A053187 -id:A053187 - cp's OEIS frontend

A000194 n appears 2n times, for n >= 1; also nearest integer to square root of n.

0, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10

Offset: 0

N. J. A. Sloane

Define the oblong root obrt(x) to be the (larger) solution of y * (y+1) = x; i.e., obrt(x) = sqrt(x+1/4) - 1/2. So obrt(x) is an integer iff x is an oblong number (A002378). Then a(n) = ceiling(obrt(n)). - Franklin T. Adams-Watters, Jun 24 2015
From Wolfdieter Lang, Mar 12 2019: (Start)
The general Pell equation is related to the non-reduced form F(n) = Xvec^T A(n) Xvec = x^2 - D(n)*y^2 with D(n) = A000037(n) (D not a square), Xvec = (x,y)^T (T for transposed) and A(n) = matrix[[1,0], [0,-D(n)]]. The discriminant of F(n) = [1, 0, -D(n)] is 4*D(n).
The first reduced form appears after two applications of an equivalence transformation A' = R^T A R obtained with R = R(t) = matrix([0, -1], [1, t]), namely first with t = 0, leading to the still not reduced form [-D, 0, 1], and then with t = ceiling(f(4*D(n))/2 - 1), where f(4*D(n)) = ceiling(2*sqrt(D(n))). This can be shown to be a(n), which is also D(n) - n, for n >= 1 (see a formula below).
This leads to the reduced form FR(n) = [1, 2*a(n), -(D(n) - a(n)^2)] = [1, 2*a(n), -(n - a(n)*(a(n) - 1))]. Example: n = 5, a(5) = 2: D(5) = 7 and FR(5) = [1, 4, -3].  (End)

			G.f. = x + x^2 + 2*x^3 + 2*x^4 + 2*x^5 + 2*x^6 + 3*x^7 + 3*x^8 + 3*x^9 + 3*x^10 + ...

Titu Andreescu, Dorin Andrica, and Zuming Feng, 104 Number Theory Problems, Birkhäuser, 2006, 59-60.
B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 78, Entry 24.

T. D. Noe, Table of n, a(n) for n = 0..1000
Jonathan M. Borwein and others, Nearest Integer Zeta Functions, solution to Problem 10212, The American Mathematical Monthly, Vol. 101, No. 6 (1994), pp. 579-580.
G. Gutin, Problem 913 (BCC20.5), Mediated digraphs, in Research Problems from the 20th British Combinatorial Conference, Discrete Math., 308 (2008), 621-630.
M. A. Nyblom, Some curious sequences involving floor and ceiling functions, Am. Math. Monthly 109 (#6, 2002), 559-564.
Michael Somos, Sequences used for indexing triangular or square arrays.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A000037, A002024, A002061, A002378, A053187, A105209, A259351.
Partial sums of A005369.
Cf. A000196 (floor(sqrt(n))), A003059 (ceiling(sqrt(n))).

a000194 n = a000194_list !! (n-1)
a000194_list = concat $ zipWith ($) (map replicate [2,4..]) [1..]
-- Reinhard Zumkeller, Mar 18 2011

Digits := 100; f := n->round(evalf(sqrt(n))); [ seq(f(n), n=0..100) ];
# More efficient:
a := n -> isqrt(n): seq(a(n), n=0..98); # Peter Luschny, Mar 13 2019

A000194[n_] := Floor[(1 + Sqrt[4 n - 3])/2]; (* Enrique Pérez Herrero, Apr 14 2010 *)
Flatten[Table[PadRight[{}, 2 n, n], {n, 10}]] (* Harvey P. Dale, Nov 16 2011 *)
CoefficientList[Series[x QPochhammer[-x^2, x^4] QPochhammer[x^8, x^8]/(1 - x), {x, 0, 50}], x] (* Eric W. Weisstein, Jan 10 2024 *)

{a(n) = ceil( sqrtint(4*n) / 2)}; /* Michael Somos, Feb 11 2004 */

a(n)=(sqrtint(4*n) + 1)\2 \\ Charles R Greathouse IV, Jun 08 2020

apply( {A000194(n)=sqrtint(4*n)\/2}, [0..99]) \\ M. F. Hasler, Jun 22 2024

from math import isqrt
def A000194(n): return (m:=isqrt(n))+int(n-m*(m+1)>=1) # Chai Wah Wu, Jul 30 2022

a(n) = A000037(n) - n.
G.f.: x * f(x^2, x^6)/(1-x) where f(,) is Ramanujan's two-variable theta function. - Michael Somos, May 31 2000
a(n) = a(n - 2*a(n - a(n-1))) + 1. - Benoit Cloitre, Oct 27 2002
a(n+1) = a(n) + A005369(n).
a(n) = floor((1/2)*(1 + sqrt(4*n - 3))). - Zak Seidov, Jan 18 2006
a(n) = A000037(n) - n. - Jaroslav Krizek, Jun 14 2009
a(n) = floor(A027434(n)/2). - Gregory R. Bryant, Apr 17 2013
From Mikael Aaltonen, Jan 17 2015: (Start)
a(n) = floor(sqrt(n) + 1/2).
a(n) = sqrt(A053187(n)). (End)
a(0) = 0, and a(n) = k for k from the closed interval [k^2 - k + 1, k*(k+1)] = [A002061(k), A002378(k)], for k >= 1. See A053187. - Wolfdieter Lang, Mar 12 2019
a(n) = floor(2*sqrt(n)) - floor(sqrt(n)). - Ridouane Oudra, Jun 08 2020
Sum_{n>=1} 1/a(n)^s = 2*zeta(s-1), for s > 2 (Borwein, 1994). - Amiram Eldar, Oct 31 2020

Additional comments from Michael Somos, May 31 2000
Edited by M. F. Hasler, Mar 01 2014
Initial 0 added by N. J. A. Sloane, Nov 13 2017

A053188 Distance from n to nearest square.

Original entry on oeis.org

0, 0, 1, 1, 0, 1, 2, 2, 1, 0, 1, 2, 3, 3, 2, 1, 0, 1, 2, 3, 4, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4

Offset: 0

Henry Bottomley, Mar 01 2000

			a(7)=2 since 9 is the closest square to 7 and |9-7| = 2.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for sequences related to distance to nearest element of some set

Cf. A053187, A074989, A000290.

a053188 0 = 0
a053188 n = min (n - last xs) (head ys - n) where
   (xs,ys) = span (< n) a000290_list
-- Reinhard Zumkeller, Nov 28 2011

Flatten[Table[Abs[Nearest[Range[0,25]^2,n]-n],{n,0,120}]]  (* Harvey P. Dale, Mar 14 2011 *)

a(n)=abs(((sqrtint(4*n) + 1)\2)^2 - n) \\ Charles R Greathouse IV, Nov 16 2022

from math import isqrt
def A053188(n): return abs(((m:=isqrt(n))+int(n-m*(m+1)>=1))**2-n) # Chai Wah Wu, Aug 03 2022

a(n) = |floor(sqrt(n) + 1/2)^2 - n|. - Ridouane Oudra, May 01 2019

a(n) <= sqrt(n). - Charles R Greathouse IV, Nov 16 2022

A007607 Skip 1, take 2, skip 3, etc.

