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%I #45 Apr 21 2023 01:24:27
%S 1,10,19,28,37,46,55,64,73,82,91,100,109,118,127,136,145,154,163,172,
%T 181,190,199,208,217,226,235,244,253,262,271,280,289,298,307,316,325,
%U 334,343,352,361,370,379,388,397,406,415,424,433,442,451,460,469,478
%N a(n) = 9*n + 1.
%C Also all the numbers with digital root 1; A010888(a(n)) = 1. - _Rick L. Shepherd_, Jan 12 2009
%C A116371(a(n)) = A156144(a(n)); positions where records occur in A156144: A156145(n+1) = A156144(a(n)). - _Reinhard Zumkeller_, Feb 05 2009
%C If A=[A147296] 9*n^2+2*n (n>0, 11, 40, 87, ...); Y=[A010701] 3 (3, 3, 3, ...); X=[A017173] 9*n+1 (n>0, 10, 19, 28, ...), we have, for all terms, Pell's equation X^2 - A*Y^2 = 1. Example: 10^2 - 11*3^2 = 1; 19^2 - 40*3^2 = 1; 28^2 - 87*3^2 = 1. - _Vincenzo Librandi_, Aug 01 2010
%H Mohammed Yaseen, <a href="/A017173/b017173.txt">Table of n, a(n) for n = 0..10000</a>
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F G.f.: (1 + 8*x)/(1 - x)^2.
%F a(n) = 2*a(n-1) - a(n-2) with a(0)=1, a(1)=10. - _Vincenzo Librandi_, Aug 01 2010
%F E.g.f.: exp(x)*(1 + 9*x). - _Stefano Spezia_, Apr 20 2023
%t Range[1, 1000, 9] (* _Vladimir Joseph Stephan Orlovsky_, May 28 2011 *)
%t LinearRecurrence[{2,-1},{1,10},60] (* _Harvey P. Dale_, Dec 27 2014 *)
%o (Sage) [i+1 for i in range(480) if gcd(i,9) == 9] # _Zerinvary Lajos_, May 20 2009
%o (PARI) forstep(n=1,500,9,print1(n", ")) \\ _Charles R Greathouse IV_, May 28 2011
%o (Haskell)
%o a017173 = (+ 1) . (* 9)
%o a017173_list = [1, 10 ..] -- _Reinhard Zumkeller_, Feb 04 2014
%Y Cf. A093644 ((9, 1) Pascal, column m=1).
%Y Cf. A010888.
%Y Cf. A116371, A156144, A156145.
%Y Cf. A010701, A147296.
%Y Numbers with digital root m: this sequence (m=1), A017185 (m=2), A017197 (m=3), A017209 (m=4), A017221 (m=5), A017233 (m=6), A017245 (m=7), A017257 (m=8), A008591 (m=9).
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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