A000952 -id:A000952 - cp's OEIS frontend

A069894 Centered square numbers: a(n) = 4n^2 + 4n + 2.

2, 10, 26, 50, 82, 122, 170, 226, 290, 362, 442, 530, 626, 730, 842, 962, 1090, 1226, 1370, 1522, 1682, 1850, 2026, 2210, 2402, 2602, 2810, 3026, 3250, 3482, 3722, 3970, 4226, 4490, 4762, 5042, 5330, 5626, 5930, 6242, 6562, 6890, 7226, 7570, 7922, 8282

Offset: 0

Glenn B. Cox (igloos_r_us(AT)canada.com), Apr 10 2002

Any number may be substituted for y to yield similar sequences. The number set used determines values given (i.e., integer yields integer). All centered square integers in the set of integers may be found by this formula.
1/2 + 1/10 + 1/26 + ... = (Pi/4)*tanh(Pi/2) [Jolley]. - Gary W. Adamson, Dec 21 2006
For n > 0, a(n-1) is the number of triples (w, x, y) having all terms in {0, ..., n} and min(|w - x|, |x - y|) = 1. - Clark Kimberling, Jun 12 2012
Consider the primitive Pythagorean triples (x(n), y(n), z(n) = y(n) + 1) with n >= 0, and x(n) = 2*n + 1, y(n) = 2*n*(n + 1), z(n) = 2*n*(n + 1) + 1. The sequence, a(n), is 2*z(n). - George F. Johnson, Oct 22 2012
Ulam's spiral (SE corner). See the Wikipedia link. - Kival Ngaokrajang, Jul 25 2014
Conference matrix orders (A000952) of the form n-1 is a perfect square are all in this sequence. All values less than 1000 are conference matrices except for 226 which is still an open question (Balonin & Seberry 2014). - Colin Hall, Nov 21 2018
For n > 0, a(n-1) is the number of maximum number of regions into which the plane can be divided using n convex quadrilaterals. Related: A077588 A077591. - Keyang Li, Jun 17 2022

			If y = 3, then 81 + 144 = 225; if y = 4, then 12^2 + 16^2 = 20^2; 7^2 + 24^2 = 25^2 = 15^2 + 20^2.

L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, p. 176.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
N. A. Balonin and Jennifer Seberry, A Review and New Symmetric Conference Matrices, Research Online, Faculty of Engineering and Information Sciences, University of Wollongong, 2014.
Keyang Li, Figure for n=1,2,3,4,5
Tintarn, n convex quadrilaterals in the plane
Wikipedia, Ulam Spiral Construction.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001105, A001844, A002522, A016754, A016825, A033996, A261377.

[4*n^2+4*n+2 : n in [0..50]]; // Wesley Ivan Hurt, Jul 26 2014

A069894:=n->4*n^2+4*n+2: seq(A069894(n), n=0..50); # Wesley Ivan Hurt, Jul 26 2014

Mathematica
```
Table[4n(n + 1) + 2, {n, 0, 45}]
```

vector(100, n, (2*n-1)^2+1); \\ Derek Orr, Jul 27 2014

[(2*n+1)^2 + 1 for n in range(50)] # G. C. Greubel, Nov 21 2018

(y*(2*x + 1))^2 + (y*(2*x^2 + 2*x))^2 = (y*(2*x^2 + 2*x + 1))^2, where y = 2. If a^2 + b^2 = c^2, then c^2 = y^2*(4*x^4 + 8*x^3 + 8*x^2 + 4*x + 1). Also 2*A001844.
a(n) = (2*n + 1)^2 + 1. - Vladimir Joseph Stephan Orlovsky, Nov 10 2008 [Corrected by R. J. Mathar, Sep 16 2009]
a(n) = 8*n + a(n-1) for n > 0, a(0)=2. - Vincenzo Librandi, Aug 08 2010
From George F. Johnson, Oct 22 2012: (Start)
G.f.: 2*(1 + x)^2/(1 - x)^3, a(0) = 2, a(1) = 10.
a(n+1) = a(n) + 4 + 4*sqrt(a(n) - 1).
a(n-1) * a(n+1) = (a(n)-4)^2 + 16.
a(n) - 1 = (2*n+1)^2 = A016754(n) for n > 0.
(a(n+1) - a(n-1))/8 = sqrt(a(n) - 1).
a(n+1) = 2*a(n) - a(n-1) + 8 for n > 2, a(0)=2, a(1)=10, a(2)=26.
a(n+1) = 3*a(n) - 3*a(n-1) + a(n-2) for n > 3; a(0)=2, a(1)=10, a(2)=26, a(3)=50.
a(n) = A033996(n) + 2 = A002522(2n + 1).
a(n)^2 = A033996(n)^2 + A016825(n)^2. (End)
a(n) = A001105(n) + A001105(n+1). - Bruno Berselli, Jul 03 2017
E.g.f.: 2*(1 + 4*x + 2*x^2)*exp(x). - G. C. Greubel, Nov 21 2018
a(n) = A261327(4*n+2). - Paul Curtz, Dec 23 2021
a(n) = 2*A001844(n) = 4*A000217(n) + 2*A002061(n+1). - Klaus Purath, Aug 13 2025