Original entry on oeis.org

2, 3, 7, 8, 9, 10, 16, 17, 18, 19, 20, 21, 29, 30, 31, 32, 33, 34, 35, 36, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130

Offset: 1

N. J. A. Sloane, Robert G. Wilson v, Mira Bernstein

Numbers k with the property that the smallest Dyck path of the symmetric representation of sigma(k) has a central peak. (Cf. A237593.) - Omar E. Pol, Aug 28 2018

Union of A317303 and A014105. - Omar E. Pol, Aug 29 2018

			From _Omar E. Pol_, Aug 29 2018: (Start)
Written as an irregular triangle in which the row lengths are the nonzero even numbers the sequence begins:
    2,   3;
    7,   8,   9,  10;
   16,  17,  18,  19,  20,  21;
   29,  30,  31,  32,  33,  34,  35,  36;
   46,  47,  48,  49,  50,  51,  52,  53,  54,  55;
   67,  68,  69,  70,  71,  72,  73,  74,  75,  76,  77,  78;
   92,  93,  94,  95,  96,  97,  98,  99, 100, 101, 102, 103, 104, 105;
  121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136;
...
Row sums give the nonzero terms of A317297.
Column 1 gives A130883, n >= 1.
Right border gives A014105, n >= 1.
(End)

R. Honsberger, Mathematical Gems III, M.A.A., 1985, p. 177.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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

Cf. A063656, A004202, A063657, A007606, A064801, A004202.
Complement of A007606.
Cf. A014105, A130883, A317297.
Similar to A360418.

a007607 n = a007607_list !! (n-1)
a007607_list = skipTake 1 [1..] where
   skipTake k xs = take (k + 1) (drop k xs)
                   ++ skipTake (k + 2) (drop (2*k + 1) xs)
-- Reinhard Zumkeller, Feb 12 2011

a007607_list' = f $ tail $ scanl (+) 0 [1..] where
   f (t:t':t'':ts) = [t+1..t'] ++ f (t'':ts)
-- Reinhard Zumkeller, Feb 12 2011

Flatten[ Table[i, {j, 2, 16, 2}, {i, j(j - 1)/2 + 1, j(j + 1)/2}]] (* Robert G. Wilson v, Mar 11 2004 *)
With[{t=20},Flatten[Take[TakeList[Range[(t(t+1))/2],Range[t]],{2,-1,2}]]] (* Harvey P. Dale, Sep 26 2021 *)

for(m=0,10,for(n=2*m^2+3*m+2,2*m^2+5*m+3,print1(n", "))) \\ Charles R Greathouse IV, Feb 12 2011

G.f.: 1/(1-x) * (1/(1-x) + x*Sum_{k>=1} (2k+1)*x^(k*(k+1))). - Ralf Stephan, Mar 03 2004
a(A000290(n)) = A001105(n). - Reinhard Zumkeller, Feb 12 2011
A057211(a(n)) = 0. - Reinhard Zumkeller, Dec 30 2011
a(n) = floor(sqrt(n) + 1/2)^2 + n = A053187(n) + n. - Ridouane Oudra, May 04 2019

A201053 Nearest cube.

Original entry on oeis.org

0, 1, 1, 1, 1, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64

Offset: 0

Reinhard Zumkeller, Nov 28 2011

a(n) = if n-A048763(n) < A048762(n)-n then A048762(n) else A048763(n);

apart from 0, k^3 occurs 3*n^2+1 times, cf. A056107.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A061023, A074989, A053187 (nearest square), A000578.

Cf. A048762, A048763, A056107.

a201053 n = a201053_list !! n
a201053_list = 0 : concatMap (\x -> replicate (a056107 x) (x ^ 3)) [1..]

seq(k^3 $ (3*k^2+1), k=0..10); # Robert Israel, Jan 03 2017

Module[{nn=70,c},c=Range[0,Ceiling[Surd[nn,3]]]^3;Flatten[Array[ Nearest[ c,#]&,nn,0]]] (* Harvey P. Dale, May 27 2014 *)

from sympy import integer_nthroot
def A201053(n):
    a = integer_nthroot(n,3)[0]
    return a**3 if 2*n < a**3+(a+1)**3 else (a+1)**3 # Chai Wah Wu, Mar 31 2021

G.f.: (1-x)^(-1)*Sum_{k>=0} (3*k^2+3*k+1)*x^((k+1)*(k^2+k/2+1)). - Robert Israel, Jan 03 2017

Sum_{n>=1} 1/a(n)^2 = Pi^4/30 + Pi^6/945. - Amiram Eldar, Aug 15 2022

A074294 Integers 1 to 2k followed by integers 1 to 2k + 2 and so on.

