A003059 -id:A003059 - cp's OEIS frontend

A193050 G.f. A(x) satisfies: 1 = Sum_{n>=0} x^n*A(-x)^A003059(n+1), where A003059 is defined by "n appears 2n-1 times.".

1, 1, 1, 2, 4, 8, 17, 38, 87, 204, 489, 1191, 2938, 7328, 18448, 46809, 119583, 307324, 793965, 2060770, 5371156, 14051901, 36887289, 97131351, 256488187, 679046184, 1802047427, 4792800096, 12773166908, 34106055493, 91228795961, 244427136822, 655900969465

Offset: 0

Paul D. Hanna, Jul 14 2011

nonn

Compare the g.f. to a g.f. C(x) of the Catalan numbers: 1 = Sum_{n>=0} x^n*C(-x)^(2*n+1).

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 4*x^4 + 8*x^5 + 17*x^6 + 38*x^7 +...
The g.f. satisfies:
1 = A(-x) + x*A(-x)^2 + x^2*A(-x)^2 + x^3*A(-x)^2 + x^4*A(-x)^3 + x^5*A(-x)^3 + x^6*A(-x)^3 + x^7*A(-x)^3 + x^8*A(-x)^3 + x^9*A(-x)^4 +...+ x^n*A(-x)^A003059(n+1) +...
where A003059 begins: [1, 2,2,2, 3,3,3,3,3, 4,4,4,4,4,4,4, 5,...].
The g.f. also satisfies:
1-x = (1-x)*A(-x) + x*(1-x^3)*A(-x)^2 + x^4*(1-x^5)*A(-x)^3 + x^9*(1-x^7)*A(-x)^4 + x^16*(1-x^9)*A(-x)^5 +...

Cf. A192455, A193039, A193040, A003059.

{a(n)=local(A=[1]); for(i=1, n, A=concat(A, 0); A[#A]=polcoeff(sum(m=1, #A, (-x)^m*Ser(A)^(1+sqrtint(m-1)) ), #A)); if(n<0, 0, A[n+1])}

G.f. satisfies: 1-x = Sum_{n>=1} x^(n^2) * (1 - x^(2*n-1)) * A(-x)^n.

A257217 A257213 - A003059, where A257213(n) = min{d>0 | floor(n/d) = floor(n/(d+1))}, A003059(n) = ceiling(sqrt(n)).

Original entry on oeis.org

1, 1, 1, 0, 1, 0, 1, 1, 0, 2, 0, 0, 1, 1, 1, 0, 2, 1, 0, 0, 2, 1, 1, 1, 0, 2, 1, 1, 0, 0, 2, 2, 1, 1, 1, 0, 2, 1, 1, 1, 0, 0, 2, 2, 2, 1, 1, 1, 0, 3, 1, 1, 1, 1, 0, 0, 2, 2, 2, 2, 1, 1, 1, 0, 3, 2, 1, 1, 1, 1, 0, 0, 2, 2, 2, 2, 2, 1, 1, 1, 0, 3, 2, 2, 1, 1

Offset: 0

M. F. Hasler, Apr 18 2015

nonn

One has a(n) <= a(n-1) except for n = k^2. The positive jumps occur exactly at the squares, cf. formula.

Cf. A003059, A257213.

f[n_] := Block[{d, k}, Reap@ For[k = 0, k <= n, k++, d = 1; While[Floor[k/d] != Floor[k/(d + 1)], d++]; Sow[d - Ceiling[Sqrt@ k]]] // Flatten // Rest]; f@ 85 (* Michael De Vlieger, Apr 18 2015 *)

PARI
```
A257217(n)=A257213(n)-A003059(n)
```

a(k^2-1) = 0 for k > 1. Proof: For n = k^2-1 = (k-1)(k+1), floor(n/k) = k-1 = n/(k+1) but n/(k-1) = k+1, thus A257213(n) = k = ceiling(sqrt(n)).
A257213(n) >= floor(sqrt(n))+1 = A257213(n+1) >= A257213(n) = ceiling(sqrt(n)), with strict inequality (in the second relation) when n is a square. Therefore a(n) >= 1 for all n = k^2.
a(k^2) >= d when k > d(d-1). Proof: This follows from k^2/(k+d) = k-d+d^2/(k+d), which shows that a(k) >= d when k > d(d-1).

A327724 Product of A003059 and A071797.

