A063656 -id:A063656 - cp's OEIS frontend

A158405 Triangle T(n,m) = 1+2*m of odd numbers read along rows, 0<=m

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

Offset: 1

Paul Curtz, Mar 18 2009

Row sums are n^2 = A000290(n).
The triangle sums, see A180662 for their definitions, link this triangle of odd numbers with seventeen different sequences, see the crossrefs. The knight sums Kn14 - Kn110 have been added. - Johannes W. Meijer, Sep 22 2010
A208057 is the eigentriangle of A158405 such that as infinite lower triangular matrices, A158405 * A208057 shifts the latter, deleting the right border of 1's. - Gary W. Adamson, Feb 22 2012
T(n,k) = A099375(n-1,n-k), 1<=k<=n. [Reinhard Zumkeller, Mar 31 2012]

			The triangle contains the first n odd numbers in row n:
  1;
  1,3;
  1,3,5;
  1,3,5,7;
From _Seiichi Manyama_, Dec 02 2017: (Start)
    |       a(n)        |                               | A000290(n)
   -----------------------------------------------------------------
   0|                                                      (=  0)
   1|                 1 = 1/3 * ( 3)                       (=  1)
   2|             1 + 3 = 1/3 * ( 5 +  7)                  (=  4)
   3|         1 + 3 + 5 = 1/3 * ( 7 +  9 + 11)             (=  9)
   4|     1 + 3 + 5 + 7 = 1/3 * ( 9 + 11 + 13 + 15)        (= 16)
   5| 1 + 3 + 5 + 7 + 9 = 1/3 * (11 + 13 + 15 + 17 + 19)   (= 25)
(End)

Seiichi Manyama, Rows n = 1..140, flattened
Daniel Erman, The Josephus Problem, Numberphile video (2016)
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Index entries for sequences related to the Josephus Problem

Cf. A129326, A099375, A005408.
Triangle sums (see the comments): A000290 (Row1; Kn11 & Kn4 & Ca1 & Ca4 & Gi1 & Gi4); A000027 (Row2); A005563 (Kn12); A028347 (Kn13); A028560 (Kn14); A028566 (Kn15); A098603 (Kn16); A098847 (Kn17); A098848 (Kn18); A098849 (Kn19); A098850 (Kn110); A000217 (Kn21. Kn22, Kn23, Fi2, Ze2); A000384 (Kn3, Fi1, Ze3); A000212 (Ca2 & Ze4); A000567 (Ca3, Ze1); A011848 (Gi2); A001107 (Gi3). - Johannes W. Meijer, Sep 22 2010
Cf. A063656, A063657, A208057, A292610, A292611.

a158405 n k = a158405_row n !! (k-1)
a158405_row n = a158405_tabl !! (n-1)
a158405_tabl = map reverse a099375_tabl
-- Reinhard Zumkeller, Mar 31 2012

Table[2 Range[1, n] - 1, {n, 12}] // Flatten (* Michael De Vlieger, Oct 01 2015 *)

a(n) = 2*(n-floor((-1+sqrt(8*n-7))/2)*(floor((-1+sqrt(8*n-7))/2)+1)/2)-1;
vector(100, n, a(n)) \\ Altug Alkan, Oct 01 2015

a(n) = 2*i-1, where i = n-t(t+1)/2, t = floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Feb 03 2013

a(n) = 2*A002262(n-1) + 1. - Eric Werley, Sep 30 2015

Edited by R. J. Mathar, Oct 06 2009

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

A063657 Numbers with property that truncated square root is unequal to rounded square root.

Original entry on oeis.org

3, 7, 8, 13, 14, 15, 21, 22, 23, 24, 31, 32, 33, 34, 35, 43, 44, 45, 46, 47, 48, 57, 58, 59, 60, 61, 62, 63, 73, 74, 75, 76, 77, 78, 79, 80, 91, 92, 93, 94, 95, 96, 97, 98, 99, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 133, 134, 135, 136, 137, 138, 139, 140

