A002654 -id:A002654 - cp's OEIS frontend

A145392 Number of inequivalent sublattices of index n in square lattice, where two sublattices are considered equivalent if one can be rotated by a multiple of Pi/2 to give the other.

1, 2, 2, 4, 4, 6, 4, 8, 7, 10, 6, 14, 8, 12, 12, 16, 10, 20, 10, 22, 16, 18, 12, 30, 17, 22, 20, 28, 16, 36, 16, 32, 24, 28, 24, 46, 20, 30, 28, 46, 22, 48, 22, 42, 40, 36, 24, 62, 29, 48, 36, 50, 28, 60, 36, 60, 40, 46, 30, 84, 32, 48, 52, 64, 44, 72, 34, 64, 48, 72

Offset: 1

N. J. A. Sloane, Feb 23 2009

nonn

From Andrey Zabolotskiy, Mar 12 2018: (Start)
The parent lattice of the sublattices under consideration has Patterson symmetry group p4, and two sublattices are considered equivalent if they are related via a symmetry from that group [Rutherford]. For other 2D Patterson groups, the analogous sequences are A000203 (p2), A069734 (p2mm), A145391 (c2mm), A145393 (p4mm), A145394 (p6), A003051 (p6mm).
If we count sublattices related by parent-lattice-preserving reflection as equivalent, we get A145393 instead of this sequence. If we count sublattices related by rotation of the sublattice only (but not parent lattice; equivalently, sublattices related by rotation by angle which is not a multiple of Pi/2; see illustration in links) as equivalent, we get A054345. If we count sublattices related by any rotation or reflection as equivalent, we get A054346.
Rutherford says at p. 161 that a(n) != A054345(n) only when A002654(n) > 1, but actually these two sequences differ at other terms, too, for example, at n = 15 (see illustration). (End)

Antti Karttunen, Table of n, a(n) for n = 1..16384
John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. (2009). A65, 156-163. [See Table 2; beware the typo in a(13).]
Andrey Zabolotskiy, Sublattices of the square lattice (illustrations for n = 1..6, 15, 25)
Index entries for sequences related to sublattices
Index entries for sequences related to square lattice

Cf. A000203, A002654, A069734, A145391, A145393, A145394, A003051, A054345, A054346, A000089, A157224, A004525.

A002654(n) = sumdiv(n, d, (d%4==1) - (d%4==3));
A145392(n) = ((sigma(n) + A002654(n))/2); \\ Antti Karttunen, Nov 23 2017

a(n) = (A000203(n) + A002654(n))/2. [Rutherford] - N. J. A. Sloane, Mar 13 2009
a(n) = Sum_{ m: m^2|n } A000089(n/m^2) + A157224(n/m^2) = A002654(n) + Sum_{ m: m^2|n } A157224(n/m^2). - Andrey Zabolotskiy, May 07 2018
a(n) = Sum_{ d|n } A004525(d). - Andrey Zabolotskiy, Aug 29 2019

New name from Andrey Zabolotskiy, Mar 12 2018

A145394 Number of inequivalent sublattices of index n in hexagonal lattice, where two sublattices are considered equivalent if one can be rotated by a multiple of Pi/3 to give the other.

Original entry on oeis.org

1, 1, 2, 3, 2, 4, 4, 5, 5, 6, 4, 10, 6, 8, 8, 11, 6, 13, 8, 14, 12, 12, 8, 20, 11, 14, 14, 20, 10, 24, 12, 21, 16, 18, 16, 31, 14, 20, 20, 30, 14, 32, 16, 28, 26, 24, 16, 42, 21, 31, 24, 34, 18, 40, 24, 40, 28, 30, 20, 56, 22, 32, 36, 43, 28, 48, 24, 42, 32, 48, 24, 65, 26, 38, 42, 48, 32, 56, 28, 62