Original entry on oeis.org

1, 2, 4, 6, 3, 6, 9, 12, 15, 4, 8, 12, 16, 20, 24, 28, 5, 10, 15, 20, 25, 30, 35, 40, 45, 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112, 120, 9, 18, 27, 36, 45, 54

Offset: 1

Todor Szimeonov, Sep 23 2019

nonn

Cf. A003059, A071797.

a(n) = {(1+sqrtint(n-1))*(n-sqrtint(n-1)^2)} \\ Andrew Howroyd, Sep 23 2019

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

Original entry on oeis.org

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

A115720 Triangle T(n,k) is the number of partitions of n with Durfee square k.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 3, 0, 4, 1, 0, 5, 2, 0, 6, 5, 0, 7, 8, 0, 8, 14, 0, 9, 20, 1, 0, 10, 30, 2, 0, 11, 40, 5, 0, 12, 55, 10, 0, 13, 70, 18, 0, 14, 91, 30, 0, 15, 112, 49, 0, 16, 140, 74, 1, 0, 17, 168, 110, 2, 0, 18, 204, 158, 5, 0, 19, 240, 221, 10, 0, 20, 285, 302, 20, 0, 21, 330, 407

Offset: 0

Franklin T. Adams-Watters, Mar 11 2006

T(n,k) is number of partitions of n-k^2 into parts of 2 kinds with at most k of each kind.

			Triangle starts:
  1;
  0,  1;
  0,  2;
  0,  3;
  0,  4,  1;
  0,  5,  2;
  0,  6,  5;
  0,  7,  8;
  0,  8, 14;
  0,  9, 20,  1;
  0, 10, 30,  2;
From _Gus Wiseman_, Apr 12 2019: (Start)
Row n = 9 counts the following partitions:
  (9)          (54)       (333)
  (81)         (63)
  (711)        (72)
  (6111)       (432)
  (51111)      (441)
  (411111)     (522)
  (3111111)    (531)
  (21111111)   (621)
  (111111111)  (3222)
               (3321)
               (4221)
               (4311)
               (5211)
               (22221)
               (32211)
               (33111)
               (42111)
               (222111)
               (321111)
               (2211111)
(End)

Alois P. Heinz, Rows n = 0..600, flattened
Findstat, St000183: The side length of the Durfee square of an integer partition
Eric Weisstein's World of Mathematics, Durfee Square

For a version without zeros see A115994. Row lengths are A003059. Row sums are A000041. Column k = 2 is A006918. Column k = 3 is A117485.
Cf. A008284, A115721, A115722, A257990, A325164.
Related triangles are A096771, A325188, A325189, A325192, with Heinz-encoded versions A263297, A325169, A065770, A325178.

b:= proc(n, i) option remember;
      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
    end:
T:= (n, k)-> add(b(m, k)*b(n-k^2-m, k), m=0..n-k^2):
seq(seq(T(n, k), k=0..floor(sqrt(n))), n=0..30); # Alois P. Heinz, Apr 09 2012

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i]]]]; T[n_, k_] := Sum[b[m, k]*b[n-k^2-m, k], {m, 0, n-k^2}]; Table[ T[n, k], {n, 0, 30}, {k, 0, Sqrt[n]}] // Flatten (* Jean-François Alcover, Dec 03 2015, after Alois P. Heinz *)
durf[ptn_]:=Length[Select[Range[Length[ptn]],ptn[[#]]>=#&]];
Table[Length[Select[IntegerPartitions[n],durf[#]==k&]],{n,0,10},{k,0,Sqrt[n]}] (* Gus Wiseman, Apr 12 2019 *)

T(n,k) = Sum_{i=0..n-k^2} P*(i,k)*P*(n-k^2-i), where P*(n,k) = P(n+k,k) is the number of partitions of n objects into at most k parts.

A004737 Concatenation of sequences (1,2,...,n-1,n,n-1,...,1) for n >= 1.

