A028895 5 times triangular numbers: a(n) = 5*n*(n+1)/2.

0, 5, 15, 30, 50, 75, 105, 140, 180, 225, 275, 330, 390, 455, 525, 600, 680, 765, 855, 950, 1050, 1155, 1265, 1380, 1500, 1625, 1755, 1890, 2030, 2175, 2325, 2480, 2640, 2805, 2975, 3150, 3330, 3515, 3705, 3900, 4100, 4305, 4515, 4730, 4950, 5175, 5405, 5640

Offset: 0

Author

Joe Keane (jgk(AT)jgk.org), Dec 11 1999

Keywords

• nonn
• easy

Comments

Sequence found by reading the line from 0, in the direction 0, 5, ... and the same line from 0, in the direction 0, 15, ..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. Axis perpendicular to A195142 in the same spiral. - Omar E. Pol, Sep 18 2011

Bisection of A195014. Sequence found by reading the line from 0, in the direction 0, 5, ..., and the same line from 0, in the direction 0, 15, ..., in the square spiral whose edges have length A195013 and whose vertices are the numbers A195014. This is the main diagonal of the spiral. - Omar E. Pol, Sep 25 2011

a(n) = the Wiener index of the graph obtained by applying Mycielski's construction to the complete graph K(n) (n>=2). - Emeric Deutsch, Aug 29 2013

Sum of the numbers from 2*n to 3*n for n=0,1,2,... - Wesley Ivan Hurt, Nov 27 2015

Numbers k such that the concatenation k625 is a square, where also 625 is a square. - Bruno Berselli, Nov 07 2018

From Paul Curtz, Nov 29 2019: (Start)

Main column of the pentagonal spiral for n (A001477):

50

49 30 31

48 29 15 16 32

47 28 14 5 6 17 33

46 27 13 4 0 1 7 18 34

45 26 12 3 2 8 19 35

44 25 11 10 9 20 36

43 24 23 22 21 37

42 41 40 39 38

(End)

References

• D. B. West, Introduction to Graph Theory, 2nd ed., Prentice-Hall, NJ, 2001, p. 205.

Links

• Ivan Panchenko, Table of n, a(n) for n = 0..1000
• Rangaswami Balakrishnan and S. Francis Raj, The Wiener number of powers of the Mycielskian, Discussiones Math. Graph Theory, Vol. 30, No. 3 (2010), pp. 489-498 (see Theorem 2.1).
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000217, A001068, A005891, A028896, A046092, A055998, A085787, A130520, A195013, A195014, A195142.

Cf. index to numbers of the form n*(d*n+10-d)/2 in A140090.

Cf. A000566, A005475, A005476, A033583, A085787, A147875, A192136, A326725 (all in the spiral).

Programs

Magma

[5*n*(n+1)/2 : n in [0..50]]; // Wesley Ivan Hurt, Jun 09 2014

Maple

[seq(5*binomial(n,2), n=1..45)]; # Zerinvary Lajos, Nov 24 2006

Mathematica

Table[Sum[i + 2*n - 1, {i, 2, n}], {n, 45}] (* Zerinvary Lajos, Jul 11 2009 *)
Table[5 n (n + 1)/2, {n, 0, 50}] (* Bruno Berselli, Sep 23 2016 *)
5 Accumulate[Range[0,60]] (* Harvey P. Dale, Jun 17 2025 *)

PARI

a(n)=5*n*(n+1)/2 \\ Charles R Greathouse IV, Sep 24 2015

Python

def A028895(n): return 5*n*(n+1)>>1 # Chai Wah Wu, Aug 07 2025

Formula

G.f.: 5*x/(1-x)^3.

a(n) = 5*n*(n+1)/2 = 5*A000217(n).

a(n+1) = 5*n+a(n). - Vincenzo Librandi, Aug 05 2010

a(n) = A005891(n) - 1. - Omar E. Pol, Oct 03 2011

a(n) = A130520(5n+4). - Philippe Deléham, Mar 26 2013

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. - Wesley Ivan Hurt, Nov 27 2015

a(n) = Sum_{i=0..n} A001068(4i). - Wesley Ivan Hurt, May 06 2016

E.g.f.: 5*x*(2 + x)*exp(x)/2. - Ilya Gutkovskiy, May 06 2016

a(n) = A055998(3*n) - A055998(2*n). - Bruno Berselli, Sep 23 2016

From Amiram Eldar, Feb 26 2022: (Start)

Sum_{n>=1} 1/a(n) = 2/5.

Sum_{n>=1} (-1)^(n+1)/a(n) = (2/5)*(2*log(2) - 1). (End)

Product_{n>=1} (1 - 1/a(n)) = -(5/(2*Pi))*cos(sqrt(13/5)*Pi/2). - Amiram Eldar, Feb 21 2023