A004018 -id:A004018 - cp's OEIS frontend

A125853 Squared radii of circles centered at a grid point in a square lattice hitting exactly 4 points. Indices k such that A004018(k)=4.

1, 2, 4, 8, 9, 16, 18, 32, 36, 49, 64, 72, 81, 98, 121, 128, 144, 162, 196, 242, 256, 288, 324, 361, 392, 441, 484, 512, 529, 576, 648, 722, 729, 784, 882, 961, 968, 1024, 1058, 1089, 1152, 1296, 1444, 1458, 1568, 1764, 1849, 1922, 1936, 2048, 2116, 2178, 2209

Offset: 1

Hugo Pfoertner, Jan 07 2007

nonn

From Jean-Christophe Hervé, Nov 17 2013: (Start)
Squares or double of squares that are not sum of two distinct nonzero squares.
Numbers without prime factor of form 4*k+1 and without prime factor of form 4*k+3 to an odd multiplicity.
The sequence is closed under multiplication. Primitive elements are 1, 2 and square of primes of form 4*k+3, that is union of {1, 2} and A087691.
Sequence A001481 (sum of two squares) is the union of {0}, this sequence and A004431 (sum of two distinct nonzero squares). These 4 sequences are all closed under multiplication. (End)

Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 18 2007, Table of n, a(n) for n = 1..501
G. Xiao, Two squares
Index entries for sequences related to sums of squares

Cf. A004018, A001481, A004431.

for(n=1,100000,fctrs=factor(n);c=1;for(i=1,matsize(fctrs)[1],p4=fctrs[i,1]%4;if(p4==1 || (p4==3 && fctrs[i,2]%2==1), c=0)); if(c,print1(n","))) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 17 2007

Numbers of the form 2^e0 * 3^(2*e1) * 7^(2*e2) * 11^(2*e3) * ... * qk^(2*ek) where qk is the k-th prime of the form 4*n+3 (A002145). - Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 17 2007

A205507 a(n) = Fibonacci(n) * A004018(n) for n>=1 with a(0)=1, where A004018(n) is the number of ways of writing n as a sum of 2 squares.

Original entry on oeis.org

1, 4, 4, 0, 12, 40, 0, 0, 84, 136, 440, 0, 0, 1864, 0, 0, 3948, 12776, 10336, 0, 54120, 0, 0, 0, 0, 900300, 971144, 0, 0, 4113832, 0, 0, 8713236, 0, 45623096, 0, 59721408, 193262536, 0, 0, 818673240, 1324641128, 0, 0, 0, 9079225360, 0, 0, 0, 31114968196

Offset: 0

Paul D. Hanna, Jan 28 2012

nonn

Compare to the g.f. of A004018 given by the Lambert series identity:

1 + 4*Sum_{n>=0} (-1)^n*x^(2*n+1)/(1 - x^(2*n+1)) = (1 + 2*Sum_{n>=1} x^(n^2))^2.

			G.f.: A(x) = 1 + 4*x + 4*x^2 + 12*x^4 + 40*x^5 + 84*x^8 + 136*x^9 + 440*x^10 +...
Compare the g.f to the square of the Jacobi theta_3 series:
theta_3(x)^2 = 1 + 4*x + 4*x^2 + 4*x^4 + 8*x^5 + 4*x^8 + 4*x^9 + 8*x^10 +...+ A004018(n)*x^n +...
The g.f. equals the sum:
A(x) = 1 + 4*x/(1-x-x^2) - 4*2*x^3/(1-4*x^3-x^6) + 4*5*x^5/(1-11*x^5-x^10) - 4*13*x^7/(1-29*x^7-x^14) + 4*34*x^9/(1-76*x^9-x^18) - 4*89*x^11/(1-199*x^11-x^22) + 4*233*x^13/(1-521*x^13-x^26) - 4*610*x^15/(1-1364*x^15-x^30) +...
which involves odd-indexed Fibonacci and Lucas numbers.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A205508, A204060, A203847, A004018, A000204 (Lucas).

Join[{1}, Table[Fibonacci[n]*SquaresR[2, n], {n,1,50}]] (* G. C. Greubel, Mar 05 2017 *)

{A004018(n)=polcoeff((1+2*sum(k=1,sqrtint(n+1),x^(k^2),x*O(x^n)))^2,n)}
{a(n)=if(n==0,1,fibonacci(n)*A004018(n))}

{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff((1+4*sum(m=0,n+1,(-1)^m*fibonacci(2*m+1)*x^(2*m+1)/(1-Lucas(2*m+1)*x^(2*m+1)-x^(4*m+2)+x*O(x^n)))),n)}

G.f.: 1 + 4*Sum_{n>=0} (-1)^n*Fibonacci(2*n+1)*x^(2*n+1) / (1 - Lucas(2*n+1)*x^(2*n+1) - x^(4*n+2)), where Lucas(n) = A000204(n).