Original entry on oeis.org

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

Offset: 1

R. Muller

The following sequences all have the same parity: A004737, A006590, A027052, A071028, A071797, A078358, A078446. - Jeremy Gardiner, Mar 16 2003
The ordinal transform of a sequence b_0, b_1, b_2, ... is the sequence a_0, a_1, a_2, ... where a_n is the number of times b_n has occurred in {b_0 ... b_n}.
From Artur Jasinski, Mar 07 2010: (Start)
This sequence is the even subset of A003983 for odd p=2,4,6,8,....
For the odd subset of A003983 see A004739. (End)
From Gary W. Adamson, Mar 30 2010: (Start)
Given the triangle rows: (1; 1,2,1; 1,2,3,2,1; ...) as polcoeff with offset 0:
q = (1 + 2x + x^2), r = (1 + 2x + 3x^2 + 2x^3 +x^4), etc.; then
(1 + 2x + 3x^2 + ...) = q(x) * q(x^2) * q(x^4) * q(x^8) * ...
..................... = r(x) * r(x^3) * r(x^9) * r(x^27) * ...
..................... = s(x) * s(x^4) * s(x^16)* s(x^64) * ...
... (End)
From L. Edson Jeffery, Jan 13 2012: (Start)
Let U_1(t)=1, U_2(t)=2*t, and U_r(t)=2*t*U_(r-1)(t)-U(r-2)(t), r>2, be Chebyshev polynomials of the second kind. For q>1 an integer, let N=2*q and x_k=cos((2*k-1)*Pi/N), and define the ordered column vectors V_k=[U_k(x_1), U_k(x_2), ..., U_k(x_q)]^T, k=1,...,q, where A^T denotes the transpose of matrix A. Let E_N=[V_1, V_2, ..., V_q] be the q X q matrix formed from the ordered components of the V_k. E_N contains the joint spectra of the Danzer basis (see [Jeffery]) associated with N. Let M_N=(1/q)*[E_N]^T*E_N. For the trivial case q=1, let M_2=[1]. CONJECTURE: E_N and M_N are always integral and symmetric, with M_N having diagonal entries {1,2,...} beginning at entries 1,j (j odd) in the first row and i,1 (i odd) in the first column and with zeros elsewhere. If N is allowed to increase without bound, and assuming the conjecture is true, then triangle A004737 emerges in its entirety from the successive antidiagonals containing those entries [M_N]_(i,j) such that i+j=2*v, for each v in {1,2,...,floor((q+1)/2)}. For example, for N=18 and q=9 (omitting the zeros for clarity),
M_18=[
  (1   1   1   1   1);
  (  2   2   2   2  );
  (1   3   3   3   3);
  (  2   4   4   4  );
  (1   3   5   5   5);
  (  2   4   6   6  );
  (1   3   5   7   7);
  (  2   4   6   8  );
  (1   3   5   7   9)],
from which the first five rows of the sequence can be read off in succession. (End)
T(n,k) =  min(n,k). The order of the list T(n,k) is by sides of squares from T(1,n) to T(n,n), then from T(n,n) to T(n,1). - Boris Putievskiy, Jan 13 2013
Expanded form of T(2,k) k=0,1,...,2m for ascending m-nomial triangles. - Bob Selcoe, Feb 07 2014
Terms in the first nine rows of the triangle can be duplicated by performing (111...)^2 with <= nine ones. By way of example, (11111)^2 = 123454321. - Gary W. Adamson, Mar 27 2015

			From _Boris Putievskiy_, Jan 13 2013: (Start)
The start of the sequence as a table:
  1 1 1 1 1 1 ...
  1 2 2 2 2 2 ...
  1 2 3 3 3 3 ...
  1 2 3 4 4 4 ...
  1 2 3 4 5 5 ...
  1 2 3 4 5 6 ...
  ...
The start of the sequence as an irregular triangle array read by rows:
  1;
  1,2,1;
  1,2,3,2,1;
  1,2,3,4,3,2,1;
  1,2,3,4,5,4,3,2,1;
  1,2,3,4,5,6,5,4,3,2,1;
  ...
Row number k contains 2*k-1 numbers: 1,2,...,k-1,k,k-1,...,1. (End)
The sequence of fractions A196199/A004737 = 0/1, -1/1, 0/2, 1/1, -2/1, -1/2, 0/3, 1/2, 2/1, -3/1, -2/2, -1/3, 0/4, 1/3, 2/2, 3/1, -4/4. -3/2, ... contains every rational number (infinitely often) [Laczkovich]. - _N. J. A. Sloane_, Oct 09 2013

Miklós Laczkovich, Conjecture and Proof, TypoTex, Budapest, 1998. See Chapter 10.
F. Smarandache, "Numerical Sequences", University of Craiova, 1975.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Jerry Brown et al., Problem 4619, School Science and Mathematics, USA, Vol. 97 (4), 1997, pp. 221-222.
L. E. Jeffery, Danzer matrices (definitions)
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
F. Smarandache, Collected Papers, Vol. II, Tempus Publ. Hse., Bucharest, 1996.
F. Smarandache, Sequences of Numbers Involved in Unsolved Problems, Hexis, Phoenix, 2006.
Eric Weisstein's World of Mathematics, Smarandache Sequences.

