A195039 23 times triangular numbers.

0, 23, 69, 138, 230, 345, 483, 644, 828, 1035, 1265, 1518, 1794, 2093, 2415, 2760, 3128, 3519, 3933, 4370, 4830, 5313, 5819, 6348, 6900, 7475, 8073, 8694, 9338, 10005, 10695, 11408, 12144, 12903, 13685, 14490, 15318, 16169, 17043, 17940, 18860

Offset: 0

Author

Omar E. Pol, Sep 12 2011

Keywords

• nonn
• easy

Comments

Related to the primitive Pythagorean triple [15, 8, 17].

Sequence found by reading the line from 0, in the direction 0, 23, ..., and the same line from 0, in the direction 0, 69, ..., in the Pythagorean spiral whose edges have length A195035 and whose vertices are the numbers A195036. This is the main diagonal of the square spiral.

Links

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

Crossrefs

Bisection of A195036.

Cf. A000217, A069174, A195035.

Programs

Mathematica

23*Accumulate[Range[0,40]] (* or *) LinearRecurrence[{3,-3,1},{0,23,69},50] (* Harvey P. Dale, Aug 28 2012 *)

PARI

a(n)=23*n*(n+1)/2 \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = (23*n^2 + 23*n)/2 = 23*n*(n+1)/2 = 23*A000217(n).

a(0)=0, a(1)=23, a(2)=69, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Aug 28 2012

From Elmo R. Oliveira, Dec 15 2024: (Start)

G.f.: 23*x/(1-x)^3.

E.g.f.: 23*exp(x)*x*(2 + x)/2.

a(n) = A069174(n+1) - 1. (End)