A330315 a(n) = r(n)*r(n+1), where r(n) = A004018(n) is the number of ways of writing n as a sum of two squares.

Original entry on oeis.org

4, 16, 0, 0, 32, 0, 0, 0, 16, 32, 0, 0, 0, 0, 0, 0, 32, 32, 0, 0, 0, 0, 0, 0, 0, 96, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 32, 0, 0, 0, 64, 0, 0, 0, 0, 0, 0, 0, 0, 48, 0, 0, 64, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 64, 0, 0, 0, 0, 0, 0, 0, 32, 64, 0, 0, 0, 0, 0, 0, 32, 32, 0, 0, 0, 0, 0, 0, 0, 64, 0, 0, 0, 0, 0, 0, 0, 32, 0, 0, 96

Offset: 0

N. J. A. Sloane, Dec 11 2019

nonn

a(n)=0 unless n == 0, 1 or 4 (mod 8).

H. Iwaniec. Spectral methods of automorphic forms, volume 53 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2002.

Robert Israel, Table of n, a(n) for n = 0..10000
Fernando Chamizo, Correlated sums of r(n), J. Math. Soc. Japan, 51(1):237-252, 1999.
Fernando Chamizo, and Roberto J. Miatello, Sums of squares in real quadratic fields and Hilbert modular groups, arXiv preprint arXiv:1812.10725 [math.NT], 2018.

Cf. A004018, A330316, A330317, A330318.

N:= 200: # for a(0)..a(N)
g1:= 1 + 2*add(x^(i^2),i=1..floor(sqrt(N+1))):
g2:= expand(g1^2):
R:= [seq(coeff(g2,x,i),i=0..N+1)]:
seq(R[i]*R[i+1],i=1..N+1); # Robert Israel, Jun 12 2020

a[n_] := SquaresR[2, n] SquaresR[2, n + 1]; a /@ Range[0, 100] (* Giovanni Resta, Jun 12 2020 *)

A333167 a(n) = r_2(n^2 + 1), where r_2(k) is the number of ways of writing k as a sum of 2 squares (A004018).

Original entry on oeis.org

4, 4, 8, 8, 8, 8, 8, 12, 16, 8, 8, 8, 16, 16, 8, 8, 8, 16, 24, 8, 8, 16, 16, 16, 8, 8, 8, 16, 16, 8, 16, 16, 24, 16, 16, 8, 8, 16, 24, 8, 8, 12, 16, 24, 16, 8, 16, 32, 16, 8, 16, 8, 16, 16, 8, 16, 8, 32, 16, 8, 16, 8, 16, 16, 16, 8, 8, 16, 32, 8, 24, 8, 32, 32

Offset: 0

Amiram Eldar, Mar 09 2020

nonn

			a(0) = r_2(0^2 + 1) = r_2(1) = A004018(1) = 4.

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 166.

Amiram Eldar, Table of n, a(n) for n = 0..10000
E. J. Scourfield, The divisors of a quadratic polynomial, Glasgow Mathematical Journal, Vol. 5, No. 1 (1961), pp. 8-20.

Cf. A004018, A002522, A193432, A193433, A333169.

Table[SquaresR[2, k^2 + 1], {k, 0, 100}]

a(n) = A004018(A002522(n)).

A205508 a(n) = Pell(n) * A004018(n) for n>=1 with a(0)=1, where A004018(n) is the number of ways of writing n as a sum of 2 squares.

Original entry on oeis.org

1, 4, 8, 0, 48, 232, 0, 0, 1632, 3940, 19024, 0, 0, 267688, 0, 0, 1883328, 9093512, 10976840, 0, 127955424, 0, 0, 0, 0, 15740857452, 25334527696, 0, 0, 356483857192, 0, 0, 2508054264192, 0, 29236023007504, 0, 85200014758320, 411382062287848, 0, 0, 5788584895037376

Offset: 0

Paul D. Hanna, Jan 28 2012

nonn

Compare to the g.f. of A004018 given by the Lambert series identity:

1 + 4*Sum_{n>=0} (-1)^n*x^(2*n+1)/(1 - x^(2*n+1)) = (1 + 2*Sum_{n>=1} x^(n^2))^2.

