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 1-3 of 3 results.

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

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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)

A014494 Even triangular numbers.

Original entry on oeis.org

0, 6, 10, 28, 36, 66, 78, 120, 136, 190, 210, 276, 300, 378, 406, 496, 528, 630, 666, 780, 820, 946, 990, 1128, 1176, 1326, 1378, 1540, 1596, 1770, 1830, 2016, 2080, 2278, 2346, 2556, 2628, 2850, 2926, 3160, 3240, 3486, 3570, 3828, 3916, 4186, 4278, 4560
Offset: 0

Views

Author

Mohammad K. Azarian

Keywords

Comments

Even numbers of the form n*(n+1)/2.
Even generalized hexagonal numbers. - Omar E. Pol, Apr 24 2016
The sequence terms occur as the exponents in the expansion of (1 - q^6) * Product_{n >= 1} (1 - q^(16*n-6))*(1 - q^(16*n))*(1 - q^(16*n+6)) = Sum_{n in Z} (-1)^n * q^(2*n*(4*n+1)) = 1 - q^6 - q^10 + q^28 + q^36 - q^66 - q^78 + + - - . - Peter Bala, Dec 23 2024

Links

Crossrefs

Cf. A000217, A000796, A014493, A056575, A074378, A193867.
See also similar sequences listed in A299645.

Programs

  • Magma
    [1/2*(2*n+1)*(2*n+1-(-1)^n): n in [0..50]]; // Vincenzo Librandi, Aug 18 2011
  • Mathematica
    Table[2Ceiling[n/2]*(2n + 1), {n, 0, 47}] (* Robert G. Wilson v, Nov 05 2004 *)
1/2 (2#+1)(2#+1-(-1)^#) &/@Range[0,47] (* Ant King, Nov 18 2010 *)
Select[1/2 #(#+1) &/@Range[0,95],EvenQ] (* Ant King, Nov 18 2010 *)
  • PARI
    a(n)=(2*n+1)*(2*n+1-(-1)^n)/2 \\ Charles R Greathouse IV, Oct 07 2015
  • Python
    def A014494(n): return (2*n+1)*(n+n%2) # Chai Wah Wu, Mar 11 2022

Formula

From Ant King, Nov 18 2010: (Start)
a(n) = (2*n+1)*(2*n+1-(-1)^n)/2.
a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5). (End)
G.f.: -2*x*(3*x^2+2*x+3)/((x+1)^2*(x-1)^3). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 10 2009
a(n) = A000217(A014601(n)). - Reinhard Zumkeller, Oct 04 2004
a(n) = A014493(n+1)-(2n+1)*(-1)^n. - R. J. Mathar, Sep 15 2009
a(n) = A193867(n+1) - 1. - Omar E. Pol, Aug 17 2011
Sum_{n>=1} 1/a(n) = 2 - Pi/2. - Robert Bilinski, Jan 20 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*log(2)-2. - Amiram Eldar, Mar 06 2022
E.g.f.: x*(5 + 2*x)*cosh(x) + (1 + x)*(1 + 2*x)*sinh(x). - Stefano Spezia, Dec 24 2024

Extensions

More terms from Erich Friedman

A056532 Bond percolation series for square lattice near a wall.

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 17, 26, 47, 72, 129, 194, 348, 516, 929, 1351, 2456, 3506, 6471, 8929, 17029, 22579, 44707, 55969, 117836, 137313, 311654, 324989, 833496, 756309, 2242031, 1623709, 6176873, 3240757, 17192674, 4663165, 49481888, 1180046, 144593684, -40561669, 439929287, -230303695, 1351358555, -1116634980, 4353263697
Offset: 0

Views

Author

N. J. A. Sloane, Aug 27 2000

Keywords

Links

Crossrefs

Cf. A056575, A006731.
Showing 1-3 of 3 results.

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

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