A001108 -id:A001108 - cp's OEIS frontend

A322699 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where A(n,k) is 1/2 * (-1 + Sum_{j=0..k} binomial(2k,2j)(n+1)^(k-j)n^j).

0, 0, 0, 0, 1, 0, 0, 8, 2, 0, 0, 49, 24, 3, 0, 0, 288, 242, 48, 4, 0, 0, 1681, 2400, 675, 80, 5, 0, 0, 9800, 23762, 9408, 1444, 120, 6, 0, 0, 57121, 235224, 131043, 25920, 2645, 168, 7, 0, 0, 332928, 2328482, 1825200, 465124, 58080, 4374, 224, 8, 0

Offset: 0

Seiichi Manyama, Dec 23 2018

			Square array begins:
   0, 0,   0,    0,      0,       0,        0, ...
   0, 1,   8,   49,    288,    1681,     9800, ...
   0, 2,  24,  242,   2400,   23762,   235224, ...
   0, 3,  48,  675,   9408,  131043,  1825200, ...
   0, 4,  80, 1444,  25920,  465124,  8346320, ...
   0, 5, 120, 2645,  58080, 1275125, 27994680, ...
   0, 6, 168, 4374, 113568, 2948406, 76545000, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Wikipedia, Chebyshev polynomials.
Index entries for sequences related to Chebyshev polynomials.

Columns 0-5 give A000004, A001477, A033996, A322675, A322677, A322745.
Rows 0-9 give A000004, A001108, A132596, A007654(n+1), A132584, A322707, A322708, A322709, A132592, A132593.
Main diagonal gives A322746.
Cf. A173175 (A(n,2*n)), A322790.

Unprotect[Power]; 0^0 := 1; Protect[Power]; Table[(-1 + Sum[Binomial[2 k, 2 j] (# + 1)^(k - j)*#^j, {j, 0, k}])/2 &[n - k], {n, 0, 9}, {k, n, 0, -1}] // Flatten (* Michael De Vlieger, Jan 01 2019 *)
nmax = 9; row[n_] := LinearRecurrence[{4n+3, -4n-3, 1}, {0, n, 4n(n+1)}, nmax+1]; T = Array[row, nmax+1, 0]; A[n_, k_] := T[[n+1, k+1]];
Table[A[n-k, k], {n, 0, nmax}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Jan 06 2019 *)

def ncr(n, r)
  return 1 if r == 0
  (n - r + 1..n).inject(:*) / (1..r).inject(:*)
end
def A(k, n)
  (0..n).map{|i| (0..k).inject(-1){|s, j| s + ncr(2 * k, 2 * j) * (i + 1) ** (k - j) * i ** j} / 2}
end
def A322699(n)
  a = []
  (0..n).each{|i| a << A(i, n - i)}
  ary = []
  (0..n).each{|i|
    (0..i).each{|j|
      ary << a[i - j][j]
    }
  }
  ary
end
p A322699(10)

sqrt(A(n,k)+1) + sqrt(A(n,k)) = (sqrt(n+1) + sqrt(n))^k.
sqrt(A(n,k)+1) - sqrt(A(n,k)) = (sqrt(n+1) - sqrt(n))^k.
A(n,0) = 0, A(n,1) = n and A(n,k) = (4*n+2) * A(n,k-1) - A(n,k-2) + 2*n for k > 1.
A(n,k) = (T_{k}(2*n+1) - 1)/2 where T_{k}(x) is a Chebyshev polynomial of the first kind.
T_1(x) = x. So A(n,1) = (2*n+1-1)/2 = n.

A008843 Squares of NSW numbers (A002315): x^2 such that x^2 - 2y^2 = -1 for some y.

Original entry on oeis.org

1, 49, 1681, 57121, 1940449, 65918161, 2239277041, 76069501249, 2584123765441, 87784138523761, 2982076586042449, 101302819786919521, 3441313796169221281, 116903366249966604049, 3971273138702695316401, 134906383349641674153601, 4582845760749114225906049, 155681849482120242006652081

Offset: 0

N. J. A. Sloane

Also indices of triangular numbers (A000217) which are also centered octagonal numbers (A016754). - Colin Barker, Jan 16 2015

a(n)-th triangular number is a square; subsequence of A001108. - Jaroslav Krizek, Aug 05 2016

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 256.
P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 288.
P. F. Teilhet, Note #2094, L'Intermédiaire des Mathématiciens, 10 (1903), pp. 235-238.