Edited by Robert G. Wilson v, Apr 11 2002

Offset corrected by Charles R Greathouse IV, Jul 25 2010

A084109 n is congruent to 1 (mod 4) and is not the sum of two squares.

Original entry on oeis.org

21, 33, 57, 69, 77, 93, 105, 129, 133, 141, 161, 165, 177, 189, 201, 209, 213, 217, 237, 249, 253, 273, 285, 297, 301, 309, 321, 329, 341, 345, 357, 381, 385, 393, 413, 417, 429, 437, 453, 465, 469, 473, 489, 497

Offset: 1

William P. Orrick, Jun 18 2003

Alternatively, n is congruent to 1 (mod 4) with at least 2 distinct prime factors congruent to 3 (mod 4) in the squarefree part of n. - Comment corrected by Jean-Christophe Hervé, Oct 25 2015
Applications to the theory of optimal weighing designs and maximal determinants: An (n+1) X (n+1) conference matrix is impossible.
The upper bound of Ehlich/Wojtas on the determinant of a (0,1) matrix of order congruent to 1 (mod 4) cannot be achieved for n X n matrices.
The bound of Ehlich/Wojtas on the determinant of a (-1,1) matrix of order congruent to 2 (mod 4) cannot be achieved for (n+1) X (n+1) matrices.
Numbers with only odd prime factors, of which a strictly positive even number are raised to an odd power and congruent to 3 (mod 4). - Jean-Christophe Hervé, Oct 24 2015

			a(1) = 3*7 = 21, a(2) = 3*11 = 33, a(3) = 3*19 = 57, a(14) = 3^3*7 = 189.

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 56.

Jean-Christophe Hervé, Table of n, a(n) for n = 1..1000
H. Ehlich, Determinantenabschätzungen für binäre Matrizen, Math. Z. 83 (1964) 123-132.
D. Raghavarao, Some aspects of weighing designs, Ann. Math. Stat. 31 (1960) 878-884.

Cf. A000952, A003432, A003433, A022544, A001481.

N:= 1000: # to get all entries <= N
S:= {seq(i,i=1..N,4)} minus
   {seq(seq(i^2+j^2, j=1..floor(sqrt(N-i^2)),2),i=0..floor(sqrt(N)),2)}:
sort(convert(S,list)); # Robert Israel, Oct 25 2015

a[m_] := Complement[Range[1, m, 4], Union[Flatten[Table[j^2+k^2, {j, 1, Sqrt[m], 2}, {k, 0, Sqrt[m], 2}]]]]

is(n)=if(n%4!=1, return(0)); my(f=factor(n)); for(i=1, #f~, if(f[i, 1]%4==3 && f[i, 2]%2, return(1))); 0 \\ Charles R Greathouse IV, Jul 01 2016

A286636 Even numbers that are a sum of two squares plus 1.

Original entry on oeis.org

2, 6, 10, 14, 18, 26, 30, 38, 42, 46, 50, 54, 62, 66, 74, 82, 86, 90, 98, 102, 110, 114, 118, 122, 126, 138, 146, 150, 154, 158, 170, 174, 182, 186, 194, 198, 206, 222, 226, 230, 234, 242, 246, 258, 262, 266, 270, 278, 282, 290, 294, 306, 314, 318, 326, 334, 338, 350, 354, 362, 366, 370, 374, 378, 390, 398

Offset: 1

Jean-François Alcover, May 11 2017

nonn

The first 13 terms coincide with A000952.
If the conjecture in A000952 is true, the two sequences are the same. - R. J. Mathar, May 18 2017
Numbers that are the sum of two centered square numbers (A001844). - Ilya Gutkovskiy, Jun 03 2017

Cf. A000952, A001481, A166687.