			 G.f.: A(x) = 1 + 4*x + 8*x^2 + 48*x^4 + 232*x^5 + 1632*x^8 + 3940*x^9 + 19024*x^10 +...
Compare the g.f to the square of the Jacobi theta_3 series:
theta_3(x)^2 = 1 + 4*x + 4*x^2 + 4*x^4 + 8*x^5 + 4*x^8 + 4*x^9 + 8*x^10 +...+ A004018(n)*x^n +...
The g.f. equals the sum:
A(x) = 1 + 4*x/(1-2*x-x^2) - 4*5*x^3/(1-14*x^3-x^6) + 4*29*x^5/(1-82*x^5-x^10) - 4*169*x^7/(1-478*x^7-x^14) + 4*985*x^9/(1-2786*x^9-x^18) - 4*5741*x^11/(1-16238*x^11-x^22) + 4*33461*x^13/(1-94642*x^13-x^26) - 4*195025*x^15/(1-551614*x^15-x^30) +...
which involves odd-indexed Pell and companion Pell numbers.

Cf. A205507, A204384, A204270, A004018, A000129 (Pell), A002203.

{Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)), n)}
{A002203(n)=Pell(n-1)+Pell(n+1)}
{a(n)=polcoeff((1+4*sum(m=0,n+1,(-1)^m*Pell(2*m+1)*x^(2*m+1)/(1-A002203(2*m+1)*x^(2*m+1)-x^(4*m+2)+x*O(x^n))))^(1/1),n)}

G.f.: 1 + 4*Sum_{n>=0} (-1)^n*Pell(2*n+1)*x^(2*n+1) / (1 - A002203(2*n+1)*x^(2*n+1) - x^(4*n+2)), where A002203 is the companion Pell numbers.

A222882 Decimal expansion of Sierpiński's second constant, K2 = lim_{n->oo} ((1/n) * (Sum_{i=1..n} A004018(i^2)) - 4/Pi * log(n)).

Original entry on oeis.org

2, 2, 5, 4, 9, 2, 2, 4, 6, 2, 8, 8, 8, 2, 6, 4, 7, 6, 6, 2, 6, 8, 1, 8, 4, 7, 5, 9, 5, 2, 8, 7, 2, 3, 5, 5, 7, 8, 7, 1, 6, 6, 1, 5, 9, 8, 6, 0, 5, 3, 5, 1, 8, 8, 9, 1, 3, 8, 3, 1, 1, 6, 1, 8, 8, 5, 9, 1, 7, 2, 9, 2, 8, 9, 5, 9, 7, 1, 3, 9, 3, 4, 1, 0, 5, 8

Offset: 1

Ant King, Mar 11 2013

Sierpiński introduced three constants in his 1908 doctoral thesis. The first, K, is very well known, bears his name and its decimal expansion is given in A062089. However, the second and third of these constants appear to have been largely forgotten. This sequence gives the decimal expansion of the second one, K2, and A222883 gives the decimal expansion of the third , K3. The formula given below show that K2 is related to several other, naturally occurring constants.

			K2 = 2.25492246288826476626818475952872355787166159860535188913831...

Steven R. Finch, Mathematical Constants, Encyclopaedia of Mathematics and its Applications, Cambridge University Press (2003), p.123. Corrigenda in the link below.

G. C. Greubel, Table of n, a(n) for n = 1..10000
Steven R. Finch, Errata and Addenda to Mathematical Constants, (June 2012), pp. 15-16.
A. Schinzel, Wacław Sierpiński’s papers on the theory of numbers, Acta Arithmetica XXI, (1972), pp. 7-13. Corrigenda in "Dzieje Matematyki Polskiej" (Wrocław 2012), p.228 (in Polish).

Cf. A001620, A004018, A014549, A062089, A062539, A074962, A222883.

Take[Flatten[RealDigits[N[4(12 Log[Gamma[3/4]]-9 Log[Pi]+72 Log[Glaisher]-5 Log[2]+3 EulerGamma-3)/(3 Pi),100]]],86]

4/Pi*(log(exp(3*Euler-1)/(2^(2/3)/agm(sqrt(2),1)^2)) - 12/Pi^2*zeta'(2)) \\ Charles R Greathouse IV, Dec 12 2013

K2 = 4 / Pi * (eulergamma + K / Pi - 12 / Pi^2 * zeta'(2) + log(2) / 3 -1), where K  is Sierpiński's first constant (A062089) and eulergamma is the Euler-Mascheroni constant (A001620).
K2 = 4 * (12 * log(Gamma(3/4)) - 9*log(Pi) + 72*log(A) - 5*log(2) + 3 * eulergamma - 3) / (3 * Pi), where A is the Glaisher-Kinkelin constant (A074962).
K2 = 4 * (12 * log(Gamma(3/4)) + log(A^72 * e^(3*eulergamma - 3) / (32 * Pi^9))) / (3 * Pi).
K2 = 4 / Pi * (log(e^(3*eulergamma - 1) / (2^(2/3) * G^2)) - 12 / Pi^2 * zeta'(2)), where G is Gauss’ AGM constant (A014549).
K2 = 4 / Pi * (log(Pi^2 * e^(3*eulergamma - 1) / (2^(2/3) * L^2)) - 12 / Pi^2 * zeta'(2)), where L is Gauss’ lemniscate constant (A062539).