Offset: 1

Floor van Lamoen, Jul 24 2001

Also: skip 1, take 0, skip 2, take 1, skip 3, take 2, ...
Integers for which the periodic part of the continued fraction for the square root of n begins with a 1. - Robert G. Wilson v, Nov 01 2001
a(n) belongs to the sequence if and only if a(n) > floor(sqrt(a(n))) * ceiling(sqrt(a(n))), i.e. a(n) in (k*(k+1),k^2), k >= 0. - Daniel Forgues, Apr 17 2011
Any integer between (k - 1/2)^2 and k^2 exclusive, for k > 1, is in this sequence. If we take this sequence and remove each term that is one more than the previous term, we obtain the central polygonal numbers (A002061). If instead we remove each term that is one less than the next term, we obtain numbers that are one less than squares (A005563). - Alonso del Arte, Dec 28 2013

			7 is in the sequence because its square root is 2.64575..., which truncates to 2 but rounds to 3.
8 is in the sequence because its square root is 2.828427..., which also truncates to 2 but rounds to 3.
9 is not in the sequence because its square root is 3 exactly, which truncates and rounds the same.
Here is the example per Lamoen's skip n, take n - 1 process: starting at 0, we skip one integer (0) but take zero integers for our sequence. Then we skip two integers (1 and 2) and take one integer (3) for our sequence. Then we skip three integers (4, 5, 6) and take two integers for our sequence (7 and 8, so the sequence now stands as 3, 7, 8). Then we skip four integers (9, 10, 11, 12) and so on and so forth.
From _Seiichi Manyama_, Sep 19 2017: (Start)
See R. B. Nelsen's paper.
   k|            A063656(n)         |            a(n)
   -------------------------------------------------------------------
   0|                             0
   1|                        1 +  2 =  3
   2|                   4 +  5 +  6 =  7 +  8
   3|              9 + 10 + 11 + 12 = 13 + 14 + 15
   4|        16 + 17 + 18 + 19 + 20 = 21 + 22 + 23 + 24
    | ...
(End)
The triangle begins as:
   3;
   7,  8;
  13, 14, 15;
  21, 22, 23, 24;
  31, 32, 33, 34, 35;
  43, 44, 45, 46, 47, 48;
  57, 58, 59, 60, 61, 62, 63;
  73, 74, 75, 76, 77, 78, 79, 80;
  91, 92, 93, 94, 95, 96, 97, 98, 99;
  ... - _Stefano Spezia_, Oct 20 2024

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)
Roger B. Nelsen, Proof without Words: Consecutive sums of consecutive integers, Mathematics Magazine, Vol. 63, No. 1 (1990), p. 25.

Cf. A002620, A004201-A004202, A063656, A189151.

Cf. A005563 (main diagonal), A059270 (row sums), A217575.

a063657 n = a063657_list !! n
a063657_list = f 0 [0..] where
   f k (_:xs) = us ++ f (k + 1) (drop (k + 1) vs) where
                        (us, vs) = splitAt k xs
-- Reinhard Zumkeller, Jun 20 2015

A063657:=n->`if`(floor(floor(sqrt(n+1)) * (1+floor(sqrt(n+1)))/(n+1))=1, NULL, n+1); seq(A063657(n), n=1..200); # Wesley Ivan Hurt, Dec 28 2013

Select[ Range[200], Floor[ Sqrt[ # ]] != Floor[ Sqrt[ # ] + 1/2] & ] (* or *) Select[ Range[200], First[ Last[ ContinuedFraction[ Sqrt[ # ]]]] == 1 & ]

{ n=0; for (m=0, 10^9, if (sqrt(m)%1 > .5, write("b063657.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 27 2009

a(n) = A217575(n) + 1. - Reinhard Zumkeller, Jun 20 2015
From Stefano Spezia, Oct 20 2024: (Start)
As a triangle:
T(n,k) = n^2 + n + k with 1 <= k <= n.
G.f.: x*y*(3 + x^2*(1 - 4*y) - x*(2 + y) + x^3*y*(1 + 2*y))/((1 - x)^3*(1 - x*y)^3). (End)

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

A292564 Take 1, skip 3 * 1 - 1, take 2, skip 3 * 2 - 1, take 3, skip 3 * 3 - 1, ...

Original entry on oeis.org

0, 3, 4, 10, 11, 12, 21, 22, 23, 24, 36, 37, 38, 39, 40, 55, 56, 57, 58, 59, 60, 78, 79, 80, 81, 82, 83, 84, 105, 106, 107, 108, 109, 110, 111, 112, 136, 137, 138, 139, 140, 141, 142, 143, 144, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 210, 211, 212

Offset: 0

Seiichi Manyama, Sep 19 2017

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Michael Boardman, Proof Without Words: Pythagorean Runs, Math. Mag., 73 (2000), 59.