M. A. Gruber, Artemas Martin, A. H. Bell, J. H. Drummond, A. H. Holmes and H. C. Wilkes, Problem 47, Amer. Math. Monthly, 4 (1897), 25-28.
D. H. Lehmer, Lacunary recurrence formulas for the numbers of Bernoulli and Euler, Annals Math., 36 (1935), 637-649.
Index entries for sequences related to Bernoulli numbers.
Index entries for linear recurrences with constant coefficients, signature (35,-35,1).

Cf. A000217, A002315, A016754, A146313, A253826.

LinearRecurrence[{35,-35,1},{1,49,1681},17] (* Stefano Spezia, Aug 17 2024 *)

a(n) = 34*a(n-1) - a(n-2) + 16 = A002315(n)^2 = 2*A001653(n)^2 - 1 = 2*A008844(n) - 1 = floor(A046176(n)*sqrt(2)) = 6*A055792(n+1) - a(n-1) + 4 = (6*A055792(n+2) + a(n-1) - 20)/35. - Henry Bottomley, Nov 13 2001
a(n) = A001108(2n+1). - Ira M. Gessel, Nov 05 2014
a(n) = Sum_{k=1..2*n+1} 2^(k-1)*binomial(4*n+2, 2*k). - Zoltan Zachar (zachar(AT)fellner.sulinet.hu), Oct 03 2003
O.g.f.: -(1+14*x+x^2)/((-1+x)*(1-34*x+x^2)). - R. J. Mathar, Nov 23 2007
a(n) = -(cosh((2*n - 1)*arctanh(sqrt(2))))^2 = -1 - (sinh((2*n - 1)*arctanh(sqrt(2))))^2. - Artur Jasinski, Oct 30 2008
a(n) = Sum_{k=0..4n+1} A000129(k), see Santana and Diaz-Barrero link at A002315. - Ivan N. Ianakiev, Jul 15 2022

a(14)-a(17) from Stefano Spezia, Aug 17 2024

A124124 Nonnegative integers n such that 2n^2 + 2n - 3 is square.

Original entry on oeis.org

1, 2, 6, 13, 37, 78, 218, 457, 1273, 2666, 7422, 15541, 43261, 90582, 252146, 527953, 1469617, 3077138, 8565558, 17934877, 49923733, 104532126, 290976842, 609257881, 1695937321, 3551015162, 9884647086, 20696833093, 57611945197, 120629983398, 335787024098

Offset: 1

John W. Layman, Nov 29 2006

First differences are apparently in A143608. [R. J. Mathar, Jul 17 2009]

Alternative definition: T_n and (T_n - 1)/2 are triangular numbers. - Raphie Frank, Sep 06 2012

Michael De Vlieger, Table of n, a(n) for n = 1..2613
Jeremiah Bartz, Bruce Dearden, and Joel Iiams, Classes of Gap Balancing Numbers, arXiv:1810.07895 [math.NT], 2018.
Jeremiah Bartz, Bruce Dearden, and Joel Iiams, Counting families of generalized balancing numbers, The Australasian Journal of Combinatorics (2020) Vol. 77, Part 3, 318-325.
Hermann Stamm-Wilbrandt, 4 interlaced bisections
Index entries for linear recurrences with constant coefficients, signature (1,6,-6,-1,1).

Cf. A001108, A001652, A001653, A006452, A008844, A046090, A046172, A077442, A098790, A216134.

A124124 := proc(n)
coeftayl(x*(1+x-2*x^2+x^3+x^4)/((1-x)*(x^2-2*x-1)*(x^2+2*x-1)), x=0, n);
end proc:
seq(A124124(n), n=1..20); # Wesley Ivan Hurt, Aug 04 2014
# Alternative:
a[1]:= 1: a[2]:= 2: a[3]:= 6:
for n from 4 to 1000 do
a[n]:= (3 + 2*(n mod 2))*(a[n-1]-a[n-2])+a[n-3]
od:
seq(a[n],n=1..100); # Robert Israel, Aug 13 2014

