A128218 -id:A128218 - cp's OEIS frontend

A130883 a(n) = 2*n^2 - n + 1.

1, 2, 7, 16, 29, 46, 67, 92, 121, 154, 191, 232, 277, 326, 379, 436, 497, 562, 631, 704, 781, 862, 947, 1036, 1129, 1226, 1327, 1432, 1541, 1654, 1771, 1892, 2017, 2146, 2279, 2416, 2557, 2702, 2851, 3004, 3161, 3322, 3487, 3656, 3829, 4006, 4187, 4372, 4561

Offset: 0

Mohammad K. Azarian, Jul 26 2007

Maximum number of regions determined by n bent lines (or angular sectors). See Concrete Mathematics reference.
A "bent line" may also be regarded as a "long-legged letter V", meaning a letter V with both line segments extended to infinity.  See A117625 for the analogous sequence for a long-legged Z. - N. J. A. Sloane, Jun 18 2025
a(n)*Pi is the total length of half circle spiral after n rotations. It is formed as irregular spiral with two center points. At the 2nd stage, there are two alternatives: (1) select 2nd half circle radius, r2  = 2, the sequence will be A014105 or (2) select r2 = 0, the sequence will be A130883. See illustration in links. - Kival Ngaokrajang, Jan 19 2014
A128218(a(n)) = 2*n+1 and A128218(m) != 2*n+1 for m < a(n). - Reinhard Zumkeller, Jun 20 2015

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, 2nd ed., Addison-Wesley, Reading, MA, 1994, pp. 7-8, and Problem 1.18, pages 19 and 500.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Dmitry Efimov, Hafnian of two-parameter matrices, arXiv:2101.09722 [math.CO], 2021.
Guo-Niu Han, Enumeration of Standard Puzzles
Guo-Niu Han, Enumeration of Standard Puzzles [Cached copy]
Ângela Mestre and José Agapito, Square Matrices Generated by Sequences of Riordan Arrays, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
Kival Ngaokrajang, Illustration of irregular spirals (center points: 1, 2)
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
Franck Ramaharo, A generating polynomial for the pretzel knot, arXiv:1805.10680 [math.CO], 2018.
N. J. A. Sloane, Illustration for a(3) = 16
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000124, A000384, A014105, A084849, A128218, A128918, A152947, A270109.
See also A117625.
A row of the array in A386478.

a130883 = a128918 . (* 2)  -- Reinhard Zumkeller, Oct 27 2013

[2*n^2 - n + 1 : n in [0..50]]; // Wesley Ivan Hurt, Mar 25 2020

a[n_]:=2*n^2-n+1; (* or *) Array[ -#*(1-#*2)+1&,5!,0] (* Vladimir Joseph Stephan Orlovsky, Dec 21 2008 *)
LinearRecurrence[{3,-3,1},{1,2,7},50] (* Harvey P. Dale, Jul 20 2011 *)

a(n)=2*n^2-n+1 \\ Charles R Greathouse IV, Sep 24 2015

def A130883(n): return n*(2*n - 1) + 1 # Chai Wah Wu, May 24 2022

a(n) = a(n-1) + 4*n - 3 for n > 0, a(0)=1. - Vincenzo Librandi, Nov 23 2010
a(n) = A000124(2*n) - 2*n. - Geoffrey Critzer, Mar 30 2011
O.g.f.: (4*x^2-x+1)/(1-x)^3. - Geoffrey Critzer, Mar 30 2011
a(n) = 2*a(n-1) - a(n-2) + 4. - Eric Werley, Jun 27 2011
a(0)=1, a(1)=2, a(2)=7; for n > 2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Jul 20 2011
a(n) = A128918(2*n). - Reinhard Zumkeller, Oct 27 2013
a(n) = 1 + A000384(n). - Omar E. Pol, Apr 27 2017
E.g.f.: (2*x^2 + x + 1)*exp(x). - G. C. Greubel, Jul 14 2017
a(n) = A152947(2*n+1). - Franck Maminirina Ramaharo, Jan 10 2018

A152271 a(n)=1 for even n and (n+1)/2 for odd n.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 1, 10, 1, 11, 1, 12, 1, 13, 1, 14, 1, 15, 1, 16, 1, 17, 1, 18, 1, 19, 1, 20, 1, 21, 1, 22, 1, 23, 1, 24, 1, 25, 1, 26, 1, 27, 1, 28, 1, 29, 1, 30, 1, 31, 1, 32, 1, 33, 1, 34, 1, 35, 1, 36, 1, 37, 1, 38, 1, 39, 1, 40, 1, 41, 1, 42, 1, 43, 1, 44