Cf. A004738, A008967, A003983, A196199.

Cf. A242357, A000290 (row sums).

import Data.List (inits)
a004737 n = a004737_list !! (n-1)
a004737_list = concatMap f $ tail $ inits [1..]
   where f xs = xs ++ tail (reverse xs)
-- Reinhard Zumkeller, May 11 2014, Mar 26 2011

Table[Min[n - #^2, (# + 1)^2 - n + 1] &@ Floor[Sqrt[n - 1]], {n, 105}] (* or *)
Table[Floor@ # - Abs[n - Floor[#]^2 - Floor@ # - 1] + 1 &@ Sqrt[n - 1], {n, 105}] (* Michael De Vlieger, Oct 21 2016 *)
Table[Join[Range[n],Range[n-1,1,-1]],{n,20}]//Flatten (* Harvey P. Dale, Dec 27 2019 *)

a(n) = n--;my(m=sqrtint(n));m+1-abs(n-m^2-m) \\ David A. Corneth, Oct 18 2016

a(A002061(n)) = n; a(A000290(n)) = a(A002522(n)) = 1. - Reinhard Zumkeller, Mar 10 2006
a(n) = if n<3 then 1 else (if a(n-1)=1 then 1 + 0^(a(n-2)-1) else a(n-1) - 0^X + (a(n-1)-a(n-2))*(1 - 0^X)), where X = A003059(n-1)-a(n-1). - Reinhard Zumkeller, Mar 10 2006
Let b(n) = floor(sqrt(n-1)). Then a(n) = min(n - b(n)^2, (b(n)+1)^2 - n + 1). - Franklin T. Adams-Watters, Jun 09 2006
Ordinal transform of A004741. - Franklin T. Adams-Watters, Aug 28 2006
If the sequence is read as a triangular array, beginning [1]; [1,2,1]; [1,2,3,2,1]; ..., then the o.g.f. is (1+qx)/((1-x)(1-qx)(1-q^2x)) = 1 + x(1 + 2q + q^2) + x^2(1 + 2q + 3q^2 + 2q^3 +q^4) + .... The row polynomials for this triangle are (1 + q + ... + q^n)^2 =[n,2]A008967).%20-%20_Peter%20Bala">q + q[n-1,2]_q, where [n,2]_q are Gaussian polynomials (see A008967). - _Peter Bala, Sep 23 2007
a(n) = floor(sqrt(n-1)) - |n - floor(sqrt(n-1))^2 - floor(sqrt(n-1)) - 1| + 1. - Boris Putievskiy, Jan 13 2013
Read as a triangular array, then T(n,k) = n - |n-k-1|; T(n,0) = 1; T(n,n-1) = n. - Juan Pablo Herrera P., Oct 17 2016

More terms from Patrick De Geest, Jun 15 1998

A087811 Numbers k such that ceiling(sqrt(k)) divides k.

Original entry on oeis.org

1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36, 42, 49, 56, 64, 72, 81, 90, 100, 110, 121, 132, 144, 156, 169, 182, 196, 210, 225, 240, 256, 272, 289, 306, 324, 342, 361, 380, 400, 420, 441, 462, 484, 506, 529, 552, 576, 600, 625, 650, 676, 702, 729, 756, 784, 812, 841

Offset: 1

Reinhard Zumkeller, Oct 16 2003

Essentially the same as the quarter-squares A002620.
Nonsquare terms of this sequence are given by A002378. - Max Alekseyev, Nov 27 2006
This also gives the number of ways to make change for "c" cents using only pennies, nickels and dimes. You must first set n=floor(c/5), to account for the 5-repetitive nature of the task. - Adam Sasson, Feb 09 2011
These are the segment boundaries of Oppermann's conjecture (1882): n^2-n < p < n^2 < p < n^2+n. - Fred Daniel Kline, Apr 07 2011
a(n) is the number of triples (w,x,y) having all terms in {0..n} and w=2*x+y. - Clark Kimberling, Jun 04 2012
a(n+1) is also the number of points with integer coordinates inside a rectangle isosceles triangle with hypotenuse [0,n] (see A115065 for an equilateral triangle). - Michel Marcus, Aug 05 2013
a(n) = degree of generating polynomials of Galois numbers in (n+1)-dimensional vector space, defined as total number of subspaces in (n+1) space over GF(n) (see Mathematica procedure), when n is a power of a prime. - Artur Jasinski, Aug 31 2016, corrected by Robert Israel, Sep 23 2016
Also number of pairs (x,y) with 0 < x <= y <= n, x + y > n. - Ralf Steiner, Jan 05 2020

Vincenzo Librandi, Table of n, a(n) for n = 1..300
Ya-Ping Lu, Illustration of the Terms in A087811 on the Square Spiral
Wikipedia, Oppermann's conjecture.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A002378, A002620, A003059, A024206, A110835, A316841, A059260.