Minor edits by Vaclav Kotesovec, Nov 14 2014

A222883 Decimal expansion of Sierpiński's third constant, K3 = lim_{n->oo} ((1/n) * (Sum_{i=1..n} (A004018(i))^2) - 4* log(n)).

Original entry on oeis.org

8, 0, 6, 6, 4, 8, 6, 1, 8, 2, 9, 3, 3, 6, 3, 2, 4, 6, 1, 0, 5, 1, 1, 8, 7, 4, 3, 8, 8, 6, 0, 4, 6, 1, 7, 0, 5, 8, 0, 0, 7, 3, 6, 7, 1, 0, 0, 9, 4, 5, 8, 9, 9, 2, 2, 4, 4, 3, 6, 7, 7, 1, 3, 3, 7, 9, 1, 2, 5, 7, 3, 6, 6, 4, 6, 4, 7, 3, 1, 1, 4, 9, 0, 2, 1, 6, 5, 4, 0, 5, 5, 9, 3, 2, 2, 4, 7, 2, 1, 6, 7, 8, 1, 5, 1

Offset: 1

Ant King, Mar 11 2013

Sierpiński introduced three constants in his 1908 doctoral thesis. The first, K, is very well known, bears his name and its decimal expansion is given in A062089. However, the second and third of these constants appear to have been largely forgotten. This sequence gives the decimal expansion of the third one, K3, and A222882 gives the decimal expansion of the second one, K2. The formula given below show that K3 is related to several other, naturally occurring constants including K and K2.

			K3 = 8.066486182933632461051187438860461705800736710094589922443677...

Steven R. Finch, Mathematical Constants, Encyclopaedia of Mathematics and its Applications, Cambridge University Press (2003), p.123. Corrigenda in the link below.

G. C. Greubel, Table of n, a(n) for n = 1..10000
Steven R. Finch, Errata and Addenda to Mathematical Constants, (June 2012), pp. 15-16.
A. Schinzel, Wacław Sierpiński’s papers on the theory of numbers, Acta Arithmetica XXI, (1972), pp. 7-13. Corrigenda in "Dzieje Matematyki Polskiej" (Wrocław 2012), p.228 (in Polish).

Cf. A001620, A004018, A014549, A062089, A062539, A074962, A222882.

Take[RealDigits[N[4/3 (24*Log[Gamma[3/4]] - 12*Log[Pi] + 72*Log[Glaisher] - 5*Log[2] + 6*EulerGamma - 3), 100]][[1]], 86]

4*log(exp(5*Euler-1)/(2^(5/3)/agm(sqrt(2),1)^4))-48/Pi^2*zeta'(2) - 4*Euler \\ Charles R Greathouse IV, Dec 12 2013

K3 = 8*K / Pi - 48 / Pi^2 * zeta'(2) + 4 * log(2) / 3 - 4, where K  is Sierpinski's first constant (A062089).
K3 = 4 / 3 * log(A^72 * e^(6 * eulergamma - 3)*( Gamma(3/4))^24 / (32 * pi^12)), where A is the Glaisher-Kinkelin constant (A074962) and eulergamma is the Euler-Mascheroni constant (A001620).
K3 = 4*log(exp(5*eulergamma - 1) / (2^(5 / 3) * G^4)) - 48 / Pi^2 * zeta'(2) - 4* eulergamma, where G is Gauss’ AGM constant (A014549).
K3 = 4*log(Pi^4 * e^(5*eulergamma - 1) / (2^(5 / 3) * L^4)) - 48 / Pi^2 * zeta'(2) - 4* eulergamma, where L is Gauss’ lemniscate constant (A062539).
K3 = 4*K / Pi + Pi * K2 - 4 * eulergamma, where K2  is Sierpiński's second constant (A222882).
1 / 4 * K3 - 1 / 4 * Pi * K2 - log(pi^2 / (2 * L^2)) = eulergamma.
1 / 4 * K3 - 1 / 4 * Pi * K2 + log(2 * G^2) = eulergamma.

