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 A000340 a(0)=1, a(n) = 3*a(n-1) + n + 1. (Formerly M3882 N1592) 28
 1, 5, 18, 58, 179, 543, 1636, 4916, 14757, 44281, 132854, 398574, 1195735, 3587219, 10761672, 32285032, 96855113, 290565357, 871696090, 2615088290, 7845264891, 23535794695, 70607384108, 211822152348, 635466457069 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS From Johannes W. Meijer, Feb 20 2009: (Start) Second right hand column (n-m=1) of the A156920 triangle. The generating function of this sequence enabled the analysis of the polynomials A156921 and A156925. (End) Partial sums of A003462, and thus the second partial sums of A000244 (3^n). Also column k=2 of A106516. - John Keith, Jan 04 2022 REFERENCES F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 260. 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). LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 389 Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009. Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992 László Tóth, On Schizophrenic Patterns in b-ary Expansions of Some Irrational Numbers, arXiv:2002.06584 [math.NT], 2020. Mentions this sequence. See also Proc. Amer. Math. Soc. 148 (2020), 461-469. Index entries for linear recurrences with constant coefficients, signature (5,-7,3). FORMULA G.f.: 1/((1-3*x)*(1-x)^2). a(n) = (3^(n+2) - 2*n - 5)/4. a(n) = Sum_{k=0..n+1} (n-k+1)*3^k = Sum_{k=0..n+1} k*3^(n-k+1). - Paul Barry, Jul 30 2004 a(n) = Sum_{k=0..n} binomial(n+2, k+2)*2^k. - Paul Barry, Jul 30 2004 a(-1)=0, a(0)=1, a(n) = 4*a(n-1) - 3*a(n-2) + 1. - Miklos Kristof, Mar 09 2005 a(n) = right term of M^(n+1) * [1,0,0], where M = the 3 X 3 matrix [1,0,0; 1,1,0; 1,1,3]. E.g., a(4) = 179 since M^5 = [1, 5, 179]. - Gary W. Adamson, Dec 28 2006 a(n) = 5*a(n-1) - 7*a(n-2) + 3*a(n-3) with a(0)=1, a(1)=5 and a(2)=18. - Johannes W. Meijer, Feb 20 2009 a(-2 - n) = 3^-n * A014915(n). - Michael Somos, May 28 2014 EXAMPLE G.f. = 1 + 5*x + 18*x^2 + 58*x^3 + 179*x^4 + 543*x^5 + 1636*x^6 + ... MAPLE a[ -1]:=0:a[0]:=1:for n from 1 to 50 do a[n]:=4*a[n-1]-3*a[n-2]+1 od: seq(a[n], n=0..50); # Miklos Kristof, Mar 09 2005 A000340:=-1/(3*z-1)/(z-1)**2; # conjectured by Simon Plouffe in his 1992 dissertation MATHEMATICA a[ n_] := MatrixPower[ {{1, 0, 0}, {1, 1, 0}, {1, 1, 3}}, n + 1][[3, 1]]; (* Michael Somos, May 28 2014 *) RecurrenceTable[{a[0]==1, a[n]==3a[n-1]+n+1}, a, {n, 30}] (* or *) LinearRecurrence[{5, -7, 3}, {1, 5, 18}, 30] (* Harvey P. Dale, Jan 31 2017 *) PROG (Magma) [(3^(n+2)-2*n-5)/4: n in [0..30]]; // Vincenzo Librandi, Aug 15 2011 CROSSREFS From Johannes W. Meijer, Feb 20 2009: (Start) Cf. A156921, A156925, A156927, A156933. Other columns A156922, A156923, A156924. Equals A156920 second right hand column. Equals A142963 second right hand column divided by 2^n Equals A156919 second right hand column divided by 2. (End) Cf. A014915. Equals column k=1 of A008971 (shifted). - Jeremy Dover, Jul 11 2021 Cf. A000340, A003462 (first differences), A106516. Sequence in context: A190163 A335551 A235612 * A301880 A034567 A133648 Adjacent sequences: A000337 A000338 A000339 * A000341 A000342 A000343 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified December 4 16:09 EST 2022. Contains 358558 sequences. (Running on oeis4.)