt(m:=1<Chai Wah Wu, Jun 18 2024

a(2n-1) = 2^n, n > 0. - M. F. Hasler, Feb 20 2012

A001521 a(1) = 1; thereafter a(n+1) = floor(sqrt(2a(n)(a(n)+1))).

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 13, 19, 27, 38, 54, 77, 109, 154, 218, 309, 437, 618, 874, 1236, 1748, 2472, 3496, 4944, 6992, 9888, 13984, 19777, 27969, 39554, 55938, 79108, 111876, 158217, 223753, 316435, 447507, 632871, 895015, 1265743, 1790031, 2531486, 3580062, 5062972

Offset: 1

N. J. A. Sloane

Graham and Pollak give an elementary proof of the following result: For given m, define a(n) by a(1) = m and a(n+1) = floor(sqrt(2*a_n*(a_n + 1))), n >= 1. Then a(n) = tau_m(2^((n-1)/2) + 2^((n-2)/2)) where tau_m is the m-th smallest element of {1, 2, 3, ... } union { sqrt(2), 2*sqrt(2), 3*sqrt(2), ... }. For m=1 it follows as a curious corollary that a(2n+1) - 2*a(2n-1) is exactly the n-th bit in the binary expansion of sqrt(2) (A004539).

a(n) is also the curvature (rounded down) of the circle inscribed in the n-th 45-45-90 triangle arranged in a spiral as shown in the illustration in the links section. - Kival Ngaokrajang, Aug 21 2013

R. L. Graham, D. E. Knuth and O. Pataschnic, Concrete Mathematics, Addison-Wesley, Reading (1994) 2nd Ed., Ex. 3.46.
F. K. Hwang, and Shen Lin. "An analysis of Ford and Johnson's sorting algorithm." In Proc. Third Annual Princeton Conf. on Inform. Sci. and Systems, pp. 292-296. 1969.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Harvey P. Dale, Table of n, a(n) for n = 1..5000 (first 200 terms from T. D. Noe)
R. L. Graham and H. O. Pollak, Note on a nonlinear recurrence related to sqrt(2), Mathematics Magazine, Volume 43, Pages 143-145, 1970. Zbl 201.04705.
R. L. Graham and H. O. Pollak, Note on a nonlinear recurrence related to sqrt(2) (annotated and scanned copy)
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
Kival Ngaokrajang, Illustration for some initial terms
S. Rabinowitz and P. Gilbert, A nonlinear recurrence yielding binary digits, Math. Mag. 64 (1991), no. 3, 168-171.
Th. Stoll, On Families of Nonlinear Recurrences Related to Digits, Journal of Integer Sequences, Vol. 8 (2005), Article 05.3.2.

Cf. A000196.

First, second, and third differences give A017911, A190660, A241576.

a001521 n = a001521_list !! (n-1)
a001521_list = 1 : (map a000196 $ zipWith (*)
                    (map (* 2) a001521_list) (map (+ 1) a001521_list))
-- Reinhard Zumkeller, Dec 16 2013

[Floor(Sqrt(2)^(n-1)+Sqrt(2)^(n-2)): n in [1..45]]; // Vincenzo Librandi, May 24 2015

Digits:=200;
f:=proc(n) option remember;
if n=1 then 1 else floor(sqrt(2*f(n-1)*(f(n-1)+1))); fi; end;
[seq(f(n),n=1..200)];

With[{c=Sqrt[2]},Table[Floor[c^(n-1)+c^(n-2)],{n,1,50}]] (* Harvey P. Dale, May 11 2011 *)
NestList[Floor[Sqrt[2#(#+1)]]&,1,50] (* Harvey P. Dale, Aug 28 2013 *)

a(n)=if(n>1, sqrtint(2^(n-1)) + sqrtint(2^(n-2)), 1) \\ Charles R Greathouse IV, Nov 27 2016

first(n)=my(v=vector(n)); v[1]=1; for(k=2,n, v[k]=sqrtint(2*(v[k-1]+1)*v[k-1])); v \\ Charles R Greathouse IV, Jan 23 2020