LinearRecurrence[{1,6,-6,-1,1},{1,2,6,13,37},40] (* Harvey P. Dale, Nov 05 2011 *)
CoefficientList[Series[(1 + x - 2*x^2 + x^3 + x^4)/((1 - x)*(x^2 - 2*x - 1)*(x^2 + 2*x - 1)), {x, 0, 30}], x] (* Wesley Ivan Hurt, Aug 04 2014 *)

for(n=1,10^10,if(issquare(2*n^2+2*n-3),print1(n,", "))) \\ Derek Orr, Aug 13 2014

It appears that a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) if n is even, a(n) = 5*a(n-1)-5*a(n-2)+a(n-3) if n is odd. Can anyone confirm this?
Corrected and confirmed (using the g.f.) by Robert Israel, Aug 27 2014
2*a(n) = sqrt(7+2*A077442(n-1)^2)-1. - R. J. Mathar, Dec 03 2006
a(n) = a(n-1)+6*a(n-2)-6*a(n-3)-a(n-4)+a(n-5). G.f.: -x*(1+x-2*x^2+x^3+x^4)/((x-1)*(x^2-2*x-1)*(x^2+2*x-1)). [R. J. Mathar, Jul 17 2009]
For n>0, a(2n-1) = 2*A001653(n) - A046090(n-1) and a(2n) = 2*A001653(n) + A001652(n-1). - Charlie Marion, Jan 03 2012
From Raphie Frank, Sep 06 2012: (Start)
If y = A006452(n), then a(n) = 2y + ((sqrt(8y^2 - 7) - 1)/2 - (1 - sgn(n))).
Also see A216134 [a(n) = y + ((sqrt(8y^2 - 7) - 1)/2 - (1 - sgn(n)))].
(End)
From Hermann Stamm-Wilbrandt, Aug 27 2014: (Start)
a(2*n+2) = A098586(2*n).
a(2*n+1) = A098790(2*n).
a(n) = 7*a(n-2) - 7*a(n-4) + a(n-6), for n>6. (End)
a(2*n+1)^2 + (a(2*n+1)+1)^2 = A038761(n)^2 + 2^2. - Hermann Stamm-Wilbrandt, Aug 31 2014

More terms from Harvey P. Dale, Feb 07 2011

More terms from Wesley Ivan Hurt, Aug 04 2014

A064784 Difference between n-th triangular number t(n) and the largest square <= t(n).

Original entry on oeis.org

0, 2, 2, 1, 6, 5, 3, 0, 9, 6, 2, 14, 10, 5, 20, 15, 9, 2, 21, 14, 6, 28, 20, 11, 1, 27, 17, 6, 35, 24, 12, 44, 32, 19, 5, 41, 27, 12, 51, 36, 20, 3, 46, 29, 11, 57, 39, 20, 0, 50, 30, 9, 62, 41, 19, 75, 53, 30, 6, 66, 42, 17, 80, 55, 29, 2, 69, 42, 14, 84, 56, 27, 100, 71, 41, 10, 87, 56

Offset: 1

Jonathan Ayres (jonathan.ayres(AT)btinternet.com), Oct 20 2001

The second differences of a(n) - (a(n)-a(n-1))-(a(n-1)-a(n-2)) - give 2, -2, -1, 6, -6, -1, -1, 12, -12, -1, 16, -16, -1 ... 82k+2, 82k-2, -1, 82k+6, 82k-6, -1, -1, 82k+12, 82k-12, -1, 82k+16, -82k-16, -1, 82k+20, -82k-20, -1, -1, 82k+26, -82k-26, -1, 82k+30, -82k-30, -1, -1, 82k+36, -82k-36, -1, 82k+40, -82k-40, -1, 82k+44, -82k-44, -1, -1, 82k+50, -82k-50, -1, 82k+54, -82k-54, -1, -1, 82k+60, -82k-60, -1, 82k+64, -82k-64, -1, -1, 82k+70, -82k-70, -1, 82k+74, -82k-74, -1, 82k+78, -82k-78, -1, -1, ...

			n = 5: A000217(5) = 28, largest square below that is 25, so a(5) = 28 - 25 = 3.

Harry J. Smith, Table of n, a(n) for n = 1..1000
J. C. Lagarias, E. M. Rains and N. J. A. Sloane, The EKG sequence, Exper. Math. 11 (2002), 437-446.
J. C. Lagarias, E. M. Rains and N. J. A. Sloane, The EKG sequence, arXiv:math/0204011 [math.NT], 2002.
Index entries for sequences related to EKG sequence

Cf. A001108, A076816, A128549, A230038. Unique values are in A230044.

seq(n*(n+1)/2-floor(sqrt(n*(n+1)/2))^2,n=0..100);

f[n_]:=n*(n+1)/2-Floor[Sqrt[n*(n+1)/2]]^2; lst={}; Do[AppendTo[lst,f[n]],{n,0,6!}]; lst (* Vladimir Joseph Stephan Orlovsky, Feb 17 2010 *)
#-Floor[Sqrt[#]]^2&/@Accumulate[Range[100]] (* Harvey P. Dale, Oct 15 2014 *)

{ default(realprecision, 100); for (n=1, 1000, t=n*(n + 1)/2; a=t - floor(sqrt(t))^2; write("b064784.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 25 2009

from math import isqrt
def A064784(n): return (m:=n*(n+1)>>1)-isqrt(m)**2 # Chai Wah Wu, Jun 01 2024

a(n) = n*(n+1)/2 - floor(sqrt(n*(n+1)/2))^2.
a(n) = A053186(A000217(n)). - R. J. Mathar, Sep 10 2016
a(A001108(n)) = 0. - Hugo Pfoertner, Jun 01 2024

Definition corrected by Harry J. Smith, Sep 25 2009

Terms corrected by Harry J. Smith, Sep 25 2009

A123737 Partial sums of (-1)^floor(n*sqrt(2)).

Original entry on oeis.org

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

Offset: 1

T. D. Noe, Oct 11 2006

Conjecture: A001652(n) is the index of the first occurrence of n in sequence A123737, A001108(n) is the index of the first occurrence of -n in sequence A123737. - Vaclav Kotesovec, Jun 02 2015

T. D. Noe, Table of n, a(n) for n = 1..10000
Henk Bruin and Robbert Fokkink, On the records and zeros of a deterministic random walk, arXiv:2503.11734 [math.DS], 2025. See p. 2.
Kevin O'Bryant, Bruce Reznick and Monika Serbinowska, Almost alternating sums, arXiv:math/0308087 [math.NT], 2003-2205; Amer. Math. Monthly, Vol. 113 (October 2006), pp. 673-688.

Cf. A123724 (sum for 2^(1/3)), A123738 (sum for Pi), A123739 (sum for e).

Cf. A001652, A001951, A228639.

[&+[(-1)^Floor(j*Sqrt(2)): j in [1..n]]: n in [1..130]]; // G. C. Greubel, Sep 05 2019

ListTools:-PartialSums([seq((-1)^floor(n*sqrt(2)),n=1..100)]); # Robert Israel, Jun 02 2015

Rest[FoldList[Plus,0,(-1)^Floor[Sqrt[2]*Range[120]]]]
Accumulate[(-1)^Floor[Range[120]Sqrt[2]]] (* Harvey P. Dale, Jan 16 2012 *)
(* The positions of the first occurrences of n and -n in this sequence: *) stab = Rest[FoldList[Plus,0,(-1)^Floor[Sqrt[2]*Range[1000000]]]]; Print[Table[FirstPosition[stab,n][[1]],{n,1,8}]]; Print[Table[FirstPosition[stab,-n][[1]],{n,1,8}]]; (* Vaclav Kotesovec, Jun 02 2015 *)

a(n)=sum(i=1,n,(-1)^sqrtint(2*i^2)) \\ Charles R Greathouse IV, Feb 07 2013

[sum((-1)^floor(j*sqrt(2)) for j in (1..n)) for n in (1..130)] # G. C. Greubel, Sep 05 2019

O'Bryant, Reznick, & Serbinowska show that |a(n)| <= k log n + 1, with k = 1/(2 log (1 + sqrt(2))), and further -a(n) > k log n + 0.78 infinitely often. - Charles R Greathouse IV, Feb 07 2013

A216162 Sequences A006452 and A216134 interlaced.