Original entry on oeis.org

1, 2, 1, 2, 3, 4, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 1, 2

Offset: 1

Michael Somos, Aug 20 2002

From Cino Hilliard, Sep 13 2004: (Start)
Also the numerator of the fraction in the continued fraction expansion of sqrt(n) for nonsquare n = 2,3,5,6,7... . E.g., for n = 7,
sqrt(7).=.2.+.3................
...............4..+.3..........
.....................4..+.3....
...........................4.....
3 is the 5th entry in the table. sqrt(1) and sqrt(4) are not included because 1 and 4 are squares." (End)
A074294 is the natural fractal sequence of A002061; the corresponding natural interspersion is A194011; see A194029 for definitions. - Clark Kimberling, Aug 17 2011
It appears that this is also a triangle read by rows in which row n lists the first 2*n positive integers, n >= 1 (see example). - Omar E. Pol, May 29 2012

			From _Omar E. Pol_, May 29 2012: (Start)
Written as a triangle the sequence begins:
1, 2;
1, 2, 3, 4;
1, 2, 3, 4, 5, 6;
1, 2, 3, 4, 5, 6, 7, 8;
1, 2, 3, 4, 5, 6, 7, 8, 9, 10;
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12;
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14;
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16;
Row n has length 2*n = A005843(n). (End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A002061, A194011.
Cf. A071797.
Cf. A000194, A053187.

import Data.List (inits)
a074294 n = a074294_list !! (n-1)
a074294_list = f $ inits [1..] where
   f (xs:_:xss) = xs ++ f xss
-- Reinhard Zumkeller, Apr 14 2014

seq(seq((j-n^2-n),j=n^2+n+1..(n+1)^2+n+1),n=0..20); # Robert Israel, Jan 05 2015

A074294[n_] := n - 2*Binomial[Floor[1/2 + Sqrt[n]], 2] (* Enrique Pérez Herrero, Apr 14 2010 *)
Table[Range[2n],{n,10}]//Flatten (* Harvey P. Dale, Oct 20 2018 *)

{a(n) = n - 2 * binomial( floor( 1/2 + sqrt(n)), 2)}

c(n) = for(x=2,n,if(issquare(x)==0,a=floor(sqrt(x));print1(x-a^2", "))) /* Cino Hilliard, Sep 13 2004 */

from math import isqrt
def A074294(n): return n+(k:=(m:=isqrt(n))+(n>m*(m+1)))*(1-k) # Chai Wah Wu, Jun 06 2025

a(n) = n - 2*binomial(floor(1/2 + sqrt(n)), 2).
a(n^2 + n) = 2*n.
a(n) = n - 2 - floor(sqrt(n)+3/2)*floor(sqrt(n)-3/2). - Mikael Aaltonen, Jan 02 2015
G.f.: x/(1-x)^2 - (2*x/(1-x))*Sum_{k>=1} k*x^(k^2+k). That sum is related to Jacobi theta functions. - Robert Israel, Jan 05 2015
a(n) = n + A000194(n) - A053187(n). - Robert Israel, Jan 05 2015

A061023 Difference between the closest square and the closest cube to n.

Original entry on oeis.org

0, 0, 0, 3, 3, 4, 4, 1, 1, 1, 1, 1, 1, 8, 8, 8, 8, 8, 11, 11, 11, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 22, 22, 22, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 17, 17, 17, 17, 17, 17

Offset: 0

Hareendra Yalamanchili (hyalaman(AT)mit.edu), May 24 2001

a(A201217(n)) = 0.

			a(46)=15 because the nearest square is 49 and the nearest cube is 64 and 64 - 49 = 15.

