A001550 a(n) = 1^n + 2^n + 3^n. (Formerly M2580 N1020)

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

Offset

0,1

Comments

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

References

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).

Links

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).

Formula

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)

Maple

A001550:=-(3-12*z+11*z^2)/(z-1)/(3*z-1)/(2*z-1); # Simon Plouffe in his 1992 dissertation.

Mathematica

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 *)

Prog

(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

Crossrefs

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

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

Additional terms from Michael Somos

Attribute "conjectured" removed from Simon Plouffe's g.f. by R. J. Mathar, Mar 11 2009

Status

approved