A047521 Numbers that are congruent to {0, 7} mod 8.

0, 7, 8, 15, 16, 23, 24, 31, 32, 39, 40, 47, 48, 55, 56, 63, 64, 71, 72, 79, 80, 87, 88, 95, 96, 103, 104, 111, 112, 119, 120, 127, 128, 135, 136, 143, 144, 151, 152, 159, 160, 167, 168, 175, 176, 183, 184, 191, 192, 199, 200, 207, 208, 215, 216, 223, 224, 231, 232

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Numbers such that the n-th triangular number is divisible by 4. - Charles R Greathouse IV, Apr 07 2011

Except for 0, numbers whose binary reflected Gray code (A014550) ends with 00. - Amiram Eldar, May 17 2021

Links

• David Lovler, Table of n, a(n) for n = 1..10000
• Lars Pos, Met kleine stapjes grote sprongen make, Pythagoras 61-4. Solutions of returning to the origin after steps of increasing width 1,2,3,.. in the 4 directions on a square grid (in Dutch).
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Crossrefs

Union of A008590 and A004771.

Cf. A014550, A030308, A274406.

Programs

Mathematica

{#,#+7}&/@(8*Range[0,30])//Flatten (* or *) LinearRecurrence[{1,1,-1},{0,7,8},60] (* Harvey P. Dale, Oct 30 2016 *)

PARI

a(n) = 4*n - 5/2 + 3*(-1)^n/2; \\ David Lovler, Jul 25 2022

R

kmax <- 10 # by choice
a <- c(0,7)
for(k in 3:kmax) a <- c(a, a + 2^k)
a
# Yosu Yurramendi, Jan 18 2022

Formula

a(n) = 8*n - a(n-1) - 9 (with a(1)=0). - Vincenzo Librandi, Aug 06 2010

From R. J. Mathar, Oct 08 2011: (Start)

a(n) = 3*(-1)^n/2 - 5/2 + 4*n.

G.f.: x^2*(7+x) / ( (1+x)*(x-1)^2 ). (End)

a(n+1) = Sum_{k>=0} A030308(n,k)*b(k) with b(0)=7 and b(k)=2^(k+2) for k > 0. - Philippe Deléham, Oct 17 2011

Sum_{n>=2} (-1)^n/a(n) = log(2)/2 + sqrt(2)*log(sqrt(2)+1)/8 - (sqrt(2)+1)*Pi/16. - Amiram Eldar, Dec 18 2021

E.g.f.: 1 + ((8*x -5)*exp(x) + 3*exp(-x))/2. David Lovler, Aug 22 2022

Extensions

More terms from Vincenzo Librandi, Aug 06 2010