Offset: 1

N. J. A. Sloane, Feb 23 2009

nonn

Also, apparently a(n) is the number of nonequivalent (up to lattice-preserving affine transformation) triangles on 2D square lattice of area n/2 [Karpenkov]. - Andrey Zabolotskiy, Jul 06 2017
From Andrey Zabolotskiy, Jan 18 2018: (Start)
The parent lattice of the sublattices under consideration has Patterson symmetry group p6, and two sublattices are considered equivalent if they are related via a symmetry from that group [Rutherford]. For other 2D Patterson groups, the analogous sequences are A000203 (p2), A069734 (p2mm), A145391 (c2mm), A145392 (p4), A145393 (p4mm), A003051 (p6mm).
If we count sublattices related by parent-lattice-preserving reflection as equivalent, we get A003051 instead of this sequence. If we count sublattices related by rotation of the sublattice only (but not parent lattice; equivalently, sublattices related by rotation by angle which is not a multiple of Pi/3; see illustration in links) as equivalent, we get A054384. If we count sublattices related by any rotation or reflection as equivalent, we get A300651.
Rutherford says at p. 161 that a(n) != A054384(n) only when A002324(n) > 1, but actually these two sequences differ at other terms, too, for example, at n = 14 (see illustration). (End)

Antti Karttunen, Table of n, a(n) for n = 1..16384
Oleg Karpenkov, Elementary notions of lattice trigonometry, Mathematica Scandinavica, vol.102, no.2, pp.161-205, (2008) [See page 203].
Oleg Karpenkov, Geometry of Lattice Angles, Polygons, and Cones, Thesis, Technische Universität Graz, 2009.
Christian Kassel and Christophe Reutenauer, The zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1505.07229v3 [math.AG], 2015. [Note that a later version of this paper has a different title and different contents, and the number-theoretical part of the paper was moved to the publication which is next in this list.]
Christian Kassel and Christophe Reutenauer, Complete determination of the zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1610.07793 [math.NT], 2016.
John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. (2009). A65, 156-163. [See Table 2.]
Andrey Zabolotskiy, Sublattices of the hexagonal lattice (illustrations for n = 1..7, 14)
Andrey Zabolotskiy, Coweight lattice A^*_n and lattice simplices, arXiv:2003.10251 [math.CO], 2020.
Index entries for sequences related to sublattices
Index entries for sequences related to A2 = hexagonal = triangular lattice

Cf. A054384, A000203, A069734, A145391, A145392, A145393, A003051, A002324, A002654, A069735, A145390, A300651, A000086, A157227, A008611.

a[n_] := (DivisorSigma[1, n] + 2 DivisorSum[n, Switch[Mod[#, 3], 1, 1, 2, -1, 0, 0] &])/3; Array[a, 80] (* Jean-François Alcover, Dec 03 2015 *)

A002324(n) = if( n<1, 0, sumdiv(n, d, (d%3==1) - (d%3==2)));
A000203(n) = if( n<1, 0, sigma(n));
a(n) = (A000203(n) + 2 * A002324(n)) / 3;
\\ Joerg Arndt, Oct 13 2013

a(n) = (A000203(n) + 2 * A002324(n))/3. [Rutherford] - N. J. A. Sloane, Mar 13 2009
a(n) = Sum_{ m: m^2|n } A000086(n/m^2) + A157227(n/m^2) = A002324(n) + Sum_{ m: m^2|n } A157227(n/m^2). [Rutherford] - Andrey Zabolotskiy, Apr 23 2018
a(n) = Sum_{ d|n } A008611(d-1). - Andrey Zabolotskiy, Aug 29 2019

New name from Andrey Zabolotskiy, Dec 14 2017

A322143 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n, d==1 (mod 4)} d^k - Sum_{d|n, d==3 (mod 4)} d^k.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, -2, 1, 1, 1, -8, 1, 2, 1, 1, -26, 1, 6, 0, 1, 1, -80, 1, 26, -2, 0, 1, 1, -242, 1, 126, -8, -6, 1, 1, 1, -728, 1, 626, -26, -48, 1, 1, 1, 1, -2186, 1, 3126, -80, -342, 1, 7, 2, 1, 1, -6560, 1, 15626, -242, -2400, 1, 73, 6, 0, 1, 1, -19682, 1, 78126, -728, -16806, 1, 703, 26, -10, 0

Offset: 1

Ilya Gutkovskiy, Nov 28 2018

			Square array begins:
  1,  1,   1,    1,    1,     1,  ...
  1,  1,   1,    1,    1,     1,  ...
  0, -2,  -8,  -26,  -80,  -242,  ...
  1,  1,   1,    1,    1,     1,  ...
  2,  6,  26,  126,  626,  3126,  ...
  0, -2,  -8,  -26,  -80,  -242,  ...