[floor(sqrt(2)^(n-1))+ floor(sqrt(2)^(n-2)) for n in (1..50)] # Bruno Berselli, May 25 2015

a(n) = floor( sqrt(2)^(n-1) ) + floor( sqrt(2)^(n-2) ), n>1. - Ralf Stephan, Sep 18 2004

k * sqrt(2)^n - 2 < a(n) < k * sqrt(2)^n, where k = (1 + sqrt(2))/2 = A174968 = 1.2071.... Probably the first inequality can be improved (!). - Charles R Greathouse IV, Jan 23 2020

Additional comments from Torsten Sillke, Apr 06 2001

A329278 Irregular table read by rows. The n-th row is the permutation of {0, 1, 2, ..., 2^n-1} given by T(n,k) = k(k+1)/2 (mod 2^n).

Original entry on oeis.org

0, 0, 1, 0, 1, 3, 2, 0, 1, 3, 6, 2, 7, 5, 4, 0, 1, 3, 6, 10, 15, 5, 12, 4, 13, 7, 2, 14, 11, 9, 8, 0, 1, 3, 6, 10, 15, 21, 28, 4, 13, 23, 2, 14, 27, 9, 24, 8, 25, 11, 30, 18, 7, 29, 20, 12, 5, 31, 26, 22, 19, 17, 16, 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 2

Offset: 0

Peter Kagey, Nov 11 2019

Conjecture: for n > 0, the n-th row has 2^(n-1)-1 descents.

T(n,k) = A000217(k) for 0 <= k <= A017911(n+1), and T(n,2^n-1) = 2^(n-1).

			Table begins:
  0;
  0, 1;
  0, 1, 3, 2;
  0, 1, 3, 6,  2,  7, 5,  4;
  0, 1, 3, 6, 10, 15, 5, 12, 4, 13, 7, 2, 14, 11, 9, 8;
  ...

Peter Kagey, Table of n, a(n) for n = 0..8190 (first 12 rows)

Cf. A000217, A000225, A017911, A053645, A105332, A330766, A331105, A363674.

T(n,2^n-1) gives A131577.

T:= (n, k)-> irem(k*(k+1)/2, 2^n):
seq(seq(T(n, k), k=0..2^n-1), n=0..6);  # Alois P. Heinz, Jan 08 2020

A001675 a(n) = round(sqrt( 2*Pi )^n).

Original entry on oeis.org

1, 3, 6, 16, 39, 99, 248, 622, 1559, 3907, 9793, 24546, 61529, 154230, 386598, 969056, 2429064, 6088760, 15262259, 38256810, 95895601, 240374624, 602529829, 1510318305, 3785806568, 9489609784, 23786924201, 59624976768, 149457652642, 374634777972

Offset: 0

N. J. A. Sloane

nonn

Cf. A001674 (floor sqrt(2 Pi)^n), A001698 (ceiling sqrt(2 Pi)^n).

Cf. A017911 (round sqrt(2)), A000227 (round e^n), A002160 (round Pi^n).

Table[Floor[Sqrt[2*Pi]^n + 1/2], {n, 0, 50}] (* T. D. Noe, Aug 09 2012 *)
Round[(Sqrt[2*Pi])^Range[0,30] ] (* Harvey P. Dale, Jun 05 2018 *)

apply( a(n)=(2*Pi)^(n/2)\/1, [0..40]) \\ M. F. Hasler, May 29 2018

A190660 Number of triangular numbers T(k) between powers of 2, 2^(n-1) < T(k) <= 2^n.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 3, 5, 7, 9, 13, 19, 27, 37, 53, 75, 106, 150, 212, 300, 424, 600, 848, 1200, 1697, 2399, 3393, 4799, 6786, 9598, 13573, 19195, 27146, 38390, 54292, 76780, 108584, 153560, 217167, 307121, 434334, 614242, 868669, 1228483, 1737338, 2456966

Offset: 0

John W. Nicholson, May 16 2011

nonn

Count of triangular numbers between powers of 2.
a(n)/a(n-1) converges to sqrt(2) (A002193). [John W. Nicholson, May 16 2011]
Essentially first differences of A017911. - Jeremy Gardiner, Aug 11 2013. Also second differences of A001521. - N. J. A. Sloane, Apr 27 2014

			Between 2^(6-1)=32 and 2^6=64 are T(8)=36, T(9)=45, T(10)=55 so A190660(6)=3.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A001521, A002193, A017911.

