A002798 a(n) = a(n-1) + a(n-2) - a(n-3).

18, 45, 69, 96, 120, 147, 171, 198, 222, 249, 273, 300, 324, 351, 375, 402, 426, 453, 477, 504, 528, 555, 579, 606, 630, 657, 681, 708, 732, 759, 783, 810, 834, 861, 885, 912, 936, 963, 987, 1014, 1038, 1065, 1089

Offset: 1

Author

N. J. A. Sloane and Simon Plouffe

Keywords

• nonn

Comments

The old definition was a(n) = a(n-2)+a(n-3)-a(n-5).

The following applies to this sequence and also to all sequences of the form a(n) = a(n-1) + a(n-2) - a(n-3), regardless of initial values: (a(n+3i) + a(n))/(a(n+2i) + a(n+i)) = 1, as long as a(n+2i) + a(n+i) != 0. - Klaus Purath, Jun 05 2024

References

• 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

• E. Ehrhart, Sur un problème de géométrie diophantienne linéaire I, (Polyèdres et réseaux), J. Reine Angew. Math. 226 1967 1-29. MR0213320 (35 #4184). [Annotated scanned copy of pages 16 and 22 only]
• E. Ehrhart, Sur un problème de géométrie diophantienne linéaire II. Systemes diophantiens lineaires, J. Reine Angew. Math. 227 1967 25-49. [Annotated scanned copy of pages 47-49 only]
• E. Ehrhart, Sur un problème de géométrie diophantienne linéaire II, (Systèmes diophantiens linéaires), J. Reine Angew. Math. 227 1967 25-49.
• 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.
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Crossrefs

Cf. A007310, A047208.

Programs

Maple

A002798:=3*(6+9*z+2*z**2)/(z+1)/(z-1)**2; # Simon Plouffe in his 1992 dissertation

Mathematica

LinearRecurrence[{1,1,-1},{18,45,69},50] (* Harvey P. Dale, Sep 17 2023 *)

Formula

a(n) = 6*A007310(n) + 3*A047208(n).

a(n) = (51*n - 12)/2 - 3*(1 - (-1)^n)/4 = 2*a(n-1) - a(n-2) + 3(-1)^n. - Klaus Purath, Jun 05 2024

Extensions

Definition simplified by Ray Chandler. - N. J. A. Sloane, Mar 07 2024