More terms from Robert G. Wilson v, Oct 19 2013

A330316 a(n) = r(n)*r(n+1)/4, where r(n) = A004018(n) is the number of ways of writing n as a sum of two squares.

Original entry on oeis.org

1, 4, 0, 0, 8, 0, 0, 0, 4, 8, 0, 0, 0, 0, 0, 0, 8, 8, 0, 0, 0, 0, 0, 0, 0, 24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 0, 8, 16, 0, 0, 0, 0, 0, 0, 8, 8, 0, 0, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 24

Offset: 0

N. J. A. Sloane, Dec 11 2019

nonn

H. Iwaniec. Spectral methods of automorphic forms, volume 53 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2002.

Fernando Chamizo, Correlated sums of r(n), J. Math. Soc. Japan, 51(1):237-252, 1999.
Fernando Chamizo, and Roberto J. Miatello, Sums of squares in real quadratic fields and Hilbert modular groups, arXiv preprint arXiv:1812.10725 [math.NT], 2018.

Cf. A004018, A330315, A330317, A330318.

Times@@#/4&/@Partition[SquaresR[2,Range[0,110]],2,1] (* Harvey P. Dale, May 30 2020 *)

A330317 a(n) = Sum_{i=0..n} r(i)*r(i+1), where r(n) = A004018(n) is the number of ways of writing n as a sum of two squares.

Original entry on oeis.org

4, 20, 20, 20, 52, 52, 52, 52, 68, 100, 100, 100, 100, 100, 100, 100, 132, 164, 164, 164, 164, 164, 164, 164, 164, 260, 260, 260, 260, 260, 260, 260, 260, 260, 260, 260, 292, 292, 292, 292, 356, 356, 356, 356, 356, 356, 356, 356, 356, 404, 404, 404, 468, 468, 468, 468, 468, 468, 468, 468, 468, 468, 468, 468, 532, 532, 532

Offset: 0

N. J. A. Sloane, Dec 11 2019

nonn

H. Iwaniec. Spectral methods of automorphic forms, volume 53 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2002.

Fernando Chamizo, Correlated sums of r(n), J. Math. Soc. Japan, 51(1):237-252, 1999.
Fernando Chamizo, and Roberto J. Miatello, Sums of squares in real quadratic fields and Hilbert modular groups, arXiv preprint arXiv:1812.10725 [math.NT], 2018.

Cf. A004018, A330316, A330318.

Partial sums of A330315.

A330318 a(n) = Sum_{i=0..n} r(i)*r(i+1)/4, where r(n) = A004018(n) is the number of ways of writing n as a sum of two squares.

Original entry on oeis.org

1, 5, 5, 5, 13, 13, 13, 13, 17, 25, 25, 25, 25, 25, 25, 25, 33, 41, 41, 41, 41, 41, 41, 41, 41, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 73, 73, 73, 73, 89, 89, 89, 89, 89, 89, 89, 89, 89, 101, 101, 101, 117, 117, 117, 117, 117, 117, 117, 117, 117, 117

Offset: 0

N. J. A. Sloane, Dec 11 2019

nonn

H. Iwaniec. Spectral methods of automorphic forms, volume 53 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2002.

Fernando Chamizo, Correlated sums of r(n), J. Math. Soc. Japan, 51(1):237-252, 1999.
Fernando Chamizo, and Roberto J. Miatello, Sums of squares in real quadratic fields and Hilbert modular groups, arXiv preprint arXiv:1812.10725 [math.NT], 2018.

Cf. A004018, A330315, A330317.

Partial sums of A330316.

cp's OEIS Frontend

A125853 Squared radii of circles centered at a grid point in a square lattice hitting exactly 4 points. Indices k such that A004018(k)=4.

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A205507 a(n) = Fibonacci(n) * A004018(n) for n>=1 with a(0)=1, where A004018(n) is the number of ways of writing n as a sum of 2 squares.

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A330315 a(n) = r(n)*r(n+1), where r(n) = A004018(n) is the number of ways of writing n as a sum of two squares.

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A333167 a(n) = r_2(n^2 + 1), where r_2(k) is the number of ways of writing k as a sum of 2 squares (A004018).

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A205508 a(n) = Pell(n) * A004018(n) for n>=1 with a(0)=1, where A004018(n) is the number of ways of writing n as a sum of 2 squares.

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A222882 Decimal expansion of Sierpiński's second constant, K2 = lim_{n->oo} ((1/n) * (Sum_{i=1..n} A004018(i^2)) - 4/Pi * log(n)).

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A222883 Decimal expansion of Sierpiński's third constant, K3 = lim_{n->oo} ((1/n) * (Sum_{i=1..n} (A004018(i))^2) - 4* log(n)).

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