TriangularIndex[n_] := (-1 + Sqrt[1 + 8*n])/2; Differences[Table[Floor[TriangularIndex[2^n]], {n, -1, 50}]] (* T. D. Noe, May 19 2011 *)

a(n) = if (n==0, 1, sum(i=2^(n-1)+1, 2^n, ispolygonal(i, 3))); \\ Michel Marcus, Apr 28 2014

Extended by T. D. Noe, May 19 2011

A017929 Powers of sqrt(8) rounded to nearest integer.

Original entry on oeis.org

1, 3, 8, 23, 64, 181, 512, 1448, 4096, 11585, 32768, 92682, 262144, 741455, 2097152, 5931642, 16777216, 47453133, 134217728, 379625062, 1073741824, 3037000500, 8589934592, 24296004000, 68719476736

Offset: 0

N. J. A. Sloane

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..2000

[Round(Sqrt(8)^n): n in [0..40]]; // Vincenzo Librandi, Nov 19 2011

Floor[(Sqrt[8]^Range[0,40]+1/2)] (* Vincenzo Librandi, Nov 19 2011 *)

a(n)=round(sqrt(8)^n) \\ Charles R Greathouse IV, Nov 18 2011

a(n) = A017911(3n). - R. J. Mathar, Apr 28 2008

A241576 Third differences of A001521.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 1, 2, 2, 2, 4, 6, 8, 10, 16, 22, 31, 44, 62, 88, 124, 176, 248, 352, 497, 702, 994, 1406, 1987, 2812, 3975, 5622, 7951, 11244, 15902, 22488, 31804, 44976, 63607, 89954, 127213, 179908, 254427, 359814, 508855, 719628, 1017709

Offset: 1

N. J. A. Sloane, Apr 27 2014

nonn

Cf. A001521, A017911, A190660.

Differences[NestList[Floor[Sqrt[2#(#+1)]]&,1,50],3] (* Harvey P. Dale, Dec 17 2019 *)

A305289 Powers of 2*Pi, rounded to the nearest integer.

Original entry on oeis.org

1, 6, 39, 248, 1559, 9793, 61529, 386598, 2429064, 15262259, 95895601, 602529829, 3785806568, 23786924201, 149457652642, 939070127125, 5900351625162, 37073002638414, 232936545470713, 1463583480006755, 9195966217409213, 57779959822545404, 363042194606444109, 2281061383037441740

Offset: 0

M. F. Hasler, May 29 2018

nonn

Index entries of sequences related to powers of irrational constants.

Cf. A001674 (floor(sqrt(2 Pi)^n)), A001675 (round sqrt(2 Pi)^n), A001698 (ceiling sqrt(2 Pi)^n), A017911 (round sqrt(2)^n), A000227 (round e^n), A002160 (round Pi^n).

Round[(2Pi)^Range[0,30]] (* Harvey P. Dale, Aug 12 2021 *)

PARI
```
apply( a(n)=(2*Pi)^n\/1, [0..40])
```

a(n) = round((2 Pi)^n) = A001675(2*n) >= A001674(2n).

cp's OEIS Frontend

A057048 a(n) = A017911(n+1) = round(sqrt(2)^(n+1)).

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A001521 a(1) = 1; thereafter a(n+1) = floor(sqrt(2*a(n)*(a(n)+1))).

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A329278 Irregular table read by rows. The n-th row is the permutation of {0, 1, 2, ..., 2^n-1} given by T(n,k) = k(k+1)/2 (mod 2^n).

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A001675 a(n) = round(sqrt( 2*Pi )^n).

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A190660 Number of triangular numbers T(k) between powers of 2, 2^(n-1) < T(k) <= 2^n.

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A017929 Powers of sqrt(8) rounded to nearest integer.

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A241576 Third differences of A001521.

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A001521 a(1) = 1; thereafter a(n+1) = floor(sqrt(2a(n)(a(n)+1))).