A062725 Write 0, 1, 2, 3, 4, ... in a triangular spiral, then a(n) is the sequence found by reading the terms along the line from 0 in the direction 0, 7, ...
0, 7, 23, 48, 82, 125, 177, 238, 308, 387, 475, 572, 678, 793, 917, 1050, 1192, 1343, 1503, 1672, 1850, 2037, 2233, 2438, 2652, 2875, 3107, 3348, 3598, 3857, 4125, 4402, 4688, 4983, 5287, 5600, 5922, 6253, 6593, 6942, 7300, 7667, 8043, 8428, 8822, 9225, 9637, 10058
Offset: 0
Examples
The spiral begins:
.
15
/ \
16 14
/ \
17 3 13
/ / \ \
18 4 2 12
/ / \ \
19 5 0---1 11
/ / \
20 6---7---8---9--10
.
Links
- Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
- Amelia Carolina Sparavigna, The groupoid of the Triangular Numbers and the generation of related integer sequences, Politecnico di Torino, Italy (2019).
- Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
Programs
-
Haskell
a062725 n = n * (9 * n + 5) `div` 2 -- Reinhard Zumkeller, Jul 17 2014
-
Mathematica
s=0;lst={s};Do[s+=n++ +7;AppendTo[lst, s], {n, 0, 7!, 9}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 16 2008 *) CoefficientList[Series[x (7 + 2 x)/(1 - x)^3, {x, 0, 45}], x] (* Michael De Vlieger, Jan 11 2020 *) -
PARI
a(n) = n*(9*n+5)/2 \\ Charles R Greathouse IV, Apr 30 2015
Formula
a(n) = n*(9*n+5)/2.
a(n) = 9*n + a(n-1) - 2 with a(0)=0. - Vincenzo Librandi, Aug 07 2010
From Colin Barker, Jul 07 2012: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: x*(7+2*x)/(1-x)^3. (End)
a(n) = A218470(9*n+6). - Philippe Deléham, Mar 27 2013
a(n) = a(n-1) + A017245(n-1), a(0)=0. - Gionata Neri, Apr 30 2015
E.g.f.: exp(x)*x*(14 + 9*x)/2. - Elmo R. Oliveira, Dec 12 2024
Extensions
Formula that confused indices corrected by R. J. Mathar, Jun 04 2010
Comments