Original entry on oeis.org

1, 0, 1, 1, 2, 4, 4, 9, 11, 26, 23, 55, 64, 154, 134, 323, 373, 900, 781, 1885, 2174, 5248, 4552, 10989, 12671, 30590, 26531, 64051, 73852, 178294, 154634, 373319, 430441, 1039176, 901273, 2175865, 2508794, 6056764, 5253004, 12681873, 14622323, 35301410

Offset: 0

Raphie Frank, Sep 07 2012

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,1,0,6,0,-6,0,-1,0,1)

Cf. A000129.
For some k in n:
a(2n) = A006452 (k^2 - 1 is triangular).
a(2n + 1) = A216134 (T_k and 2T_k + 1 are triangular).
a(2n + 1) - a(2n) = A006451 (T_k + 1 is square).
a(2n + 1) + a(2n) = A124124 (T_k and (T_k - 1)/2 are triangular).
a(4n + 1) + a(4n + 2) = A001108 (T_k is square).
a(4n + 3) + a(4n + 4) = A001652 (T_k and 2T_k are triangular).
Sum(a(n)) - 1 = A048776 for even n (the second partial summation of the Pell numbers).

Vec((-1-x^3+5*x^4-3*x^5-2*x^6+x^7-2*x^8+x^9)/((x-1)*(1+x)*(x^4-2*x^2-1)*(x^4+2*x^2-1))+O(x^99)) \\ Charles R Greathouse IV, Jun 12 2015

(a(2n) + a(2n - 1)) - (a(2n - 2) + a(2n - 3)) = A000129(n); n>1.
It follows that sqrt(2) = lim n --> infinity ((a(2n + 2) + a(2n + 1)) - (a(2n - 2) + a(2n - 3)))/((a(2n + 2) + a(2n + 1)) - (a(2n) + a(2n - 1))).
G.f. ( -1-x^3+5*x^4-3*x^5-2*x^6+x^7-2*x^8+x^9 ) / ( (x-1)*(1+x)*(x^4-2*x^2-1)*(x^4+2*x^2-1) ). - R. J. Mathar, Sep 08 2012

Edited by N. J. A. Sloane, May 24 2021

A309507 Number of ways the n-th triangular number T(n) = A000217(n) can be written as the difference of two positive triangular numbers.

Original entry on oeis.org

0, 1, 1, 1, 3, 3, 1, 2, 5, 3, 3, 3, 3, 7, 3, 1, 5, 5, 3, 7, 7, 3, 3, 5, 5, 7, 7, 3, 7, 7, 1, 3, 7, 7, 11, 5, 3, 7, 7, 3, 7, 7, 3, 11, 11, 3, 3, 5, 8, 11, 7, 3, 7, 15, 7, 7, 7, 3, 7, 7, 3, 11, 5, 3, 15, 7, 3, 7, 15, 7, 5, 5, 3, 11, 11, 7, 15, 7, 3, 9, 9, 3, 7

Offset: 1

Alois P. Heinz, Aug 05 2019

nonn

Equivalently, a(n) is the number of triples [n,k,m] with k>0 satisfying the Diophantine equation n*(n+1) + k*(k+1) - m*(m+1) = 0. Any such triple satisfies a triangle inequality, n+k > m. The n for which there is a triple [n,n,m] are listed in A053141. - Bradley Klee, Mar 01 2020; edited by N. J. A. Sloane, Mar 31 2020

			a(5) = 3: T(5) = T(6)-T(3) = T(8)-T(6) = T(15)-T(14).
a(7) = 1: T(7) = T(28)-T(27).
a(8) = 2: T(8) = T(13)-T(10) = T(36)-T(35).
a(9) = 5: T(9) = T(10)-T(4) = T(11)-T(6) = T(16)-T(13) = T(23)-T(21) = T(45)-T(44).
a(49) = 8: T(49) = T(52)-T(17) = T(61)-T(36) = T(94)-T(80) = T(127)-T(117) = T(178)-T(171) = T(247)-T(242) = T(613)-T(611) = T(1225)-T(1224).
The triples with n <= 16 are:
2, 2, 3
3, 5, 6
4, 9, 10
5, 3, 6
5, 6, 8
5, 14, 15
6, 5, 8
6, 9, 11
6, 20, 21
7, 27, 28
8, 10, 13
8, 35, 36
9, 4, 10
9, 6, 11
9, 13, 16
9, 21, 23
9, 44, 45
10, 8, 13
10, 26, 28
10, 54, 55
11, 14, 18
11, 20, 23
11, 65, 66
12, 17, 21
12, 24, 27
12, 77, 78
13, 9, 16
13, 44, 46
13, 90, 91
14, 5, 15
14, 11, 18
14, 14, 20
14, 18, 23
14, 33, 36
14, 51, 53
14, 104, 105
15, 21, 26
15, 38, 41
15, 119, 120
16, 135, 136. - _N. J. A. Sloane_, Mar 31 2020

Alois P. Heinz, Table of n, a(n) for n = 1..20000
J. S. Myers, R. Schroeppel, S. R. Shannon, N. J. A. Sloane, and P. Zimmermann, Three Cousins of Recaman's Sequence, arXiv:2004:14000 [math.NT], April 2020.
M. A. Nyblom, On the representation of the integers as a difference of nonconsecutive triangular numbers, Fibonacci Quarterly 39:3 (2001), pp. 256-263.

