The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A002452 a(n) = (9^n - 1)/8. (Formerly M4733 N2025) 80
 0, 1, 10, 91, 820, 7381, 66430, 597871, 5380840, 48427561, 435848050, 3922632451, 35303692060, 317733228541, 2859599056870, 25736391511831, 231627523606480, 2084647712458321, 18761829412124890, 168856464709124011, 1519708182382116100, 13677373641439044901, 123096362772951404110 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS From David W. Wilson: Numbers triangular, differences square. To be precise, the differences are the squares of the powers of three with positive indices. Hence a(n+1) - a(n) = (A000244(n+1))^2 = A001019(n+1). [Added by Ant King, Jan 05 2011] Partial sums of A001019. This is m-th triangular number, where m is partial sums of A000244. a(n) = A000217(A003462(n)). - Lekraj Beedassy, May 25 2004 With offset 0, binomial transform of A003951. - Philippe Deléham, Jul 22 2005 Numbers in base 9: 1, 11, 111, 1111, 11111, 111111, 1111111, etc. - Zerinvary Lajos, Apr 26 2009 Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=9, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det(A). - Milan Janjic, Feb 21 2010 Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=10, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n >= 2, a(n-1) = (-1)^n*charpoly(A,1). - Milan Janjic, Feb 21 2010 From Hieronymus Fischer, Jan 30 2013: (Start) Least index k such that A052382(k) >= 10^(n-1), for n > 0. Also index k such that A052382(k) = (10^n-1)/9, n > 0. A052382(a(n)) is the least zerofree number with n digits, for n > 0. For n > 1: A052382(a(n)-1) is the greatest zerofree number with n-1 digits. (End) For n > 0, 4*a(n) is the total number of holes in a certain triangle fractal (start with 9 triangles, 4 holes) after n iterations. See illustration in links. - Kival Ngaokrajang, Feb 21 2015 For n > 0, a(n) is the sum of the numerators and denominators of the reduced fractions 0 < (b/3^(n-1)) < 1 plus 1. Example for n=3 gives fractions 1/9, 2/9, 1/3, 4/9, 5/9, 2/3, 7/9, and 8/9 plus 1 has sum of numerators and denominators +1 = a(3) = 91. - J. M. Bergot, Jul 11 2015 Except for 0 and 1, all terms are Brazilian repunits numbers in base 9, so belong to A125134. All these terms are composite because a(n) is the ((3^n - 1)/2)-th triangular number. - Bernard Schott, Apr 23 2017 These are also the second steps after the junctions of the Collatz trajectories of 2^(2k-1)-1 and 2^2k-1. - David Rabahy, Nov 01 2017 REFERENCES A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 112. J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217. N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). T. N. Thiele, Interpolationsrechnung. Teubner, Leipzig, 1909, p. 36. LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..300 Kival Ngaokrajang, Illustration of initial terms Vladimir Pletser, Congruence conditions on the number of terms in sums of consecutive squared integers equal to squared integers, arXiv:1409.7969 [math.NT], 2014. Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992. Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992 A. G. Shannon, Letter to N. J. A. Sloane, Dec 06 1974 M. Ward, Note on divisibility sequences, Bull. Amer. Math. Soc., 42 (1936), 843-845. Eric Weisstein's World of Mathematics, Repunit Index entries for linear recurrences with constant coefficients, signature (10,-9). FORMULA From Philippe Deléham, Mar 13 2004: (Start) a(n) = 9*a(n-1) + 1; a(1) = 1. G.f.: x / ((1-x)*(1-9*x)). (End) a(n) = 10*a(n-1) - 9*a(n-2). - Ant King, Jan 05 2011 a(n) = floor(A000217(3^n)/4) - A033113(n-1). - Arkadiusz Wesolowski, Feb 14 2012 E.g.f.: 1/6*sin(x)^3 = Sum_{n>0} a(n)*(-1)^(n+1)*x^(2*n+1)/(2*n+1)!. - Vladimir Kruchinin, Sep 30 2012 a(n) = A011540(A217094(n-1)) - A217094(n-1) + 2, n > 0. - Hieronymus Fischer, Jan 30 2013 a(n) = 10^(n-1) + 2 - A217094(n-1). - Hieronymus Fischer, Jan 30 2013 a(n) = det(|v(i+2,j+1)|, 1 <= i,j <= n-1), where v(n,k) are central factorial numbers of the first kind with odd indices (A008956) and n > 0. - Mircea Merca, Apr 06 2013 a(n) = Sum_{k=0..n-1} 9^k. - Doug Bell, May 26 2017 EXAMPLE a(4) = (9^4 - 1)/8 = 820 = 1111_9 = (1/2) * 40 * 41 is the ((3^4 - 1)/2)-th = 40th triangular number. - Bernard Schott, Apr 23 2017 MAPLE A002452 := 1/(9*z-1)/(z-1); # Simon Plouffe in his 1992 dissertation MATHEMATICA (9^# & /@ Range[0, 18] // Accumulate) (* Ant King, Jan 06 2011 *) LinearRecurrence[{10, -9}, {0, 1}, 30] (* Harvey P. Dale, Sep 23 2018 *) PROG (MAGMA) [(9^n - 1)/8 : n in [0..25]]; // Vincenzo Librandi, Jun 01 2011 (PARI) a(n)=9^n>>3 \\ Charles R Greathouse IV, Jul 25 2011 (Maxima) A002452(n):=floor((9^n-1)/8)\$ makelist(A002452(n), n, 0, 30); /* Martin Ettl, Nov 05 2012 */ CROSSREFS Right-hand column 1 in triangle A008958. Cf. A217094, A011540, A052382, A125857. Cf. A125134, A000217. Sequence in context: A002739 A079928 A231412 * A096261 A015455 A110410 Adjacent sequences:  A002449 A002450 A002451 * A002453 A002454 A002455 KEYWORD nonn,easy AUTHOR EXTENSIONS More terms from Pab Ter (pabrlos(AT)yahoo.com), May 08 2004 Offset changed from 1 to 0 and added 0 by Vincenzo Librandi, Jun 01 2011 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified April 20 17:56 EDT 2021. Contains 343135 sequences. (Running on oeis4.)