cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 32 results. Next

A094136 Values x of smallest positive pair (x,y) satisfying x^2 - 2*y^2 = -+d, where d=A058529(n).

Original entry on oeis.org

1, 1, 1, 5, 1, 3, 7, 1, 1, 9, 9, 3, 1, 5, 11, 3, 1, 5, 13, 1, 13, 3, 7, 1, 5, 15, 3, 7, 1, 17, 5, 17, 17, 1, 7, 9, 5, 3, 1, 7, 19, 19, 5, 3, 1, 7, 21, 9, 21, 21, 1, 11, 7, 23, 5, 9, 3, 1, 23, 23, 7, 5, 3, 1, 25, 7, 11, 25, 25, 5, 13, 9, 1, 7, 11, 5, 3, 9, 1, 27, 29, 7, 11, 29, 3, 1, 29, 29, 7, 29, 5
Offset: 1

Views

Author

Lekraj Beedassy, May 04 2004

Keywords

Comments

Here (x,y) is considered smaller than (u,v) iff x+y < u+v or (x+y = u+v and x < u).

Examples

			A058529(6) = 41; (3, 5), (7, 2), (11, 9), (13, 8), ... are pairs satisfying x^2 - 2*y^2 = -+41; (3, 5) is the smallest one, so a(6) = 3.

Crossrefs

Cf. A058529, A094137, A094138.

Programs

  • PARI
    {sp(d)=local(m,b,z,x,y);m=d+2;b=1;z=1;while(b&&z1&&v[k]<7,b=0));if(b>0,print1(sp(n)[1],",")))}

Extensions

Edited, corrected and extended by Klaus Brockhaus May 31 2004

A094137 Values y of smallest positive pair (x,y) satisfying x^2 - 2*y^2 = -+d, where d=A058529(n).

Original entry on oeis.org

1, 2, 3, 1, 4, 5, 1, 5, 6, 2, 1, 7, 7, 8, 2, 8, 8, 9, 3, 9, 1, 10, 11, 10, 11, 1, 11, 12, 11, 4, 12, 3, 2, 12, 13, 14, 13, 13, 13, 14, 2, 1, 14, 14, 14, 15, 4, 16, 2, 1, 15, 17, 16, 5, 16, 17, 16, 16, 2, 1, 17, 17, 17, 17, 4, 18, 19, 3, 2, 18, 20, 19, 18, 19, 20, 19, 19, 20, 19, 1, 7, 20, 21
Offset: 1

Views

Author

Lekraj Beedassy, May 04 2004

Keywords

Comments

Here (x,y) is considered smaller than (u,v) iff x+y < u+v or (x+y = u+v and x < u).

Examples

			A058529(5) = 31; (1, 4), (7, 3), (9, 5), (13, 10), ... are pairs satisfying x^2 - 2*y^2 = -+31; (1, 4) is the smallest one, so a(5) = 4.

Crossrefs

Cf. A058529, A094136, A094138.

Programs

  • PARI
    {for(n=1,765,fac=factor(n);v=vector(matsize(fac)[1],j,fac[j,1])%8;b=1;for(k=1,length(v),if(v[k]>1&&v[k]<7,b=0));if(b>0,print1(sp(n)[2],",")))}. For sp(n) see A094136.

Extensions

Edited, corrected and extended by Klaus Brockhaus May 31 2004

A094138 Values x+y of smallest pair (x,y) satisfying x^2 - 2*y^2=-+d, where d=A058529(n).

Original entry on oeis.org

2, 3, 4, 6, 5, 8, 8, 6, 7, 11, 10, 10, 8, 13, 13, 11, 9, 14, 16, 10, 14, 13, 18, 11, 16, 16, 14, 19, 12, 21, 17, 20, 19, 13, 20, 23, 18, 16, 14, 21, 21, 20, 19, 17, 15, 22, 25, 25, 23, 22, 16, 28, 23, 28, 21, 26, 19, 17, 25, 24, 24, 22, 20, 18, 29, 25, 30, 28, 27, 23, 33, 28, 19
Offset: 1

Views

Author

Lekraj Beedassy, May 04 2004

Keywords

Comments

Here (x,y) is considered smaller than (u,v) iff x+y < u+v or (x+y = u+v and x < u).

Examples

			A058529(14) = 103; (5, 8), (11, 3), (17, 14), (21, 13), ... are pairs satisfying x^2 - 2*y^2 = -+103; (5, 8) is the smallest one, so a(14) = 5+8 = 13.

Crossrefs

Cf. A058529, A094136, A094137.

Programs

  • PARI
    {for(n=1,690,fac=factor(n);v=vector(matsize(fac)[1],j,fac[j,1])%8;b=1;for(k=1,length(v),if(v[k]>1&&v[k]<7,b=0));if(b>0,print1(sp(n)[3],",")))} \\ For sp(n) see A094136.

Extensions

Edited, corrected and extended by Klaus Brockhaus May 31 2004

A094140 Smallest inradius of primitive Pythagorean triangle whose legs differ by A058529(n).