Select[Range[2, 400, 2], SquaresR[2, # - 1] != 0 &]

A061344 Numbers of form p^m + 1, p odd prime, m >= 1.

Original entry on oeis.org

4, 6, 8, 10, 12, 14, 18, 20, 24, 26, 28, 30, 32, 38, 42, 44, 48, 50, 54, 60, 62, 68, 72, 74, 80, 82, 84, 90, 98, 102, 104, 108, 110, 114, 122, 126, 128, 132, 138, 140, 150, 152, 158, 164, 168, 170, 174, 180, 182, 192, 194, 198, 200, 212, 224, 228, 230, 234, 240

Offset: 1

Hans Dieter Lueke (lueke(AT)ient.rwth-aachen.de), Jun 08 2001

Lengths of almost-binary sequences with perfect odd-periodic autocorrelation function.

As J. Arndt points out, each element of this sequence leads to a conference matrix (cf. link to Wikipedia and A000952). - M. F. Hasler, Mar 14 2008

H. D. Lueke, Binary odd-periodic complementary sequences. IEEE Trans. Inform. Theory, 43, pp. 365-367, 1997.

M. F. Hasler, Table of n, a(n) for n = 1..10000
Wikipedia, Conference matrix.

Equals A061345 + 1. Cf. A000952.

A061344(n)= local(m=1,p); for(c=1,n, until( isprime(m+=2) || ispower(m,[null], && p) && isprime(p),); /*print(c," ",m+1)*/); m+1 \\ - M. F. Hasler, Mar 14 2008

from sympy import primepi, integer_nthroot
def A061344(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+x-sum(primepi(integer_nthroot(x,k)[0])-1 for k in range(1,x.bit_length())))
    return bisection(f,n+1,n+1)+1 # Chai Wah Wu, Feb 03 2025

More terms from Larry Reeves (larryr(AT)acm.org), Jun 12 2001

Edited by M. F. Hasler, Mar 14 2008

A123220 a(n)=the (1,1)-term of M^(n-1), where M=matrix(5,5, [3,-1,-1,-1,-1; 1,3,-1,-1,-1; 1,1,3,-1,-1; 1,1,1,3,-1; 1,1,1,1,3]).

Original entry on oeis.org

1, 3, 5, -9, -99, -297, 389, 8655, 46573, 122823, -120491, -3003393, -15885315, -40246281, 50400229, 1040606127, 5296630541, 12512952615, -22872751243, -368600380833, -1789336379619, -3926384911017, 9502037022725, 129579396089871, 602116408170541, 1219711972804743

Offset: 1

Roger L. Bagula, Oct 05 2006

sign

Cf. A000952, A074872, A085340.

a[1]:=1:a[2]:=3:a[3]:=5:a[4]:=-9:a[5]:=-99: for n from 6 to 26 do a[n]:=15*a[n-1]-100*a[n-2]+360*a[n-3]-680*a[n-4]+528*a[n-5] od: seq(a[n],n=1..26); with(linalg): M[1]:=matrix(5,5,[3, -1, -1, -1, -1, 1, 3, -1, -1, -1, 1, 1, 3, -1, -1, 1, 1, 1, 3, -1, 1, 1, 1, 1, 3]): for n from 2 to 25 do M[n]:=multiply(M[1],M[n-1]) od: 1,seq(M[n][1,1],n=1..25);

M = {{3, -1, -1, -1, -1}, {1, 3, -1, -1, -1}, {1, 1, 3, -1, -1}, {1, 1, 1, 3, -1}, {1, 1, 1, 1, 3}}; w[1] = {1, 0, 0, 0, 0}; w[n_] := w[n] = M.w[n - 1]; a = Table[w[n][[1]], {n, 1, 30}]

a(1) = 1; a(2) = 3; a(3) = 5; a(4) = -9; a(5) = -99; a(n) = 15a(n-1)-100a(n-2)+360a(n-3)-680a(n-4)+528a(n-5) for n>= 6. The minimal polynomial of M is x^5-15x^4+100x^3-360x^2+680x-528, the coefficients of which yield the coefficients of the recurrence relation.