Index entries for sequences mentioned by Glaisher

Columns k=0..12 give A002654, A050457, A002173, A050459, A050456, A321821, A321822, A321823, A321824, A321825, A321826, A321827, A321828.

Cf. A109974, A322081, A322082, A322083, A322084.

Table[Function[k, SeriesCoefficient[Sum[(-1)^(j - 1) (2 j - 1)^k x^(2 j - 1)/(1 - x^(2 j - 1)), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten

G.f. of column k: Sum_{j>=1} (-1)^(j-1)*(2*j - 1)^k*x^(2*j-1)/(1 - x^(2*j-1)).

A193583 Number of fixed points under iteration of sum of squares of digits in base b.

Original entry on oeis.org

1, 3, 1, 3, 1, 5, 3, 3, 1, 3, 3, 7, 1, 3, 1, 7, 5, 3, 1, 7, 3, 7, 1, 3, 1, 7, 3, 3, 3, 7, 5, 7, 3, 3, 1, 7, 5, 3, 1, 5, 3, 11, 3, 3, 3, 15, 3, 3, 3, 3, 3, 7, 1, 7, 1, 15, 3, 3, 3, 3, 3, 7, 3, 3, 1, 7, 7, 3, 5, 3, 7, 15, 1, 7, 3, 7, 3, 3, 3, 7, 5, 15, 1, 3, 3

Offset: 2

Martin Renner, Jul 31 2011

If b>=2 and a>=b^2 then S(a,2,b)

From Christian N. K. Anderson, Apr 22 2013: (Start)

It can be shown that no fixed point has more than 2 digits in base b, and that the two-digit number A+Bb must satisfy the condition that (2A-1)^2+(2B-b)^2=1+b^2. The number of ways of writing (1+b^2) as the sum of two squares is d(1+b^2)-1, where d(n) is the number of divisors of n. (Beardon, 1998, Theorem 3.1)

From the above chain of logic follows:

- The value of the fixed points can be determined by investigating only 8*A002654(n^2+1) pairs of possibilities.

- a(n) = A000005(n^2+1)-1

- a(n) = A193432(n)-1

a(n)=1 iff n^2+1 is prime, and the value of that single fixed point is 1.

The only odd value of n for which a(n)=9 is n=239.

Several values of a(n) occur very infrequently. For example, a(1068)=13 is the only occurrence of 13 for n < 10000. (End)

			In the decimal system all integers go to (1) or (4, 16, 37, 58, 89, 145, 42, 20) under the iteration of sum of squares of digits, hence there is one fixed point and one cycle. Therefore a(10) = 1.
a(5)=3 because 1 is always a fixed point; also in base 5, decimal 13 -> 23 and 2^2+3^2 = 13; decimal 18 -> 33 and 3^2+3^2 = 18. - _Christian N. K. Anderson_, Apr 22 2013

Martin Renner and Christian N. K. Anderson, Table of n, a(n) for n = 2..10000 (first 299 values from Martin Renner)
Alan F. Beardon, Sums of Squares of Digits, The Mathematical Gazette, 82(1998), 379-388.
H. G. Grundman, E. A. Teeple, Generalized Happy Numbers, Fibonacci Quarterly 39 (2001), nr. 5, p. 462-466.

Equals A193432-1.
Cf. A007770.
Cf. A002654, A000005.

library(gmp); y=rep(0,10000)
for(B in 1:10000) y[B]=prod(table(as.numeric(factorize(1+as.bigz(B)^2)))+1)-1; y # Christian N. K. Anderson, Apr 22 2013

A014200 Number of solutions to x^2 + y^2 <= n, excluding (0,0), divided by 4.

