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

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

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

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).

Crossrefs

Cf. A000051, A000079, A000244, A007689, A034472.

Cf. A001576, A001579, A034513, A074501 - A074580.

Column 3 of array A103438.

Programs

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

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 *)

PARI

a(n)=1+2^n+3^n \\ Charles R Greathouse IV, Jun 10 2011

Python

def A001550(n): return 3**n+(1<Chai Wah Wu, Nov 01 2024

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)

Extensions

Additional terms from Michael Somos

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