Cf. A000217, A001108, A046079 (the same for squares), A068194, A100821 (the same for primes for n>1), A309332.
See also A053141. The monotonic triples [n,k,m] with n <= k <= m are counted in A333529.
Cf. A092517, A063440.

with(numtheory): seq(tau(n*(n+1))-tau(n*(n+1)/2)-1, n=1..80); # Ridouane Oudra, Dec 08 2023

TriTriples[TNn_] := Sort[Select[{TNn, (TNn + TNn^2 - # - #^2)/(2 #),
      (TNn + TNn^2 - # + #^2)/(2 #)} & /@
    Complement[Divisors[TNn (TNn + 1)], {TNn}],
   And[And @@ (IntegerQ /@ #), And @@ (# > 0 & /@ #)] &]]
Length[TriTriples[#]] & /@ Range[100]
(* Bradley Klee, Mar 01 2020 *)

a(n) = 1 <=> n in { A068194 } \ { 1 }.
a(n) is even <=> n in { A001108 } \ { 0 }.
a(n) = number of odd divisors of n*(n+1) (or, equally, of T(n)) that are greater than 1. - N. J. A. Sloane, Apr 03 2020
a(n) = A092517(n) - A063440(n) - 1. - Ridouane Oudra, Dec 08 2023

A001953 a(n) = floor((n + 1/2) * sqrt(2)).

Original entry on oeis.org

0, 2, 3, 4, 6, 7, 9, 10, 12, 13, 14, 16, 17, 19, 20, 21, 23, 24, 26, 27, 28, 30, 31, 33, 34, 36, 37, 38, 40, 41, 43, 44, 45, 47, 48, 50, 51, 53, 54, 55, 57, 58, 60, 61, 62, 64, 65, 67, 68, 70, 71, 72, 74, 75, 77, 78, 79, 81, 82, 84, 85, 86, 88, 89, 91, 92, 94, 95

Offset: 0

N. J. A. Sloane

nonn

Let s(n) = zeta(3) - Sum_{k = 1..n} 1/k^3. Conjecture: for n >= 1, s(a(n)) < 1/n^2 < s(a(n)-1), and the difference sequence of A049473 consists solely of 0's and 1's, in positions given by the nonhomogeneous Beatty sequences A001954 and A001953, respectively. - Clark Kimberling, Oct 05 2014

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..10000
Ian G. Connell, A generalization of Wythoff's game, Canad. Math. Bull. 2 (1959) 181-190.
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)

Complement of A001954.

Cf. A000217 (T), A136119, A001108.

[Floor((2*n+1)/Sqrt(2)): n in [0..100]]; // G. C. Greubel, Nov 14 2019

seq( floor((2*n+1)/sqrt(2)), n=0..100); # G. C. Greubel, Nov 14 2019

Table[Floor[(n + 1/2) Sqrt[2]], {n, 0, 100}] (* T. D. Noe, Aug 17 2012 *)

PARI
```
a(n)=floor((n+1/2)*sqrt(2))
```

a(n)={sqrtint(2*n*(n+1))} \\ Andrew Howroyd, Oct 24 2019

[floor((2*n+1)/sqrt(2)) for n in (0..100)] # G. C. Greubel, Nov 14 2019

From Ralf Steiner, Oct 23 2019: (Start)
a(n) = floor(2*sqrt(A000217(n))).
a(n) = A136119(n + 1) - 1.
a(n + 1) - a(n) is in {1,2}.
a(n + 3) - a(n) is in {4,5}. (End)

More terms from Michael Somos, Apr 26 2000.

A175035 Offsets i such that i + n*(n+1)/2 is a perfect square for some positive integer n.

