A157466 -id:A157466 - cp's OEIS frontend

A003059 k appears 2k-1 times. Also, square root of n, rounded up.

1, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 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, 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, 10, 10, 10, 10, 10, 10, 10

Offset: 1

N. J. A. Sloane

n+1 first appears in the sequence at the A002522(n)-th entry (since the ultimate occurrence of n is n^2). a(n) refers to the greatest minimal length of monotone subsequence (i.e.either increasing or decreasing) contained within any sequence of n distinct numbers,according to the Erdős-Szekeres theorem. - Lekraj Beedassy, May 20 2003
With offset 0, apparently the least k such that binomial(2n,n-k) < (1/e) binomial(2n,n). - T. D. Noe, Mar 12 2009
a(n) is the number of nonnegative integer solutions of equation x + y^2 = n - 1. - Ran Pan, Oct 02 2015
Also the burning number of the cycle graph C_n (for n >= 4) and the path graph (for n >= 1). - Eric W. Weisstein, Jan 10 2024

T. D. Noe, Table of n, a(n) for n = 1..1000
Michael Somos, Sequences used for indexing triangular or square arrays.
Eric Weisstein's World of Mathematics, Burning Number
Eric Weisstein's World of Mathematics, Cycle Graph
Eric Weisstein's World of Mathematics, Path Graph

Cf. A000196, A000290, A002162, A002522, A010052, A157466.

a003059 n = a003059_list !! (n-1)
a003059_list = concat $ zipWith ($) (map replicate [1,3..]) [1..]
-- Reinhard Zumkeller, Mar 18 2011

[Ceiling(Sqrt(n)): n in [1..100]]; // G. C. Greubel, Nov 14 2018

A003059:=n->ceil(sqrt(n)); seq(A003059(k), k=1..100); # Wesley Ivan Hurt, Nov 08 2013

Table[ Table[n, {2n - 1}], {n, 1, 10}] // Flatten (* Jean-François Alcover, Jun 10 2013 *)
Ceiling[Sqrt[Range[100]]] (* G. C. Greubel, Nov 14 2018 *)
Table[PadRight[{},2k-1,k],{k,10}]//Flatten (* Harvey P. Dale, Jun 07 2020 *)

PARI
```
a(n)=if(n<1,0,1+sqrtint(n-1))
```

from math import isqrt
def A003059(n): return isqrt(n-1)+1 # Chai Wah Wu, Nov 14 2022

[ceil(sqrt(n)) for n in (1..100)] # G. C. Greubel, Nov 14 2018

a(n) = ceiling(sqrt(n)).
G.f.: (Sum_{n>=0} x^(n^2)) * x/(1-x). - Michael Somos, May 03 2003
a(n) = Sum_{k=0..n-1} A010052(k). - Reinhard Zumkeller, Mar 01 2009
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2) (A002162). - Amiram Eldar, Sep 29 2022

Name edited by M. F. Hasler, Nov 13 2018

A157464 Number of distinct squares when inserting at most one digit in decimal representation of n.

Original entry on oeis.org

4, 3, 1, 1, 3, 1, 3, 0, 1, 2, 1, 1, 1, 0, 1, 0, 3, 0, 0, 2, 0, 1, 1, 0, 1, 4, 1, 0, 1, 3, 0, 1, 1, 0, 1, 0, 2, 0, 0, 0, 1, 2, 0, 0, 3, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 2, 1, 0, 1, 0, 2, 1, 0, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 2, 0, 1, 1, 0, 2, 0, 0, 3, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 2, 0, 0, 0, 3, 0, 1, 0, 1

Offset: 0

Reinhard Zumkeller, Mar 01 2009

a(A157465(n)) > 0; a(A000290(n)) > 0.

			a(25) = #{25=5^2, 225=15^2, 256=16^2, 625=25^2} = 4;
a(26) = #{256=16^2} = 1;
a(27) = #{} = 0;
a(28) = #{289=17^2} = 1;
a(29) = #{289=17^2, 529=23^2, 729=27^2} = 3.

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

A157466.

Join[{4},Table[Count[Union[Flatten[Table[FromDigits[Insert[IntegerDigits[n],d,m]],{d,0,9},{m,IntegerLength[n]+1}]]],?(IntegerQ[Sqrt[#]]&)],{n,110}]] (* _Harvey P. Dale, Sep 12 2021 *)

A157465 Numbers seen as squares with at most one missing digit in decimal representation.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 14, 16, 19, 21, 22, 24, 25, 26, 28, 29, 31, 32, 34, 36, 40, 41, 44, 48, 49, 52, 56, 57, 59, 61, 62, 64, 65, 66, 67, 69, 72, 74, 76, 78, 79, 81, 84, 89, 90, 91, 96, 100, 102, 104, 108, 109, 115, 116, 121, 122, 124, 125, 126, 129, 136

Offset: 1

Reinhard Zumkeller, Mar 01 2009

A157464(a(n)) > 0; A000290(n) is a subsequence;

A157466 gives numbers of these numbers <= n, A157466(a(n+1))=A157466(a(n))+1.

			Insert 5 into 200: 2500=50^2, therefore 200 is a term;
insert 4 into 201: 2401=49^2, therefore 201 is a term;
append 5 to 202: 2025=45^2, therefore 202 is a term.

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

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A003059 k appears 2k-1 times. Also, square root of n, rounded up.

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A157464 Number of distinct squares when inserting at most one digit in decimal representation of n.

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A157465 Numbers seen as squares with at most one missing digit in decimal representation.

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