T. D. Noe, Table of n, a(n) for n = 0..10000

Cf. A000290, A000578.

a061023 n = abs (a053187 n - a201053 n)
a061023_list = map a061023 [0..]
-- Reinhard Zumkeller, Nov 28 2011

dsc[n_]:=Module[{s=Floor[Sqrt[n]],c=Floor[Power[n, (3)^-1]],ns,nc}, ns= Nearest[{s^2,(s+1)^2},n]; nc=Nearest[{c^3,(c+1)^3},n];Abs[nc-ns]]; Flatten[ Array[dsc,100,0]] (* Harvey P. Dale, Aug 19 2011 *)

{ for (n=0, 10000, x=n^(1/2); s=floor(x)^2; t=ceil(x)^2; if (n-s > t-n, s=t); x=n^(1/3); c=floor(x)^3; d=ceil(x)^3; if (n-c > d-n, c=d); write("b061023.txt", n, " ", abs(c-s)) ) } \\ Harry J. Smith, Jul 16 2009

a(n) = abs(A053187(n) - A201053(n)). [Reinhard Zumkeller, Nov 28 2011]

More terms from Harvey P. Dale, Aug 19 2011

A038759 a(n) = ceiling(sqrt(n))*floor(sqrt(n)).

Original entry on oeis.org

0, 1, 2, 2, 4, 6, 6, 6, 6, 9, 12, 12, 12, 12, 12, 12, 16, 20, 20, 20, 20, 20, 20, 20, 20, 25, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 36, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 49, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 64, 72, 72, 72, 72, 72, 72, 72

Offset: 0

Henry Bottomley, May 03 2000

a(n) = n iff n is a square or a pronic (or heteromecic) number of form k(k+1). The sequence interleaves individual squares with 2k copies of each pronic.

			a(31) = 30 since 6 and 5 are on either side of the square root of 31 and 6*5 = 30.

Cf. A000196, A002378, A002620, A003059, A038760, A053187, A072691.

a[n_] := Ceiling[Sqrt[n]]*Floor[Sqrt[n]]; Array[a, 70, 0] (* Amiram Eldar, Dec 04 2022 *)

a(n) = my(r,s=sqrtint(n,&r)); if(r, n-r+s, n); \\ Kevin Ryde, Jul 30 2022

from math import isqrt
def A038759(n): return m+n+k if (m:=(k:=isqrt(n))**2-n) else n # Chai Wah Wu, Jul 28 2022

a(n) = A003059(n)*A000196(n) = n - A038760(n).
a(A002620(n)) = A002620(n). - Bernard Schott, Nov 06 2022
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/12 (A072691). - Amiram Eldar, Dec 04 2022

A198954 Expansion of the rotational partition function for a heteronuclear diatomic molecule.

Original entry on oeis.org

1, 3, 0, 5, 0, 0, 7, 0, 0, 0, 9, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0, 15, 0, 0, 0, 0, 0, 0, 0, 17, 0, 0, 0, 0, 0, 0, 0, 0, 19, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 23, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 25, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 27, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Michael Somos, Oct 31 2011

The partition function of a heteronuclear diatomic molecule is Sum_{J>=0} (2*J + 1) * exp( - J * (J + 1) * hbar^2 / (2 * I * k * T)) where I is the moment of inertia, hbar is reduced Planck's constant, k is Boltzmann's constant, and T is temperature. The degeneracy for the J-th energy level is 2*J + 1.
As triangle: triangle T(n,k), read by rows, given by (3,-4/3,1/3,0,0,0,0,0,0,0,...) DELTA (0,0,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 01 2011
Note that the g.f. theta_1'(0, q^(1/2)) / (2 * q^(1/8)) = 1 - 3*q  + 5*q^3 - 7*q^6 + 9*q^10 + ... which is the same as this sequence except the signs alternate. - Michael Somos, Aug 26 2015