Original entry on oeis.org

1, 3, 4, 6, 8, 9, 10, 13, 15, 16, 19, 21, 22, 24, 25, 26, 28, 30, 33, 34, 35, 36, 39, 43, 45, 46, 48, 49, 53, 54, 55, 58, 60, 61, 63, 64, 66, 71, 72, 75, 76, 78, 79, 80, 81, 85, 89, 90, 91, 93, 94, 97, 99, 100, 103, 105, 106, 108, 111, 114, 115, 116, 118, 120

Offset: 1

Ctibor O. Zizka, Nov 10 2009

The ansatz n*(n+1)/2+i=s^2 can be transformed into (2*n+1)^2-2*(2*s)^2 =1-8*i.
A necessary condition for solutions to this Diophantine equation is that D=2 is a quadratic residue of the squarefree part of 8*i-1 (see A057126).
A sufficient condition is then available by a sequence of tests on the continued fractions of a quadratic surd that originates from a solution of this congruence.
See Mollin and Matthews for details. - R. J. Mathar, Nov 16 2009

R. A. Mollin, Simple continued fraction solutions for Diophantine Equations, Exposit. Mathem. 19 (2001) 55-73.
Keith Matthews, The Diophantine equation x^2-Dy^2=N,D >0, Exposit. Mathem. 18 (4) (2000) 323-331 [MR1788328].

Cf. A001108, A006451, A154138 to A154154.

Cf. A230044.

Take[Rest[Ceiling[Sqrt[#]]^2-#&/@Accumulate[Range[1000]]//Union],70] (* Harvey P. Dale, Sep 07 2019 *)

is(n)=#bnfisintnorm(bnfinit(z^2-8),-8*n+1) /* Ralf Stephan, Oct 14 2013 */

Extended by R. J. Mathar, Nov 26 2009

A229118 Distance from the n-th triangular number to the nearest square.

Original entry on oeis.org

0, 1, 2, 1, 1, 4, 3, 0, 4, 6, 2, 3, 9, 5, 1, 8, 9, 2, 6, 14, 6, 3, 13, 11, 1, 10, 17, 6, 6, 19, 12, 1, 15, 19, 5, 10, 26, 12, 4, 21, 20, 3, 15, 29, 11, 8, 28, 20, 0, 21, 30, 9, 13, 36, 19, 4, 28, 30, 6, 19, 42, 17, 9, 36, 29, 2, 26, 42, 14, 15, 45, 27, 3, 34, 41, 10, 22, 55, 24, 9, 43, 39, 5, 30, 55, 20, 16, 53, 36, 1

Offset: 1

Ralf Stephan, Sep 14 2013

The maximum of a(n)/n appears to converge to sqrt(2)/2 (A010503), i.e. n*(n+1)/2 seems not more than n*sqrt(2)/2 distant from a square.
Some values don't seem to be in the sequence (checked up to n=10^7): 7,18,23,31,37,38...
Those values k are not in the sequence because the Pell-type equations x^2 - 8*y^2 = 8*k+1 and x^2 - 8*y^2 = -8*k+1 have no solutions. - Robert Israel, Apr 08 2019
a(A001108(n)) = 0, a(A229131(n)) = 1, a(A229083(n)) <= 1, a(A229133(n)) is square.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A229117, A000217, A000290.

dns[n_]:=Module[{a=Floor[Sqrt[n]]^2,b=Ceiling[Sqrt[n]]^2},Min[n-a, b-n]]; dns/@Accumulate[Range[90]] (* Harvey P. Dale, Nov 07 2016 *)

m=0;for(n=1, 100, t=n*(n+1)/2;s=sqrtint(t);d=min(t-s^2,(s+1)^2-t);print1(d, ","))

cp's OEIS Frontend

A322699 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where A(n,k) is 1/2 * (-1 + Sum_{j=0..k} binomial(2*k,2*j)*(n+1)^(k-j)*n^j).

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A008843 Squares of NSW numbers (A002315): x^2 such that x^2 - 2y^2 = -1 for some y.

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A124124 Nonnegative integers n such that 2n^2 + 2n - 3 is square.

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A064784 Difference between n-th triangular number t(n) and the largest square <= t(n).

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A123737 Partial sums of (-1)^floor(n*sqrt(2)).

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A216162 Sequences A006452 and A216134 interlaced.

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A309507 Number of ways the n-th triangular number T(n) = A000217(n) can be written as the difference of two positive triangular numbers.

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A322699 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where A(n,k) is 1/2 * (-1 + Sum_{j=0..k} binomial(2k,2j)(n+1)^(k-j)n^j).