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A001550
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a(n) = 1^n + 2^n + 3^n.
(Formerly M2580 N1020)
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97
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3, 6, 14, 36, 98, 276, 794, 2316, 6818, 20196, 60074, 179196, 535538, 1602516, 4799354, 14381676, 43112258, 129271236, 387682634, 1162785756, 3487832978, 10462450356, 31385253914, 94151567436, 282446313698, 847322163876
(list;
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refs;
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history;
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OFFSET
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0,1
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COMMENTS
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a(n)*(-1)^n, n>=0, gives the z-sequence for the Sheffer triangle A049458 ((signed) 3-restricted Stirling1 numbers), which is the inverse triangle of A143495 with offset [0,0] (3-restricted Stirling2 numbers). See the W. Lang link under A006232 for a- and z-sequences for Sheffer matrices. The a-sequence for each (signed) r-restricted Stirling1 Sheffer triangle is A027641/A027642 (Bernoulli numbers). - Wolfdieter Lang, Oct 10 2011
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REFERENCES
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 813.
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).
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LINKS
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T. D. Noe, Table of n, a(n) for n = 0..200
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 363
C. J. Pita Ruiz V., Some Number Arrays Related to Pascal and Lucas Triangles, J. Int. Seq. 16 (2013) #13.5.7.
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.
Kai Wang, Girard-Waring Type Formula For A Generalized Fibonacci Sequence, Fibonacci Quarterly (2020) Vol. 58, No. 5, 229-235.
Index entries for linear recurrences with constant coefficients, signature (6,-11,6).
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FORMULA
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From Michael Somos: (Start)
G.f.: (3 -12*x +11*x^2)/(1 -6*x +11*x^2 -6*x^3).
a(n) = 5*a(n-1) - 6*a(n-2) + 2. (End)
E.g.f.: exp(x) + exp(2*x) + exp(3*x). - Mohammad K. Azarian, Dec 26 2008
a(0)=3, a(1)=6, a(2)=14, a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3). - Harvey P. Dale, Apr 30 2011
a(n) = A007689(n) + 1. - Reinhard Zumkeller, Mar 01 2012
From Kai Wang, May 18 2020: (Start)
a(n) = 3*A000392(n+3) - 12*A000392(n+2) + 11*A000392(n+1).
A000392(n) = (3*a(n+1) - 12*a(n) + 10*a(n-1))/2. (End)
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MAPLE
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A001550:=-(3-12*z+11*z^2)/(z-1)/(3*z-1)/(2*z-1); # Simon Plouffe in his 1992 dissertation.
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MATHEMATICA
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Table[1^n + 2^n + 3^n, {n, 0, 30}]
CoefficientList[Series[(3-12x+11x^2)/(1-6x+11x^2-6x^3), {x, 0, 30}], x] (* or *) LinearRecurrence[{6, -11, 6}, {3, 6, 14}, 31] (* Harvey P. Dale, Apr 30 2011 *)
Total[Range[3]^#]&/@Range[0, 30] (* Harvey P. Dale, Sep 23 2019 *)
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PROG
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(PARI) a(n)=1+2^n+3^n \\ Charles R Greathouse IV, Jun 10 2011
(Haskell) a001550 n = sum $ map (^ n) [1..3] -- Reinhard Zumkeller, Mar 01 2012
(Magma) [1^n + 2^n + 3^n : n in [0..30]]; // Wesley Ivan Hurt, Jun 25 2020
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CROSSREFS
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Cf. A000051, A000079, A000244, A007689, A034472.
Cf. A001576, A001579, A034513, A074501 - A074580.
Column 3 of array A103438.
Sequence in context: A196479 A147772 A129703 * A197461 A100446 A106395
Adjacent sequences: A001547 A001548 A001549 * A001551 A001552 A001553
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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Additional terms from Michael Somos
Attribute "conjectured" removed from Simon Plouffe's g.f. by R. J. Mathar, Mar 11 2009
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STATUS
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approved
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