OFFSET
0,2
COMMENTS
a(n-1), n >= 1, gives the first column of the triangle A238475 related to the Collatz problem. - Wolfdieter Lang, Mar 12 2014
First differences of A054552. - Wesley Ivan Hurt, Jul 08 2014
An odd number is congruent to a perfect square modulo every power of 2 iff it is in this sequence. Sketch of proof: Suppose the modulus is 2^k with k at least three and note that the only odd quadratic residue (mod 8) is 1. By application of difference of squares and the fact that gcd(x-y,x+y)=2 we can show that for odd x,y, we have x^2 and y^2 congruent mod 2^k iff x is congruent to one of y, 2^(k-1)-y, 2^(k-1)+y, 2^k-y. Now when we "lift" to (mod 2^(k+1)) we see that the degeneracy between a^2 and (2^(k-1)-a)^2 "breaks" to give a^2 and a^2-2^ka+2^(2k-2). Since a is odd, the latter is congruent to a^2+2^k (mod 2^(k+1)). Hence we can form every binary number that ends with '001' by starting modulo 8 and "lifting" while adding digits as necessary. But this sequence is exactly the set of binary numbers ending in '001', so our claim is proved. - Rafay A. Ashary, Oct 23 2016
For n > 3, also the number of (not necessarily maximal) cliques in the n-antiprism graph. - Eric W. Weisstein, Nov 29 2017
Bisection of A016813. - L. Edson Jeffery, Apr 26 2022
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
Eric Weisstein's World of Mathematics, Antiprism Graph
Eric Weisstein's World of Mathematics, Clique
Index entries for linear recurrences with constant coefficients, signature (2,-1).
FORMULA
G.f.: (1+7*x)/(1-x)^2.
a(n+1) = A004768(n). - R. J. Mathar, May 28 2008
a(n) = 2*a(n-1)-a(n-2). - Vincenzo Librandi, Mar 14 2014
E.g.f.: exp(x)*(1 + 8*x). - Stefano Spezia, May 13 2021
EXAMPLE
Illustration of initial terms:
. o o o
. o o o o o o
. o o o o o o o o o
. o o o o o o o o o o o o
. o o o o o o o o o o o o o o o o o o o o o o o o o
. o o o o o o o o o o o o
. o o o o o o o o o
. o o o o o o
. o o o
--------------------------------------------------------------
. 1 9 17 25 33
- Bruno Berselli, Feb 28 2014
MAPLE
MATHEMATICA
Table[8 n + 1, {n, 0, 6!}] (* Vladimir Joseph Stephan Orlovsky, Mar 10 2010 *)
CoefficientList[Series[(1 + 7 x)/(1 - x)^2, {x, 0, 60}], x] (* Vincenzo Librandi, Mar 14 2014 *)
8 Range[0, 50] + 1 (* Wesley Ivan Hurt, Jul 08 2014 *)
LinearRecurrence[{2, -1}, {9, 17}, {0, 20}] (* Eric W. Weisstein, Nov 29 2017 *)
PROG
(Haskell)
a017077 = (+ 1) . (* 8)
a017077_list = [1, 9 ..] -- Reinhard Zumkeller, Dec 28 2012
(Magma) I:=[1, 9]; [n le 2 select I[n] else 2*Self(n-1)-Self(n-2): n in [1..60]]; // Vincenzo Librandi, Mar 14 2014
(Magma) [8*n+1 : n in [0..50]]; // Wesley Ivan Hurt, Jul 08 2014
(PARI) a(n)=8*n+1 \\ Charles R Greathouse IV, Jul 10 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved