A201053 -id:A201053 - cp's OEIS frontend

A056107 Third spoke of a hexagonal spiral.

1, 4, 13, 28, 49, 76, 109, 148, 193, 244, 301, 364, 433, 508, 589, 676, 769, 868, 973, 1084, 1201, 1324, 1453, 1588, 1729, 1876, 2029, 2188, 2353, 2524, 2701, 2884, 3073, 3268, 3469, 3676, 3889, 4108, 4333, 4564, 4801, 5044, 5293, 5548, 5809, 6076, 6349

Offset: 0

Henry Bottomley, Jun 09 2000

a(n+1) is the number of lines crossing n cells of an n X n X n cube. - Lekraj Beedassy, Jul 29 2005
Equals binomial transform of [1, 3, 6, 0, 0, 0, ...]. - Gary W. Adamson, May 03 2008
Each term a(n), with n>1 represents the area of the right trapezoid with bases whose values are equal to hex number A003215(n) and A003215(n+1)and height equal to 1. The right trapezoid is formed by a rectangle with the sides equal to A003215(n) and 1 and a right triangle whose area is 3*n with the greater cathetus equal to the difference A003215(n+1)-A003215(n). - Giacomo Fecondo, Jun 11 2010
2*a(n)^2 is of the form x^4+y^4+(x+y)^4. In fact, 2*a(n)^2 = (n-1)^4+(n+1)^4+(2n)^4. - Bruno Berselli, Jul 16 2013
Numbers m such that m+(m-1)+(m-2) is a square. - César Aguilera, May 26 2015
After 4, twice each term belongs to A181123: 2*a(n) = (n+1)^3 - (n-1)^3. - Bruno Berselli, Mar 09 2016
This is a subsequence of A003136: a(n) = (n-1)^2 + (n-1)*(n+1) + (n+1)^2. - Bruno Berselli, Feb 08 2017
For n > 3, also the number of (not necessarily maximal) cliques in the n X n torus grid graph. - Eric W. Weisstein, Nov 30 2017

Edward J. Barbeau, Murray S. Klamkin and William O. J. Moser, Five Hundred Mathematical Challenges, MAA, Washington DC, 1995, Problem 444, pp. 42 and 195.
Ben Hamilton, Brainteasers and Mindbenders, Fireside, 1992, p. 107.

Nathaniel Johnston, Table of n, a(n) for n = 0..5000
Henry Bottomley, Illustration of initial terms
A. J. C. Cunningham, Factorisation of N and N' = (x^n -+ y^n) / (x -+ y) [when x-y=n], Messenger Math., 54 (1924), 17-21 [Incomplete annotated scanned copy]
Gabriele Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2.
A. L. Rubinoff and Leo Moser, Solution to Problem E773, The American Mathematical Monthly, Vol. 55, No. 2 (Feb., 1948), p. 99.
Eric Weisstein's World of Mathematics, Clique.
Eric Weisstein's World of Mathematics, Torus Grid Graph.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A002648 (prime terms), A201053.
Cf. A000578, A003136, A132111, A181123.
Other spokes: A003215, A056105, A056106, A056108, A056109.
Other spirals: A054552.

List([0..40], n -> 3*n^2 + 1); # G. C. Greubel, Dec 02 2018

[3*n^2 + 1: n in [0..40]]; // G. C. Greubel, Dec 02 2018

seq(3*n^2+1, n=0..46); # Nathaniel Johnston, Jun 26 2011

Table[3 n^2 + 1, {n, 100}] (* Vladimir Joseph Stephan Orlovsky, Jun 26 2011 *)
LinearRecurrence[{3, -3, 1}, {1, 4, 13}, 47] (* Michael De Vlieger, Feb 08 2017 *)
CoefficientList[Series[(1 + x + 4 x^2)/(1 - x)^3, {x, 0, 46}], x] (* Michael De Vlieger, Feb 08 2017 *)
1 + 3 Range[0, 20]^2 (* Eric W. Weisstein, Nov 30 2017 *)

for(n=0,1000,if(issquare(n+(n-1)+(n-2)),print1(n", "))); \\ César Aguilera, May 26 2015

