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

%I A001550 M2580 N1020 #94 Jan 05 2025 19:51:32
%S A001550 3,6,14,36,98,276,794,2316,6818,20196,60074,179196,535538,1602516,
%T A001550 4799354,14381676,43112258,129271236,387682634,1162785756,3487832978,
%U A001550 10462450356,31385253914,94151567436,282446313698,847322163876
%N A001550 a(n) = 1^n + 2^n + 3^n.
%C A001550 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 A001550 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 A001550 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001550 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001550 T. D. Noe, <a href="/A001550/b001550.txt">Table of n, a(n) for n = 0..200</a>
%H A001550 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 A001550 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=363">Encyclopedia of Combinatorial Structures 363</a>
%H A001550 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 A001550 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 A001550 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H A001550 Kai Wang, <a href="https://web.archive.org/web/2024*/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 A001550 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-11,6).
%F A001550 From _Michael Somos_: (Start)
%F A001550 G.f.: (3 -12*x +11*x^2)/(1 -6*x +11*x^2 -6*x^3).
%F A001550 a(n) = 5*a(n-1) - 6*a(n-2) + 2. (End)
%F A001550 E.g.f.: exp(x) + exp(2*x) + exp(3*x). - _Mohammad K. Azarian_, Dec 26 2008
%F A001550 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 A001550 a(n) = A007689(n) + 1. - _Reinhard Zumkeller_, Mar 01 2012
%F A001550 From _Kai Wang_, May 18 2020: (Start)
%F A001550 a(n) = 3*A000392(n+3) - 12*A000392(n+2) + 11*A000392(n+1).
%F A001550 A000392(n) = (3*a(n+1) - 12*a(n) + 10*a(n-1))/2. (End)
%p A001550 A001550:=-(3-12*z+11*z^2)/(z-1)/(3*z-1)/(2*z-1); # _Simon Plouffe_ in his 1992 dissertation.
%t A001550 Table[1^n + 2^n + 3^n, {n, 0, 30}]
%t A001550 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 A001550 Total[Range[3]^#]&/@Range[0,30] (* _Harvey P. Dale_, Sep 23 2019 *)
%o A001550 (PARI) a(n)=1+2^n+3^n \\ _Charles R Greathouse IV_, Jun 10 2011
%o A001550 (Haskell) a001550 n = sum $ map (^ n) [1..3]  -- _Reinhard Zumkeller_, Mar 01 2012
%o A001550 (Magma) [1^n + 2^n + 3^n : n in [0..30]]; // _Wesley Ivan Hurt_, Jun 25 2020
%o A001550 (Python)
%o A001550 def A001550(n): return 3**n+(1<<n)+1 # _Chai Wah Wu_, Nov 01 2024
%Y A001550 Cf. A000051, A000079, A000244, A007689, A034472.
%Y A001550 Cf. A001576, A001579, A034513, A074501 - A074580.
%Y A001550 Column 3 of array A103438.
%K A001550 nonn,easy,nice
%O A001550 0,1
%A A001550 _N. J. A. Sloane_
%E A001550 Additional terms from _Michael Somos_
%E A001550 Attribute "conjectured" removed from _Simon Plouffe_'s g.f. by _R. J. Mathar_, Mar 11 2009