A055264 Possible values of A055263; numbers equal to 0, 1, 3 or 6 modulo 9.

0, 1, 3, 6, 9, 10, 12, 15, 18, 19, 21, 24, 27, 28, 30, 33, 36, 37, 39, 42, 45, 46, 48, 51, 54, 55, 57, 60, 63, 64, 66, 69, 72, 73, 75, 78, 81, 82, 84, 87, 90, 91, 93, 96, 99, 100, 102, 105, 108, 109, 111, 114, 117, 118, 120, 123, 126, 127, 129, 132, 135, 136, 138, 141

Offset: 0

Author

Henry Bottomley, May 08 2000

Keywords

• easy
• nonn
• base

Comments

The terms are the possible digit sums of a triangular number. - Amarnath Murthy, Jan 09 2002

References

• Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See p. 190.

Links

• Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Crossrefs

Cf. A055263.

Programs

Mathematica

Select[Range[0,200],MemberQ[{0,1,3,6},Mod[#,9]]&] (* Harvey P. Dale, Apr 10 2014 *)
#+{0,1,3,6}&/@(9*Range[0,20])//Flatten (* Harvey P. Dale, Jun 03 2019 *)

Python

def A055264(n): return (0,1,3,6)[n&3]+9*(n>>2) # Chai Wah Wu, Jan 30 2023

Formula

a(n) = a(n-4) + 9 = 9*floor(n/4) + (n mod 4)*(1 + (n mod 4))/2.

G.f.: x*(1+2*x+3*x^2+3*x^3)/((1-x)^2*(1+x)*(1+x^2)). - R. J. Mathar, Sep 30 2008

E.g.f.: (3*cos(x) + (9*x - 3)*cosh(x) - sin(x) + (9*x - 4)*sinh(x))/4. - Stefano Spezia, Aug 07 2024