Subsequence of A006446.

a087811 n = (n + n `mod` 2) * (n + 2 - n `mod` 2) `div` 4
-- Reinhard Zumkeller, Oct 27 2012

[ n: n in [1..841] | n mod Ceiling(Sqrt(n)) eq 0 ]; // Bruno Berselli, Feb 09 2011

f:= gfun:-rectoproc({a(n)=n+a(n-2),a(1)=1,a(2)=2},a(n),remember):
map(f, [$1..100]); # Robert Israel, Aug 31 2016

a[1] := 1; a[2] := 2; a[n_] := n + a[n - 2]; Table[a[n], {n, 57}] (* Alonso del Arte *)
GaloisNumber[n_, q_] :=
Sum[QBinomial[n, m, q], {m, 0, n}]; aa = {}; Do[
sub = Table[GaloisNumber[m, n], {n, 0, 200}];
pp = InterpolatingPolynomial[sub, x]; pol = pp /. x -> n + 1;
coef = CoefficientList[pol, n];
AppendTo[aa, Length[coef] - 1], {m, 2, 25}]; aa (* Artur Jasinski, Aug 31 2016 *)
Select[Range[900],Divisible[#,Ceiling[Sqrt[#]]]&] (* or *) LinearRecurrence[ {2,0,-2,1},{1,2,4,6},60] (* Harvey P. Dale, Nov 06 2016 *)

a(n)=(n+n%2)*(n+2-n%2)/4 \\ Charles R Greathouse IV, Apr 03 2012

j=0;for(k=1,850,s=sqrtint(4*k+1);if(s>j,j=s;print1(k,", "))) \\ Hugo Pfoertner, Sep 17 2018

def A087811(n): return n*(n+2)+(n&1)>>2 # Chai Wah Wu, Jul 27 2022

a(n) = (n + n mod 2)*(n + 2 - n mod 2)/4.
Numbers of the form m^2 or m^2 - m. - Don Reble, Oct 17 2003
a(1) = 1, a(2) = 2, a(n) = n + a(n - 2). - Alonso del Arte, Jun 18 2005
From Bruno Berselli, Feb 09 2011: (Start)
G.f.: x/((1+x)*(1-x)^3).
a(n) = (2*n*(n+2)-(-1)^n+1)/8. (End)
G.f.: G(0)/(2*(1-x^2)*(1-x)), where G(k) = 1 + 1/(1 - x*(2*k+1)/(x*(2*k+2) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 25 2013
a(n) = (C(n+2,2) - floor((n+2)/2))/2. - Mircea Merca, Nov 23 2013
a(n) = ((-1)^n*(-1 + (-1)^n*(1 + 2*n*(2 + n))))/8. - Fred Daniel Kline, Jan 06 2015
a(n) = Product_{k=1...n-1} (1 + 2 / (k + k mod 2)), n >= 1. - Fred Daniel Kline, Oct 30 2016
E.g.f.: (1/4)*(x*(3 + x)*cosh(x) + (1 + 3*x + x^2)*sinh(x)). - Stefano Spezia, Jan 05 2020
a(n) = (n*(n+2)+(n mod 2))/4. - Chai Wah Wu, Jul 27 2022
Sum_{n>=1} 1/a(n) = Pi^2/6 + 1. - Amiram Eldar, Sep 17 2022
a(n) = A024206(n) + 1. - Ya-Ping Lu, Dec 29 2023

A134914 a(n) = ceiling(n^(1/3)).

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 1

Mohammad K. Azarian, Nov 17 2007

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

Cf. A048766, A038605, A003059.

[Ceiling(n^(1/3)): n in [1..80]]; // Vincenzo Librandi, Feb 18 2013

A134914:=n->ceil(n^(1/3)): seq(A134914(n), n=1..50); # Wesley Ivan Hurt, Jul 24 2014

Table[Ceiling[n^(1/3)], {n, 80}] (* Vincenzo Librandi, Feb 18 2013 *)

a(n)=ceil(n^(1/3)) \\ Felix Fröhlich, Jul 24 2014

[ceil(n^(1/3)) for n in range(1,106)] # Stefano Spezia, Aug 17 2024

A055228 a(n) = ceiling(sqrt(n!)).