a(n) = 3*n^2 + 1; \\ Altug Alkan, Feb 08 2017

[3*n^2 + 1 for n in range(40)] # G. C. Greubel, Dec 02 2018

a(n) = 3*n^2 + 1.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2.
G.f.: (1+x+4*x^2)/(1-x)^3.
a(n) = a(n-1) + 6*n - 3 for n>0.
a(n) = 2*a(n-1) - a(n-2) + 6 for n>1.
a(n) = A056105(n) + 2*n = A056106(n) + n.
a(n) = A056108(n) - n = A056109(n) - 2*n = A003215(n) - 3*n.
a(n) = (A000578(n+1) - A000578(n-1))/2. - Lekraj Beedassy, Jul 29 2005
a(n) = A132111(n+1,n-1) for n>1. - Reinhard Zumkeller, Aug 10 2007
E.g.f.: (1 + 3*x + 3*x^2)*exp(x). - G. C. Greubel, Dec 02 2018
From Amiram Eldar, Jul 15 2020: (Start)
Sum_{n>=0} 1/a(n) = (1 + (Pi/sqrt(3))*coth(Pi/sqrt(3)))/2.
Sum_{n>=0} (-1)^n/a(n) = (1 + (Pi/sqrt(3))*csch(Pi/sqrt(3)))/2. (End)
From Amiram Eldar, Feb 05 2021: (Start)
Product_{n>=0} (1 + 1/a(n)) = sqrt(2)*csch(Pi/sqrt(3))*sinh(sqrt(2/3)*Pi).
Product_{n>=1} (1 - 1/a(n)) = (Pi/sqrt(3))*csch(Pi/sqrt(3)). (End)

A053187 Square nearest to n.

Original entry on oeis.org

0, 1, 1, 4, 4, 4, 4, 9, 9, 9, 9, 9, 9, 16, 16, 16, 16, 16, 16, 16, 16, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64

Offset: 0

Henry Bottomley, Mar 01 2000

Apart from 0, k^2 appears 2k times from a(k^2-k+1) through to a(k^2+k) inclusive.

			a(7) = 9 since 7 is closer to 9 than to 4.
G.f. = x + x^2 + 4*x^3 + 4*x^4 + 4*x^5 + 4*x^6 + 9*x^7 + 9*x^8 + 9*x^9 + ...

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

Cf. A048760, A053188, A002483, A245552.

Cf. A061023, A201053 (nearest cube), A000290, A000194.

a053187 n = a053187_list !! n
a053187_list = 0 : concatMap (\x -> replicate (2*x) (x ^ 2)) [1..]
-- Reinhard Zumkeller, Nov 28 2011

seq(ceil((-1+sqrt(4*n+1))/2)^2, n=0..20); # Robert Israel, Jan 05 2015

nearestSq[n_] := Block[{a = Floor@ Sqrt@ n}, If[a^2 + a + 1/2 > n, a^2, a^2 + 2 a + 1]]; Array[ nearestSq, 75, 0] (* Robert G. Wilson v, Aug 01 2014 *)

from math import isqrt
def A053187(n): return ((m:=isqrt(n))+int(n>m*(m+1)))**2 # Chai Wah Wu, Jun 06 2025

a(n) = ceiling((-1 + sqrt(4*n+1))/2)^2. - Robert Israel, Aug 01 2014
G.f.: (1/(1-x))*Sum_{n>=0} (2*n+1)*x^(n^2+n+1). - Robert Israel, Aug 01 2014.  This is related to the Jacobi theta-function theta'_1(q), see A002483 and A245552.
G.f.: x / (1-x) * Sum_{k>0} (2*k - 1) * x^(k^2 - k). - Michael Somos, Jan 05 2015
a(n) = floor(sqrt(n)+1/2)^2. - Mikael Aaltonen, Jan 17 2015
Sum_{n>=1} 1/a(n)^2 = 2*zeta(3). - Amiram Eldar, Aug 15 2022
a(n) = A000194(n)^2. - Chai Wah Wu, Jun 06 2025

Title improved by Jon E. Schoenfield, Jun 09 2019

A048762 Largest cube <= n.

Original entry on oeis.org

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

Offset: 0

Charles T. Le (charlestle(AT)yahoo.com)

nonn

Krassimir T. Atanassov, On the 40th and 41st Smarandache Problems, Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 4, No. 3 (1998), 101-104.
J. Castillo, Other Smarandache Type Functions: Inferior/Superior Smarandache f-part of x, Smarandache Notions Journal, Vol. 10, No. 1-2-3 (1999), 202-204.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Florentin Smarandache, Only Problems, Not Solutions!.
Krassimir T. Atanassov, On Some of Smarandache's Problems, American Research Press, 1999, 27-32.

Cf. A048763, A201053, A000578.

a048762 n = last $ takeWhile (<= n) a000578_list
-- Reinhard Zumkeller, Nov 28 2011

A048762 := proc(n)
        floor(root[3](n)) ;
        %^3 ;
end proc: # R. J. Mathar, Nov 06 2011

Floor[Surd[Range[0,70],3]]^3 (* Harvey P. Dale, Jun 23 2013 *)

Sum_{n>=1} 1/a(n)^2 = Pi^4/30 + Pi^6/945 + 3*zeta(5). - Amiram Eldar, Aug 15 2022

A074989 Distance from n to nearest cube.

Original entry on oeis.org

0, 0, 1, 2, 3, 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, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24

Offset: 0

Zak Seidov, Oct 02 2002

nonn

a(n)=0 when n is a cube; between zeros local maxima are of form 3/2 k(k-1).

			a(3) = 2 because the nearest cube to 3 is 1 and distance from 3 to 1 is 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. A053188 (distance from n to nearest square).

