A001550 - OEIS

%I M2580 N1020 #93 Nov 01 2024 23:36:59

%S 3,6,14,36,98,276,794,2316,6818,20196,60074,179196,535538,1602516,

%T 4799354,14381676,43112258,129271236,387682634,1162785756,3487832978,

%U 10462450356,31385253914,94151567436,282446313698,847322163876

%N a(n) = 1^n + 2^n + 3^n.

%C 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

%D 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.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A001550/b001550.txt">Table of n, a(n) for n = 0..200</a>

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=363">Encyclopedia of Combinatorial Structures 363</a>

%H C. J. Pita Ruiz V., <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Pita/pita19.html">Some Number Arrays Related to Pascal and Lucas Triangles</a>, J. Int. Seq. 16 (2013) #13.5.7.

%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.

%H Kai Wang, <a href="https://www.fq.math.ca/Papers1/58-5/wang.pdf">Girard-Waring Type Formula For A Generalized Fibonacci Sequence</a>, Fibonacci Quarterly (2020) Vol. 58, No. 5, 229-235.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-11,6).

%F From _Michael Somos_: (Start)

%F G.f.: (3 -12*x +11*x^2)/(1 -6*x +11*x^2 -6*x^3).

%F a(n) = 5*a(n-1) - 6*a(n-2) + 2. (End)

%F E.g.f.: exp(x) + exp(2*x) + exp(3*x). - _Mohammad K. Azarian_, Dec 26 2008

%F 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

%F a(n) = A007689(n) + 1. - _Reinhard Zumkeller_, Mar 01 2012

%F From _Kai Wang_, May 18 2020: (Start)

%F a(n) = 3*A000392(n+3) - 12*A000392(n+2) + 11*A000392(n+1).

%F A000392(n) = (3*a(n+1) - 12*a(n) + 10*a(n-1))/2. (End)

%p A001550:=-(3-12*z+11*z^2)/(z-1)/(3*z-1)/(2*z-1); # _Simon Plouffe_ in his 1992 dissertation.

%t Table[1^n + 2^n + 3^n, {n, 0, 30}]

%t 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 *)

%t Total[Range[3]^#]&/@Range[0,30] (* _Harvey P. Dale_, Sep 23 2019 *)

%o (PARI) a(n)=1+2^n+3^n \\ _Charles R Greathouse IV_, Jun 10 2011

%o (Haskell) a001550 n = sum $ map (^ n) [1..3]  -- _Reinhard Zumkeller_, Mar 01 2012

%o (Magma) [1^n + 2^n + 3^n : n in [0..30]]; // _Wesley Ivan Hurt_, Jun 25 2020

%o (Python)

%o def A001550(n): return 3**n+(1<<n)+1 # _Chai Wah Wu_, Nov 01 2024

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

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

%Y Column 3 of array A103438.

%K nonn,easy,nice

%O 0,1

%A _N. J. A. Sloane_

%E Additional terms from _Michael Somos_

%E Attribute "conjectured" removed from _Simon Plouffe_'s g.f. by _R. J. Mathar_, Mar 11 2009