A100542 Two-color Rado numbers R(0,n).

5, 1, 9, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225, 1275, 1326, 1378, 1431, 1485, 1540, 1596

Offset

1,1

References

S. Burr, S. Loo and D. Schaal, On Rado numbers, I, preprint.

Links

G. C. Greubel, Table of n, a(n) for n = 1..5000

S. Jones and D. Schaal, Two-color Rado numbers for x + y + z = kz, Discr. Math., 289 (2004), 63-69.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

After the third term these are simply triangular numbers.

From _Colin Barker_, Jul 30 2013: (Start)

a(n) = n*(n+1)/2 for n > 3.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 6.

G.f.: x*(5-14*x+21*x^2-19*x^3+11*x^4-3*x^5)/(1-x)^3. (End)

E.g.f.: (1/2)*x*(8 - 2*x + x^2 + (2+x)*exp(x) ). - _G. C. Greubel_, Mar 27 2023

Mathematica

LinearRecurrence[{3, -3, 1}, {5, 1, 9, 10, 15, 21}, 70] (* _Harvey P. Dale_, Sep 12 2017 *)
Join[{5, 1, 9}, Binomial[Range[5, 70], 2]] (* _G. C. Greubel_, Mar 27 2023 *)

Prog

(Magma) [5, 1, 9] cat [Binomial(n+1, 2): n in [4..70]]; // _G. C. Greubel_, Mar 27 2023
(SageMath) [5, 1, 9]+[binomial(n+1, 2) for n in range(4, 71)] # _G. C. Greubel_, Mar 27 2023

Crossrefs

Cf. A000217.

Sequence in context: A147406 A147354 A134233 * A324378 A147404 A254068

Adjacent sequences: A100539 A100540 A100541 * A100543 A100544 A100545

Keyword

nonn,easy

Author

_N. J. A. Sloane_, Dec 31 2004

Status

approved