Offset: 0

Philippe Deléham, Dec 01 2008

A000012 and A000027 interleaved. - Omar E. Pol, Mar 12 2012
Run lengths in A128218. - Reinhard Zumkeller, Jun 20 2015
a(n+1) is the number of reversible binary strings of length n+1 with Hamming weight 1 or 2 such that the 1's are separated by an even number of 0's. - Christian Barrientos, Jan 28 2019
Simple continued fraction of -1 + BesselJ(1,2)/BesselJ(2,2) = 1/(1 + 1/(1 + 1/(1 + 1/(2 + 1/(1 + 1/(3 + 1/(1 + 1/(4 + 1/(1 + ... ))))))))). - Peter Bala, Oct 06 2023

			G.f. = 1 + x + x^2 + 2*x^3 + x^4 + 3*x^5 + x^6 + 4*x^7 + x^8 + ... - _Michael Somos_, Mar 26 2022

Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A000012, A000027, A008619, A057979, A128218.

import Data.List (transpose)
a152271 = a057979 . (+ 2)
a152271_list = concat $ transpose [repeat 1, [1..]]
-- Reinhard Zumkeller, Aug 11 2014

Table[If[EvenQ[n],1,(n+1)/2],{n,0,120}] (* or *) LinearRecurrence[{0,2,0,-1},{1,1,1,2},120] (* or *) Riffle[Range[60],1,{1,-1,2}] (* Harvey P. Dale, Jan 20 2018 *)

Vec((1+x-x^2)/(1-2*x^2+x^4)+O(x^99)) \\ Charles R Greathouse IV, Jan 12 2012

a(n)=gcd(n+1,(n+1)\2) \\ Charles R Greathouse IV, Mar 13 2012

def A152271(n): return n+1>>1 if n&1 else 1 # Chai Wah Wu, Jan 04 2024

a(n) = 2*a(n-2) - a(n-4) with a(0)=a(1)=a(2)=1 and a(3)=2.
a(n) = (a(n-2) + a(n-3))/a(n-1).
G.f.: (1 + x - x^2)/(1 - 2*x^2 + x^4).
a(n) = A057979(n+2).
a(n)*a(n+1) = floor((n+2)/2) = A008619(n). - Paul Barry, Feb 27 2009
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)*0^floor((n-2k)/2). - Paul Barry, Feb 27 2009
a(n) = gcd(floor((n+1)/2), (n+1)). - Enrique Pérez Herrero, Mar 13 2012
G.f.: U(0) where U(k) = 1 + x*(k+1)/(1 - x/(x + (k+1)/U(k+1))) ; (continued fraction, 3-step). - Sergei N. Gladkovskii, Oct 04 2012
E.g.f.: ((2 + x)*cosh(x) + sinh(x))/2. - Stefano Spezia, Mar 26 2022
a(n) = (-1)^n * a(-2-n) for all n in Z. - Michael Somos, Mar 26 2022

A128217 Nonnegative integers n such that the square-root of n differs from its nearest integer by less than 1/4.

Original entry on oeis.org

0, 1, 4, 5, 8, 9, 10, 15, 16, 17, 18, 23, 24, 25, 26, 27, 34, 35, 36, 37, 38, 39, 46, 47, 48, 49, 50, 51, 52, 61, 62, 63, 64, 65, 66, 67, 68, 77, 78, 79, 80, 81, 82, 83, 84, 85, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125

Offset: 1

John W. Layman, Feb 19 2007

nonn

The squares are a subsequence; apparently A052928(n-1) = number of terms between (n-1)^2 and n^2. - Reinhard Zumkeller, Jun 20 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A063656. See the first differences in A128218.

Cf. A052928, A000290, A063657.

a128217 n = a128217_list !! (n-1)
a128217_list = filter f [0..] where
   f x = 4 * abs (root - fromIntegral (round root)) < 1
         where root = sqrt $ fromIntegral x
-- Reinhard Zumkeller, Jun 20 2015

nsrQ[n_]:=Module[{sr=Sqrt[n]},Abs[First[sr-Nearest[{Floor[sr], Ceiling[sr]},sr]]]<1/4]; Select[Range[0,150],nsrQ] (* Harvey P. Dale, Aug 19 2011 *)

from itertools import count, islice
from math import isqrt
def A128217_gen(startvalue=0): # generator of terms >= startvalue
    return filter(lambda n:(m:=n<<4)<(k:=(isqrt(n)<<2)+1)**2 or m>(k+2)**2, count(max(startvalue,0)))
A128217_list = list(islice(A128217_gen(),40)) # Chai Wah Wu, Jun 06 2025

Offset changed by Reinhard Zumkeller, Jun 20 2015

cp's OEIS Frontend

A130883 a(n) = 2*n^2 - n + 1.

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A152271 a(n)=1 for even n and (n+1)/2 for odd n.

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A128217 Nonnegative integers n such that the square-root of n differs from its nearest integer by less than 1/4.

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