Original entry on oeis.org

1, 2, 3, 5, 4, 14, 7, 5, 6, 18, 9, 21, 7, 33, 22, 11, 8, 45, 39, 9, 13, 30, 60, 10, 30, 15, 33, 84, 11, 68, 60, 51, 34, 12, 91, 95, 65, 39, 13, 57, 38, 19, 70, 42, 14, 105, 84, 144, 42, 21, 15, 138, 112, 115, 80, 92, 48, 16, 46, 23, 119, 85, 51, 17, 100, 126, 209, 75, 50, 25
Offset: 1

Views

Author

Lekraj Beedassy, May 04 2004

Keywords

Comments

This is the product x*y of the smallest positive solution (x,y) to x^2 - 2*y^2 = -+d, where d=A058529(n).

References

  • Mohammad K. Azarian, Circumradius and Inradius, Problem S125, Math Horizons, Vol. 15, Issue 4, April 2008, p. 32. Solution published in Vol. 16, Issue 2, November 2008, p. 32.

Crossrefs

Cf. A058529, A094136, A094137, A094138, A094142, A094143.

Extensions

Corrected and extended by Ray Chandler, Jun 20 2004

A094142 Smallest semiperimeter of primitive Pythagorean triangle whose legs differ by A058529(n).

Original entry on oeis.org

6, 15, 28, 42, 45, 99, 72, 66, 91, 143, 110, 170, 120, 238, 195, 156, 153, 322, 304, 190, 210, 299, 437, 231, 323, 272, 350, 589, 276, 525, 493, 460, 399, 325, 660, 696, 558, 464, 378, 550, 483, 420, 627, 527, 435, 814, 725, 1025, 575, 506, 496, 1015, 897, 924
Offset: 1

Views

Author

Lekraj Beedassy, May 04 2004

Keywords

Comments

This is the product m*(m+y), with m=x+y, of the smallest positive solution pair (x, y) to x^2 - 2*y^2 = -+d, where d=A058529(n).

Crossrefs

Cf. A058529, A094136, A094137, A094138, A094140, A094143.

Extensions

Corrected and extended by Ray Chandler, Jun 20 2004

A094143 Least area of primitive Pythagorean triangle whose legs differ by A058529(n).

Original entry on oeis.org

6, 30, 84, 210, 180, 1386, 504, 330, 546, 2574, 990, 3570, 840, 7854, 4290, 1716, 1224, 14490, 11856, 1710, 2730, 8970, 26220, 2310, 9690, 4080, 11550, 49476, 3036, 35700, 29580, 23460, 13566, 3900, 60060, 66120, 36270, 18096, 4914, 31350, 18354, 7980
Offset: 1

Views

Author

Lekraj Beedassy, May 04 2004

Keywords

Examples

			This is product of A094140(n) and A094142(n).

Crossrefs

Cf. A058529, A094136, A094137, A094138, A094140, A094142.

Extensions

Corrected and extended by Ray Chandler, Jun 20 2004

A144407 A058529(n+1)^2.

Original entry on oeis.org

49, 289, 529, 961, 1681, 2209, 2401, 5041, 5329, 6241, 7921, 9409, 10609, 12769, 14161, 16129, 18769, 22801, 25921, 27889, 36481, 37249, 39601, 47089, 49729, 54289, 57121, 58081, 66049, 69169, 73441, 78961, 82369, 83521, 96721, 97969
Offset: 1

Views

Author

Paul Curtz, Sep 30 2008

Keywords

Comments

The last digit is either 1 or 9 .

Links

Crossrefs

Cf. A058529, A198437.

Programs

  • Mathematica
    Select[Range[500], Union[Abs[Mod[FactorInteger[#][[All, 1]], 8, -1]]] == {1} &]^2 // Rest (* Jean-François Alcover, Sep 21 2015, after T. D. Noe *)

A094079 Least hypotenuse of primitive Pythagorean triangle whose legs differ by A058529(n).

Original entry on oeis.org

5, 13, 25, 37, 41, 85, 65, 61, 85, 125, 101, 149, 113, 205, 173, 145, 145, 277, 265, 181, 197, 269, 377, 221, 293, 257, 317, 505, 265, 457, 433, 409, 365, 313, 569, 601, 493, 425, 365, 493, 445, 401, 557, 485, 421, 709, 641, 881, 533, 485, 481, 877, 785, 809
Offset: 1

Views

Author

Lekraj Beedassy, Apr 30 2004

Keywords

Comments

Complete Pythagorean triple is {(m-+d)/2,a(n)} where m=A094080 and d=A058529(n).

Extensions

More terms from Joshua Zucker, May 11 2006

A094080 a(n)=sqrt(2*h^2 - d^2) where h=A094079 and d=A058529(n).

Original entry on oeis.org

7, 17, 31, 47, 49, 113, 79, 71, 97, 161, 119, 191, 127, 271, 217, 167, 161, 367, 343, 199, 223, 329, 497, 241, 353, 287, 383, 673, 287, 593, 553, 511, 433, 337, 751, 791, 623, 503, 391, 607, 521, 439, 697, 569, 449, 919, 809, 1169, 617, 527, 511, 1153, 1009
Offset: 1

Views

Author

Lekraj Beedassy, Apr 30 2004

Keywords

Extensions

Corrected and extended by R. J. Mathar, Jul 27 2007

A047522 Numbers that are congruent to {1, 7} mod 8.

Original entry on oeis.org

1, 7, 9, 15, 17, 23, 25, 31, 33, 39, 41, 47, 49, 55, 57, 63, 65, 71, 73, 79, 81, 87, 89, 95, 97, 103, 105, 111, 113, 119, 121, 127, 129, 135, 137, 143, 145, 151, 153, 159, 161, 167, 169, 175, 177, 183, 185, 191, 193, 199, 201, 207, 209, 215, 217, 223, 225, 231, 233
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Also n such that Kronecker(2,n) = mu(gcd(2,n)). - Jon Perry and T. D. Noe, Jun 13 2003
Also n such that x^2 == 2 (mod n) has a solution. The primes are given in sequence A001132. - T. D. Noe, Jun 13 2003
As indicated in the formula, a(n) is related to the even triangular numbers. - Frederick Magata (frederick.magata(AT)uni-muenster.de), Jun 17 2004
Cf. property described by Gary Detlefs in A113801: more generally, these a(n) are of the form (2*h*n + (h-4)*(-1)^n-h)/4 (h,n natural numbers). Therefore a(n)^2 - 1 == 0 (mod h); in this case, a(n)^2 - 1 == 0 (mod 8). Also a(n)^2 - 1 == 0 (mod 16). - Bruno Berselli, Nov 17 2010
A089911(3*a(n)) = 2. - Reinhard Zumkeller, Jul 05 2013
S(a(n+1)/2, 0) = (1/2)*(S(a(n+1), sqrt(2)) - S(a(n+1) - 2, sqrt(2))) = T(a(n+1), sqrt(2)/2) = cos(a(n+1)*Pi/4) = sqrt(2)/2 = A010503, identically for n >= 0, where S is the Chebyshev polynomial (A049310) here extended to fractional n, evaluated at x = 0. (For T see A053120.) - Wolfdieter Lang, Jun 04 2023

References

  • L. B. W. Jolley, Summation of Series, Dover Publications, 1961, p. 16.

Links

Crossrefs

Cf. A001132, A014494, A056575, A010709, A074378, A047336, A056020, A005408, A047209, A007310, A090771, A175885, A091998, A175886, A175887, A058529, A047621, A179260, A352125.
Cf. A077221 (partial sums).
Cf. A000217, A089911, A113801.
Cf. A010503, A049310, A053120.

Programs

  • Haskell
    a047522 n = a047522_list !! (n-1)
a047522_list = 1 : 7 : map (+ 8) a047522_list
-- Reinhard Zumkeller, Jan 07 2012
  • Mathematica
    Select[Range[1, 191, 2], JacobiSymbol[2, # ]==1&]
  • PARI
    a(n)=4*n-2+(-1)^n \\ Charles R Greathouse IV, Sep 24 2015

Formula

a(n) = sqrt(8*A014494(n)+1) = sqrt(16*ceiling(n/2)*(2*n+1)+1) = sqrt(8*A056575(n)-8*(2n+1)*(-1)^n+1). - Frederick Magata (frederick.magata(AT)uni-muenster.de), Jun 17 2004
1 - 1/7 + 1/9 - 1/15 + 1/17 - ... = (Pi/8)*(1 + sqrt(2)). [Jolley] - Gary W. Adamson, Dec 16 2006
From R. J. Mathar, Feb 19 2009: (Start)
a(n) = 4n - 2 + (-1)^n = a(n-2) + 8.
G.f.: x(1+6x+x^2)/((1+x)(1-x)^2). (End)
a(n) = 8*n - a(n-1) - 8. - Vincenzo Librandi, Aug 06 2010
From Bruno Berselli, Nov 17 2010: (Start)
a(n) = -a(-n+1) = a(n-1) + a(n-2) - a(n-3).
a(n) = 8*A000217(n-1)+1 - 2*Sum_{i=1..n-1} a(i) for n > 1. (End)
E.g.f.: 1 + (4*x - 1)*cosh(x) + (4*x - 3)*sinh(x). - Stefano Spezia, May 13 2021
E.g.f.: 1 + (4*x - 3)*exp(x) + 2*cosh(x). - David Lovler, Jul 16 2022
From Amiram Eldar, Nov 22 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = sqrt(2+sqrt(2)) (A179260).
Product_{n>=2} (1 + (-1)^n/a(n)) = (Pi/8)*cosec(Pi/8) (A352125). (End)
Showing 1-10 of 32 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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