Original entry on oeis.org

1, 1, 2, 3, 5, 11, 27, 71, 201, 603, 1905, 6318, 21887, 78912, 295260, 1143536, 4574144, 18859678, 80014835, 348776577, 1559776269, 7147792819, 33526120083, 160785623546, 787685471323, 3938427356615, 20082117944246, 104349745809074, 552166953567229

Offset: 0

Henry Bottomley, Jun 21 2000

For n>0, a(n) is the least m>0 such that n! <= m^2. - Clark Kimberling, Jul 18 2012

Chai Wah Wu, Table of n, a(n) for n = 0..807 (n = 0..300 from T. D. Noe)
M. Axenovich, D. Fon-Der-Flaass and A. Kostochka, On set systems without weak 3-Delta-subsystems, Discrete Math. 138(1995), 57-62.
Bela Bollobas, To Prove and Conjecture: Paul Erdős and His Mathematics, Am. Math. Monthly, 105 (March 1998)3, p. 232.
P. Erdős and R. Rado, Intersection theorems for systems of finite sets I, J. London Math. Soc. (2) 35(1960), 85-90.
P. Erdős and R. Rado, Intersection theorems for systems of finite sets II, J. London Math. Soc. (2) 44(1969), 467-479.

Cf. A000142, A003059.

A055228:=n->ceil(sqrt(n!)); seq(A055228(n), n=0..30); # Wesley Ivan Hurt, May 06 2014

Table[Ceiling[Sqrt[n!]], {n, 0, 30}] (* Wesley Ivan Hurt, May 06 2014 *)

a(n) = ceil(sqrt(n!)) \\ Michel Marcus, Jul 30 2013

from math import isqrt, factorial
def A055228(n): return 1+isqrt(factorial(n)-1) # Chai Wah Wu, Jul 28 2022

a(n) = A003059(A000142(n)). - Jonathan Vos Post, Apr 29 2007

A comment stating that one of the terms was wrong has been deleted - the terms are correct. - T. D. Noe, Apr 22 2009

More terms from Wesley Ivan Hurt, May 06 2014

A277961 Numbers n such that 4 is the largest decimal digit of n^2.

Original entry on oeis.org

2, 12, 18, 20, 21, 32, 38, 48, 49, 102, 120, 152, 179, 180, 182, 200, 201, 210, 318, 320, 321, 332, 338, 348, 362, 380, 451, 452, 462, 480, 482, 490, 548, 549, 649, 1002, 1012, 1020, 1021, 1049, 1102, 1111, 1188, 1200, 1201, 1429, 1488, 1498, 1518, 1520

Offset: 1

Colin Barker, Nov 06 2016

The actual squares are listed in A277948. - M. F. Hasler, Nov 12 2017
Includes 2*10^n+10^m for all n <> m. - Robert Israel, Nov 13 2017
For any term of q digits, the first m digits don't exceed (2 * 10^m - 2) / 3 = 666..66 (m 6's) for 1 <= m <= q. - David A. Corneth, Nov 13 2017
A term a(n) is in the sequence if and only if a(n)*10^k is in the sequence, for all k >= 0. If a(n) = (x*10^k + y)*10^m with 2xy < 10^k, then (y*10^k+x)*10^m' is also in the sequence, for all m'. - M. F. Hasler, Nov 13 2017

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

Cf. A277948, A277959, A277960.

select(n -> max(convert(n^2,base,10))=4, [$1..10000]); # Robert Israel, Nov 13 2017

L=List(); for(n=1, 10000, if(vecmax(digits(n^2))==4, listput(L, n))); Vec(L)

a(n) = sqrt(A277948(n)), where sqrt = A000196 or A000194 or A003059. - M. F. Hasler, Nov 12 2017

cp's OEIS Frontend

A193050 G.f. A(x) satisfies: 1 = Sum_{n>=0} x^n*A(-x)^A003059(n+1), where A003059 is defined by "n appears 2n-1 times.".

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A257217 A257213 - A003059, where A257213(n) = min{d>0 | floor(n/d) = floor(n/(d+1))}, A003059(n) = ceiling(sqrt(n)).

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A327724 Product of A003059 and A071797.

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

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A115720 Triangle T(n,k) is the number of partitions of n with Durfee square k.

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A004737 Concatenation of sequences (1,2,...,n-1,n,n-1,...,1) for n >= 1.

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A087811 Numbers k such that ceiling(sqrt(k)) divides k.

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