			G.f. = 1 + 3*x + 5*x^3 + 7*x^6 + 9*x^10 + 11*x^15 + 13*x^21 + 15*x^28 + ...
G.f. = 1 + 3*q^2 + 5*q^6 + 7*q^12 + 9*q^20 + 11*q^30 + 13*q^42 + 15*q^56 + ...
Triangle begins:
   1;
   3, 0;
   5, 0, 0;
   7, 0, 0, 0;
   9, 0, 0, 0, 0;
  11, 0, 0, 0, 0, 0;
  13, 0, 0, 0, 0, 0, 0;
  15, 0, 0, 0, 0, 0, 0, 0;
  17, 0, 0, 0, 0, 0, 0, 0, 0;

G. H. Wannier, Statistical Physics, Dover Publications, 1987, see p. 215 equ. (11.13).

Antti Karttunen, Table of n, a(n) for n = 0..10011

Cf. A053187, A107270.

seq(op([2*i+1,0$i]), i=0..10); # Robert Israel, Jan 15 2015

a[ n_] := If[ n < 0, 0, With[ {m = Sqrt[8 n + 1]}, If[ IntegerQ[m], m KroneckerSymbol[ 4, m], 0]]]; (* Michael Somos, Aug 26 2015 *)

{a(n) = my(m); if( issquare( 8*n + 1, &m), m, 0)};

G.f.: Sum_{k>=0} (2*k + 1) * x^( (k^2 + k) / 2). This is related to Jacobi theta functions.
a(n) = (t*(t+1)-2*n-1)*(t-r), where t = floor(sqrt(2*(n+1))+1/2) and r = floor(sqrt(2*n)+1/2). - Mikael Aaltonen, Jan 15 2015
a(n) = A053187(2n+1) - A053187(2n). - Robert Israel, Jan 15 2015
a(n) = abs(A010816(n)). - Joerg Arndt, Jan 16 2015

A002483 Expansion of Jacobi theta function {theta_1}'(q) in powers of q^(1/4).

Original entry on oeis.org

0, 2, 0, 0, 0, 0, 0, 0, 0, -6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -14, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 18, 0, 0, 0

Offset: 0

N. J. A. Sloane

sign

			2*x - 6*x^9 + 10*x^25 - 14*x^49 + 18*x^81 - 22*x^121 + 26*x^169 - 30*x^225 + ...

J. Tannery and J. Molk, Eléments de la Théorie des Fonctions Elliptiques, Vol. 2, Gauthier-Villars, Paris, 1902; Chelsea, NY, 1972, see p. 27.
E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge Univ. Press, 4th ed., 1963, p. 464.

Antti Karttunen, Table of n, a(n) for n = 0..10201
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, p. 102.

Dividing by 2 gives (essentially) A245552.

A201217 Numbers such that (closest square) = (closest cube).

Original entry on oeis.org

0, 1, 2, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 703, 704, 705, 706, 707, 708, 709, 710, 711, 712, 713, 714, 715, 716, 717, 718, 719, 720, 721, 722, 723, 724, 725, 726, 727, 728, 729, 730, 731, 732, 733, 734, 735, 736, 737, 738, 739

Offset: 1

Reinhard Zumkeller, Nov 28 2011

nonn

A061023(a(n)) = 0; A053187(a(n)) = A201053(a(n));

A001014 is a subsequence (6th powers).

Reinhard Zumkeller, Table of n, a(n) for n = 1..6051

import Data.List (elemIndices)
a201217 n = a201217_list !! (n-1)
a201217_list = elemIndices 0 a061023_list
-- Reinhard Zumkeller, Nov 28 2011

cp's OEIS Frontend

A000194 n appears 2n times, for n >= 1; also nearest integer to square root of n.

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A053188 Distance from n to nearest square.

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A007607 Skip 1, take 2, skip 3, etc.

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A201053 Nearest cube.

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A074294 Integers 1 to 2*k followed by integers 1 to 2*k + 2 and so on.

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A061023 Difference between the closest square and the closest cube to n.

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A074294 Integers 1 to 2k followed by integers 1 to 2k + 2 and so on.