Cf. A201053, A000578.

a074989 0 = 0
a074989 n = min (n - last xs) (head ys - n) where
   (xs,ys) = span (< n) a000578_list
-- Reinhard Zumkeller, Nov 28 2011

A074989 := proc(n) local iscbr ; iroot(n,3,'iscbr') ; if iscbr then 0; else iscbr := floor(n^(1/3)) ; min((iscbr+1)^3-n, n-iscbr^3) ; end if; end proc; # R. J. Mathar, Nov 01 2009

dnc[n_]:=Module[{cr=Surd[n,3]},Min[n-Floor[cr]^3,Ceiling[cr]^3-n]]; Array[ dnc,90,0] (* Harvey P. Dale, Jan 24 2015 *)

from sympy import integer_nthroot
def A074989(n):
    a = integer_nthroot(n,3)[0]
    return min(n-a**3,(a+1)**3-n) # Chai Wah Wu, Mar 31 2021

a(0) added and offset changed by Reinhard Zumkeller, Nov 28 2011

A048763 Smallest cube >= n.

Original entry on oeis.org

0, 1, 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, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 125, 125, 125, 125, 125, 125

Offset: 0

Charles T. Le (charlestle(AT)yahoo.com)

nonn

Krassimir T. Atanassov, On the 40th and 41st Smarandache Problems, Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 4, No. 3 (1998), 101-104.
J. Castillo, Other Smarandache Type Functions: Inferior/Superior Smarandache f-part of x, Smarandache Notions Journal, Vol. 10, No. 1-2-3 (1999), 202-204.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Krassimir T. Atanassov, On Some of Smarandache's Problems, American Research Press, 1999, 27-32.
Florentin Smarandache, Only Problems, Not Solutions!.

Cf. A048762, A201053, A000578.

a048763 0 = 0
a048763 n = head $ dropWhile (< n) a000578_list
-- Reinhard Zumkeller, Nov 28 2011

A048763 := proc(n)
        ceil(root[3](n)) ;
        %^3 ;
end proc: # R. J. Mathar, Nov 06 2011

With[{nn=80},Flatten[Table[Select[Range[0,Floor[nn^(1/3)]+1]^3,#>=n&,1],{n,0,nn}]]] (* Harvey P. Dale, Aug 09 2012 *)

Sum_{n>=1} 1/a(n)^2 = Pi^4/30 + Pi^6/945 - 3*zeta(5). - Amiram Eldar, Aug 15 2022

a(65), a(66) and a(67) corrected by Reinhard Zumkeller, Nov 28 2011

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

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

A342872 Distance to nearest product of 3 consecutive numbers (three-dimensional promic number, A007531).

Original entry on oeis.org

0, 1, 2, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17

Offset: 0

Lamine Ngom, Mar 28 2021

			a(13) = 7 since 6 is the closest three-dimensional promic to 13 and 13 - 6 = 7.

Lamine Ngom, Table of n, a(n) for n = 0..10000

Cf. A007531, A342873.

Other distance to nearest: A081134, A053646, A201053.

def a(n): return min(abs(n-k*(k+1)*(k+2)) for k in range(int(n**1/3)+1))
print([a(n) for n in range(78)]) # Michael S. Branicky, Mar 28 2021

A342873 Numbers whose distance to the nearest cube equals the distance to the nearest product of 3 consecutive integers (three-dimensional oblong).

Original entry on oeis.org

0, 7, 16, 62, 92, 213, 276, 508, 616, 995, 1160, 1722, 1956, 2737, 3052, 4088, 4496, 5823, 6336, 7990, 8620, 10637, 11396, 13812, 14712, 17563, 18616, 21938, 23156, 26985, 28380, 32752, 34336, 39287, 41072, 46638, 48636, 54853, 57076, 63980, 66440, 74067

Offset: 1

Lamine Ngom, Mar 28 2021

That is, numbers k such that A074989(k) = A342872(k).
They form 2 partitions:
7, 62, 213, ... = 8*k^3 - k = k*A157914(k).
0, 16, 92,  ... = 8*k^3 + 6*k^2 + 2*k = 2*k*A033951(k).

Cf. A074989, A342872.

Cf. A157914, A033951, A074378, A201053, A000578, A007531, A081134.

def aupto(limit):
  cubes = [k**3 for k in range(int((limit+1)**1/3)+2)]
  proms = [k*(k+1)*(k+2) for k in range(int((limit+1)**1/3)+1)]
  A074989 = [min(abs(n-c) for c in cubes) for n in range(limit+1)]
  A342872 = [min(abs(n-p) for p in proms) for n in range(limit+1)]
  return [m for m in range(limit+1) if A074989[m] == A342872[m]]
print(aupto(10**4)) # Michael S. Branicky, Mar 28 2021

cp's OEIS Frontend

A056107 Third spoke of a hexagonal spiral.

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A053187 Square nearest to n.

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A048762 Largest cube <= n.

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A074989 Distance from n to nearest cube.

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A048763 Smallest cube >= n.

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

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