O.g.f.: -x*(1-12*x+60*x^2-144*x^3+136*x^4)/((3*x-1)*(176*x^4-168*x^3+64*x^2-12*x+1)). - R. J. Mathar, Dec 05 2007

Edited by N. J. A. Sloane, Oct 15 2006

A123222 Expansion of -x * (x-1) * (3x^2-1) / (9x^4-8x^3+4x-1).

Original entry on oeis.org

1, 3, 9, 31, 109, 391, 1397, 4995, 17833, 63675, 227313, 811543, 2897269, 10343647, 36928061, 131837979, 470678161, 1680380979, 5999172633, 21417807055, 76464283837, 272987183095, 974598829637, 3479441311347, 12422046335161

Offset: 1

Roger L. Bagula, Oct 05 2006

Index entries for linear recurrences with constant coefficients, signature (4,0,-8,9).

Cf. A000952, A074872, A085340, A003432.

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(-x*(x-1)*(3*x^2-1)/(9*x^4-8*x^3+4*x-1))); // G. C. Greubel, Oct 12 2018

seq(coeff(series(-x*(x-1)*(3*x^2-1)/(9*x^4-8*x^3+4*x-1),x,n+1), x, n), n = 1 .. 25); # Muniru A Asiru, Oct 13 2018

LinearRecurrence[{4,0,-8,9},{1,3,9,31},30] (* Harvey P. Dale, Jul 26 2018 *)

x='x+O('x^30); Vec(-x*(x-1)*(3*x^2-1)/(9*x^4-8*x^3+4*x-1)) \\ G. C. Greubel, Oct 12 2018

From Colin Barker, Oct 19 2012: (Start)
a(n) = 4*a(n-1) -8*a(n-3) +9*a(n-4).
G.f.: -x*(x-1)*(3*x^2-1)/(9*x^4-8*x^3+4*x-1). (End)

Sequence edited by Joerg Arndt and Colin Barker, Oct 19 2012

A352348 Maximum determinant of n X n matrix composed of {-1, 0, 1} with pairwise orthogonal rows.

Original entry on oeis.org

1, 2, 2, 16, 16, 125, 128, 4096, 4096, 59049, 59049, 2985984, 2985984, 62748517

Offset: 1

Max Alekseyev, Mar 12 2022

a(n) >= a(m)*a(n-m) for any m < n.
a(n) <= A003433(n), a bound achieved if the orthogonality requirement is dropped.
If there exists an order n Hadamard matrix, then a(n) = A003433(n) = n^(n/2).
For n == 2 (mod 4), if there exists an order n conference matrix (cf. A000952), then a(n) = (n-1)^(n/2). In particular, a(18) = 118587876497.

Arun et al., Reference request: maximal determinant of matrices with pairwise orthogonal rows and entries in {1, 0, -1}, MathOverflow, 2022.
Wikipedia, Conference matrix.
Wikipedia, Hadamard matrix.

Cf. A000952, A003432, A003433.

a(11)-a(14) from Max Alekseyev, May 20 2023

cp's OEIS Frontend

A069894 Centered square numbers: a(n) = 4*n^2 + 4*n + 2.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A084109 n is congruent to 1 (mod 4) and is not the sum of two squares.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A286636 Even numbers that are a sum of two squares plus 1.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A061344 Numbers of form p^m + 1, p odd prime, m >= 1.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

PARI

Python

Extensions

A123220 a(n)=the (1,1)-term of M^(n-1), where M=matrix(5,5, [3,-1,-1,-1,-1; 1,3,-1,-1,-1; 1,1,3,-1,-1; 1,1,1,3,-1; 1,1,1,1,3]).

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A123222 Expansion of -x * (x-1) * (3*x^2-1) / (9*x^4-8*x^3+4*x-1).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions

A352348 Maximum determinant of n X n matrix composed of {-1, 0, 1} with pairwise orthogonal rows.

Views

Author

Keywords

Comments

Links

A069894 Centered square numbers: a(n) = 4n^2 + 4n + 2.

A123222 Expansion of -x * (x-1) * (3x^2-1) / (9x^4-8x^3+4x-1).