Cf. A059255, A063656, A292565.

seq(seq(2*k^2+k+j,j=0..k),k=0..10); # Robert Israel, Sep 20 2017

a(n) = n + floor((sqrt(8*n+1)-1)/2)*(3*floor((sqrt(8*n+1)-1)/2)+1)/2. - Robert Israel, Sep 20 2017

A307508 Primes p for which the continued fraction expansion of sqrt(p) does not have a 1 in the second position.

Original entry on oeis.org

2, 5, 11, 17, 19, 29, 37, 41, 53, 67, 71, 83, 89, 101, 103, 107, 109, 127, 131, 149, 151, 173, 179, 181, 197, 199, 227, 229, 233, 239, 257, 263, 269, 271, 293, 331, 337, 367, 373, 379, 401, 409, 419, 443, 449, 457, 461, 487, 491, 499, 503, 541, 547, 577, 587, 593, 599

Offset: 1

Michel Marcus, Apr 11 2019

nonn

These are the primes that are located between a square number and the following oblong number. - Charles Kusniec, Apr 17 2020

Primes in A063656. - Charles Kusniec, Sep 04 2022

			For p = 2,  we have [1; 2, ...]; see A040000.
For p = 5,  we have [2; 4, ...]; see A040002.
For p = 11, we have [3; 3, ...]; see A040007.

Michel Marcus, Table of n, a(n) for n = 1..5000
Piotr Miska and Maciej Ulas, On consecutive 1's in continued fractions expansions of square roots of prime numbers, Experimental Mathematics, 31:1 (2022), 238-251. Also arXiv:1904.03404 [math.NT], 2019.
Index entries for continued fractions for constants

Cf. A040000, A040002, A040007, A067614, A307453, A063656.

Complement of A334163 with respect to the primes.

isok(p) = isprime(p) && contfrac(sqrt(p))[2] != 1;

A377187 Triangle read by rows: T(n,k) = numerator((n^2 + k)/(n^2 - k)).

Original entry on oeis.org

3, 11, 2, 9, 19, 5, 27, 14, 29, 3, 19, 13, 5, 41, 7, 51, 26, 53, 27, 55, 4, 33, 67, 17, 69, 35, 71, 9, 83, 14, 85, 43, 29, 44, 89, 5, 51, 103, 13, 21, 53, 107, 27, 109, 11, 123, 62, 125, 63, 127, 64, 129, 65, 131, 6, 73, 49, 37, 149, 25, 151, 19, 17, 77, 155, 13

Offset: 2

Stefano Spezia, Oct 19 2024

			The triangle begins as:
   3;
  11,   2;
   9,  19,  5;
  27,  14, 29,  3;
  19,  13,  5, 41,  7;
  51,  26, 53, 27, 55,   4;
  33,  67, 17, 69, 35,  71,  9;
  83,  14, 85, 43, 29,  44, 89,   5;
  51, 103, 13, 21, 53, 107, 27, 109, 11;
  ...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, Section 1.3, p. 14.

Cf. A001113, A026741, A063656, A164900, A377188 (denominator).

T[n_,k_]:=Numerator[(n^2+k)/(n^2-k)]; Table[T[n,k],{n,2,12},{k,2,n}]//Flatten

from math import isqrt, comb, gcd
def A377187(n): return (d:=(a:=(m:=isqrt(k:=n-1<<1))+(k>m*(m+1))+1)**2+(b:=n-comb(a-1,2)))//gcd(d,d-(b<<1)) # Chai Wah Wu, Nov 12 2024

Limit_{n->oo} Product_{k=1..n} T(n,k)/A377188(n,k) = e = A001113 (see Finch).
T(n,n) = A026741(n+1).
T(n,2) = A164900(n-1).

cp's OEIS Frontend

A158405 Triangle T(n,m) = 1+2*m of odd numbers read along rows, 0<=m

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

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A063657 Numbers with property that truncated square root is unequal to rounded square root.

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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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A292564 Take 1, skip 3 * 1 - 1, take 2, skip 3 * 2 - 1, take 3, skip 3 * 3 - 1, ...

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A307508 Primes p for which the continued fraction expansion of sqrt(p) does not have a 1 in the second position.

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A377187 Triangle read by rows: T(n,k) = numerator((n^2 + k)/(n^2 - k)).

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