Original entry on oeis.org

0, 1, 2, 2, 3, 5, 5, 5, 6, 7, 9, 9, 9, 11, 11, 11, 12, 14, 15, 15, 17, 17, 17, 17, 17, 20, 22, 22, 22, 24, 24, 24, 25, 25, 27, 27, 28, 30, 30, 30, 32, 34, 34, 34, 34, 36, 36, 36, 36, 37, 40, 40, 42, 44, 44, 44, 44, 44, 46

Offset: 0

N. J. A. Sloane

nonn

From Ant King, Mar 15 2013: (Start)
The terms of this sequence give a running total of the excess of the 4k + 1 divisors of the natural numbers (from 1 through to n) over their 4k + 3 divisors.
To see how good the approximation n * Pi/4 is to a(n), note that a(10^6) = 785387 whereas 10^6 * Pi/4 rounds to 785398. (End)

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A014198, A059851, A101455.

Partial sums of A002654.

1/4*Prepend[SquaresR[2,#]&/@Range[58],0]//Accumulate (* Ant King, Mar 15 2013 *)

a(n) = sum(k=1, n, sumdiv(k, d, kronecker(-4, k/d))); \\ Seiichi Manyama, Dec 18 2021

a(n) = A014198(n) / 4.
Limit_{n->infinity} a(n)/n = Pi/4 = A003881.
a(n) = n - floor(n/3) + floor(n/5) - floor(n/7) + floor(n/9) - floor(n/11) + ... - Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 28 2003
G.f.: (1/(1 - x))*Sum_{k>=1} x^k/(1 + x^(2*k)). - Ilya Gutkovskiy, Dec 23 2016

A113652 Expansion of (1 - theta_4(q)^2) / 4 in powers of q.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Nov 03 2005

			G.f. = x - x^2 - x^4 + 2*x^5 - x^8 + x^9 - 2*x^10 + 2*x^13 - x^16 + 2*x^17 + ...

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 114 Entry 8(v).
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 576.
J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987.
P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 28, Article 269.

G. C. Greubel, Table of n, a(n) for n = 1..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A002654, A053692, A008441, A104794, A113407, A122856, A122865, A258277, A258278.

a[ n_] := If[ n < 1, 0, -(-1)^n DivisorSum[ n, KroneckerSymbol[ -4, #] &]]; (* Michael Somos, Jun 06 2015 *)
a[ n_] := SeriesCoefficient[ (1 - EllipticTheta[4, 0, q]^2) / 4, {q, 0, n}]; (* Michael Somos, Jun 06 2015 *)
a[ n_] := With[ {m = InverseEllipticNomeQ @ -q}, SeriesCoefficient[(1 - EllipticK[m] / (Pi/2)) / 4, {q, 0, n}]]; (* Michael Somos, Jun 06 2015 *)

{a(n) = if( n<1, 0, -(-1)^n * sumdiv(n, d, kronecker( -4, d)))};

{a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, -1, p%4==1, e+1, !(e%2))))};

{a(n) = if( n<1, 0, direuler(p=2, n, if( p==2, 1 - X/(1 - X), 1 / ((1 - X) * (1 - kronecker( -4, p)*X))) )[n])};

{a(n) = my(A); if( n<1, 0, A = x * O(x^n); polcoeff( (1 - eta(x + A)^4 / eta(x^2 + A)^2) / 4, n))};

