A152773 3 times heptagonal numbers: a(n) = 3*n*(5*n-3)/2.

0, 3, 21, 54, 102, 165, 243, 336, 444, 567, 705, 858, 1026, 1209, 1407, 1620, 1848, 2091, 2349, 2622, 2910, 3213, 3531, 3864, 4212, 4575, 4953, 5346, 5754, 6177, 6615, 7068, 7536, 8019, 8517, 9030, 9558, 10101, 10659, 11232, 11820

Offset

0,2

Comments

Also the number of 6-cycles in the (n+5)-triangular honeycomb acute knight graph. - Eric W. Weisstein, Jun 25 2017

Links

Ivan Panchenko, Table of n, a(n) for n = 0..1000

Eric Weisstein's World of Mathematics, Graph Cycle.

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

Formula

a(n) = (15n^2 - 9n)/2 = A000566(n)*3.

a(n) = a(n-1)+15*n-12 with n>0, a(0)=0. - Vincenzo Librandi, Nov 26 2010

G.f.: 3*x*(1+4*x)/(1-x)^3. - Bruno Berselli, Jan 21 2011

a(0)=0, a(1)=3, a(2)=21, a(n)=3*a(n-1)-3*a(n-2)+a(n-3). - Harvey P. Dale, May 08 2012

a(n) = n + A226489(n). - Bruno Berselli, Jun 11 2013

Sum_{n>=1} 1/a(n) = tan(Pi/10)*Pi/9 - sqrt(5)*log(phi)/9 + 5*log(5)/18, where phi is the golden ratio (A001622). - Amiram Eldar, May 20 2023

Mathematica

Table[3 n (5 n - 3)/2, {n, 0, 50}] (* Harvey P. Dale, May 08 2012 *)
LinearRecurrence[{3, -3, 1}, {0, 3, 21}, 50] (* Harvey P. Dale, May 08 2012 *)
CoefficientList[Series[-((3 x^5 (1 + 4 x))/(-1 + x)^3), {x, 0, 20}], x] (* Eric W. Weisstein, Jun 25 2017 *)

Prog

(PARI) a(n)=3*n*(5*n-3)/2 \\ Charles R Greathouse IV, Sep 24 2015

Crossrefs

Cf. A000566, A001622, A135706.

3 times n-gonal numbers: A045943, A033428, A062741, A094159, A152751, A152759, A152767, A153783, A153448, A153875.

Cf. numbers of the form n*(n*k-k+6)/2, this sequence is the case k=15: see Comments lines of A226492.

Cf. A002378 (3-cycles in triangular honeycomb acute knight graph), A045943 (4-cycles), A028896 (5-cycles).

Sequence in context: A027499 A303834 A340687 * A039595 A033567 A181156

Adjacent sequences: A152770 A152771 A152772 * A152774 A152775 A152776

Keyword

nonn,easy

Author

Omar E. Pol, Dec 13 2008

Status

approved