A002798 - OEIS

A002798

a(n) = a(n-1) + a(n-2) - a(n-3), starting with 18, 45, and 69.
(Formerly M5055 N2186)

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, 1116, 1140, 1167, 1191, 1218, 1242, 1269, 1293, 1320, 1344, 1371, 1395, 1422, 1446, 1473, 1497

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OFFSET

1,1

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

The squares of the sequence are a(102*k^2+72*k+13) = a(2*k+1)^2 as well as a(102*k^2+132*k+43) = (a(2*k+2)-12)^2 for k >= 0. - Klaus Purath, Apr 29 2026

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

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

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. Systèmes diophantiens linéaires, 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).

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

From Klaus Purath, Apr 29 2026: (Start)

a(n) = 2*a(n-2) - a(n-4).

a(n) + a(n+1) + 6 = a(2*n+1).

a(2*k+1)*a(2*k+2) = a(102*k^2+126*k+32) for k >= 0.

a(2*k+2)*a(2*k+3) = a(102*k^2+228*k+122) for k >= 0. (End)

MAPLE

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