a(n) is multiplicative with a(2^e) = -1 if e>0, a(p^e) = e+1 if p == 1 (mod 4), (1 + (-1)^e)/2 if p == 3 (mod 4).
Expansion of (1 - eta(q)^4 / eta(q^2)^2) / 4 in powers of q.
Moebius transform is period 8 sequence [ 1, -2, -1, 0, 1, 2, -1, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u2 - 2*u3 + u6 - u1^2 + 3*u3^2 + 2*u1*u3 - 4*u2*u6.
G.f.: Sum_{k>0} -(-1)^k * x^((k^2 + k)/2) / (1 + x^k).
G.f.: Sum_{k>0} -(-1)^k * x^k / (1 + x^(2*k)).
G.f.: Sum_{k>0} -(-1)^k * x^(2*k - 1) / (1 + x^(2*k - 1)).
a(n) = -(-1)^n * A002654(n). a(n) = - A104794(n) / 4 unless n = 0.
a(2*n) = - A002654(n). a(3*n + 1) = A258277(n). a(3*n + 2) = - A258278(n). a(4*n + 1) = A008441(n). a(4*n + 3) = 0. a(6*n + 2) = - A122856(n). a(6*n + 4 ) = - A122856(n). - Michael Somos, Jun 06 2015
a(8*n + 1) = A113407(n). a(8*n + 5) = 2 * A053692(n). a(9*n + 3) = a(9*n + 6) = 0. - Michael Somos, Jun 06 2015

A229062 1 if n is representable as sum of two nonnegative squares, otherwise 0.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1

Offset: 0

Ralf Stephan, Sep 17 2013

Characteristic function of A001481.
a(n) = 1 if A000161(n) > 0.
a(A022544(n)) = 0.
Multiplicative because A002654 is. - Andrew Howroyd, Aug 01 2018
For positive n, m = 2*a(n) + 1 is the smallest positive integer such that m * n is not a sum of two squares. - Peter Schorn, Dec 29 2023

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for characteristic functions

Cf. A002654, A004018, A070176. Partial sums are in A102548.

Cf. A000161, A022544.

Join[{1},Table[If[SquaresR[2,n]>1,1,0],{n,120}]] (* Harvey P. Dale, Aug 25 2017 *)

a(n)=my(f=0); my(r=sqrtint(n)); forstep(i=r, 1, -1, if(issquare(n-i*i), f=1; break)); f

a(n)=if(0==n,1,(sumdiv(n, d,(d%4==1) - (d%4==3)) > 0)); \\ Andrew Howroyd, Aug 01 2018, the check for 0-argument added by Antti Karttunen, Apr 22 2022

from sympy import factorint
def A229062(n): return int(all(p & 3 != 3 or e & 1 == 0 for p, e in factorint(n).items())) # Chai Wah Wu, Jun 28 2022

a(n) = min{1, A004018(n)}. - N. J. A. Sloane, Jan 11 2020

Previous Showing 51-60 of 104 results. Next

The list on p. 260 of Cox is missing -12, the list in Theorem 7.30 on p. 149 is correct. - Andrew V. Sutherland, Sep 02 2012

Let b(k) be the number of integer solutions of f(x,y) = k, where f(x,y) is the principal binary quadratic form with discriminant d<0 (i.e., f(x,y) = x^2 - (d/4)*y^2 if 4|d, x^2 + x*y + ((1-d)/4)*y^2 otherwise), then this sequence lists |d| such that {b(k)/b(1): k>=1} is multiplicative. See Crossrefs for the actual sequences. - Jianing Song, Nov 20 2019

cp's OEIS Frontend

A321835 a(n) = Sum_{d|n, n/d==1 mod 4} d^11 - Sum_{d|n, n/d==3 mod 4} d^11.

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A071385 Number of points (i,j) on the circumference of a circle around (0,0) with squared radius A071383(n).

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A133675 Negative discriminants with form class number 1 (negated).

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A145392 Number of inequivalent sublattices of index n in square lattice, where two sublattices are considered equivalent if one can be rotated by a multiple of Pi/2 to give the other.

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A145394 Number of inequivalent sublattices of index n in hexagonal lattice, where two sublattices are considered equivalent if one can be rotated by a multiple of Pi/3 to give the other.

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A322143 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n, d==1 (mod 4)} d^k - Sum_{d|n, d==3 (mod 4)} d^k.

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A193583 Number of fixed points under iteration of sum of squares of digits in base b.

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A014200 Number of solutions to x^2 + y^2 <= n, excluding (0,0), divided by 4.

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