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%I M1114 #53 Sep 07 2023 23:34:14
%S 1,2,4,8,16,23,28,29,31,35,43,50,55,56,58,62,70,77,82,83,85,89,97,104,
%T 109,110,112,116,124,131,136,137,139,143,151,158,163,164,166,170,178,
%U 185,190,191,193,197,205,212,217,218,220,224,232,239,244,245,247,251
%N a(n+1) = a(n) + digital root (A010888) of a(n).
%C Take m, a natural number. If m == 1 (mod 6), then for every n a(m)*a(n) is in A007612. - _Ivan N. Ianakiev_, May 08 2013
%D J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 65.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A007612/b007612.txt">Table of n, a(n) for n = 1..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DigitalRoot.html">Digital Root</a>.
%F a(1) = 1, a(n+1) = a(n) + a(n) mod 9. - _Reinhard Zumkeller_, Mar 23 2003
%F First differences are [1,2,4,8,7,5] repeated. - _M. F. Hasler_, Sep 15 2009; corrected by _John Keith_, Aug 17 2022
%F n == 1, 2, 4, 8, 16, or 23 (mod 27). - _Dean Hickerson_, Mar 25 2003
%F Limit_{n->oo} a(n)/n = 9/2; A029898(n) = a(n+1) - a(n) = A010888(a(n)). - _Reinhard Zumkeller_, Feb 27 2006
%F a(6n+1)=27n+1, a(6n+2)=27n+2, a(6n+3)=27n+4, a(6n+4)=27n+8, a(6n+5)=27n+16, a(6n+6)=27n+23. - _Franklin T. Adams-Watters_, Mar 13 2006
%F G.f.: (1+4*x^4+3*x^3+x^2)/((x+1)*(x^2-x+1)*(x-1)^2). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 10 2009
%F a(n+1) = A064806(a(n)). - _Reinhard Zumkeller_, Apr 13 2013
%p A007612 := proc(n) option remember: if(n=1)then return 1: fi: return procname(n-1) + ((procname(n-1)-1) mod 9) + 1: end: seq(A007612(n), n=1..100); # _Nathaniel Johnston_, May 04 2011
%t dr[n_]:=NestWhile[Total[IntegerDigits[#]]&,n,#>9&]; NestList[#+dr[#]&, 1,60] (* _Harvey P. Dale_, Sep 24 2011 *)
%t NestList[#+Mod[#,9]&,1,60] (* _Harvey P. Dale_, Sep 14 2016 *)
%o (Haskell)
%o a007612 n = a007612_list !! (n-1)
%o a007612_list = iterate a064806 1 -- _Reinhard Zumkeller_, Apr 13 2013
%o (PARI) first(n)=my(v=vector(n)); v[1]=1; for(k=2,n, v[k]=v[k-1]+v[k-1]%9); v \\ _Charles R Greathouse IV_, Jun 25 2017
%o (PARI) a(n)=n\6*27 + [-4,1,2,4,8,16][n%6+1] \\ _Charles R Greathouse IV_, Jun 25 2017
%Y Cf. A004207, A010888, A029898, A064806.
%K nonn,base,easy,nice
%O 1,2
%A _N. J. A. Sloane_, _Robert G. Wilson v_, _Mira Bernstein_
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