OFFSET
0,3
COMMENTS
120*a(n) + 169 is a square. - Bruno Berselli, Jun 08 2018
Partial sums of A317313. - Omar E. Pol, Jul 28 2018
Generalized k-gonal numbers are second k-gonal numbers and positive terms of k-gonal numbers interleaved, k >= 5. They are also the partial sums of the sequence formed by the multiples of (k - 4) and the odd numbers (A005408) interleaved, k >= 5. In this case k = 17. - Omar E. Pol, Apr 25 2021
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).
FORMULA
From Bruno Berselli, Jun 08 2018: (Start)
G.f.: x*(1 + 13*x + x^2)/((1 + x)^2*(1 - x)^3).
a(n) = a(-n-1) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
a(n) = (30*n*(n + 1) + 11*(2*n + 1)*(-1)^n - 11)/16. Therefore:
a(n) = n*(15*n + 26)/8, if n is even, or (n + 1)*(15*n - 11)/8 otherwise.
2*(2*n - 1)*a(n) + 2*(2*n + 1)*a(n-1) - n*(15*n^2 - 13) = 0. (End)
EXAMPLE
From Omar E. Pol, Apr 24 2021: (Start)
Illustration of initial terms as vertices of a rectangular spiral:
43_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _17
| |
| 0 |
| |_ _ _ _ _ _ _ _ _ _ _ _ _|
| 1 14
|
48
More generally, all generalized k-gonal numbers can be represented with this kind of spirals, k >= 5". (End)
MATHEMATICA
With[{pp = 17, nn = 55}, {0}~Join~Riffle[Array[PolygonalNumber[pp, #] &, