The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A047522 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A047522

Numbers that are congruent to {1, 7} mod 8.
(history; published version)

#62 by N. J. A. Sloane at Sat Jun 24 13:12:44 EDT 2023

STATUS

proposed

approved

#61 by Jon E. Schoenfield at Wed Jun 07 22:39:21 EDT 2023

STATUS

editing

proposed

#60 by Jon E. Schoenfield at Wed Jun 07 22:39:19 EDT 2023

COMMENTS

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

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 For T see A053120.) - Wolfdieter Lang, Jun 04 2023

STATUS

proposed

editing

#59 by Wolfdieter Lang at Sun Jun 04 05:36:43 EDT 2023

STATUS

editing

proposed

#58 by Wolfdieter Lang at Sun Jun 04 05:36:29 EDT 2023

COMMENTS

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(n, x) is the Chebyshev polynomial (A049310) here extended to fractional n, evaluated at x = 0. (for T see A053120) - Wolfdieter Lang, Jun 03 04 2023

CROSSREFS

Cf. A010503, A049310, A053120.

#57 by Wolfdieter Lang at Sat Jun 03 17:32:35 EDT 2023

COMMENTS

S(a(n+1)/2, 0) = cos(a(n+1)*Pi/4) = sqrt(2)/2 = A010503, identically for n >= 0, where S (n, x) is the Chebyshev polynomial (A049310) extended to fractional n, evaluated at x = 0. - Wolfdieter Lang, Jun 03 2023

STATUS

proposed

editing

#56 by Wolfdieter Lang at Sat Jun 03 16:48:32 EDT 2023

STATUS

editing

proposed

#55 by Wolfdieter Lang at Sat Jun 03 16:48:20 EDT 2023

CROSSREFS

Cf. A010503, A049310.

#54 by Wolfdieter Lang at Sat Jun 03 05:40:08 EDT 2023

COMMENTS

S(a(n+1)/2, 0) = sqrt(2)/2 = A010503, identically for n >= 0, where S is the Chebyshev polynomial (A049310) extended to fractional n, evaluated at x = 0. - Wolfdieter Lang, Jun 03 2023

STATUS

approved

editing

#53 by Michael De Vlieger at Sun Jul 17 10:51:41 EDT 2022

STATUS

reviewed

approved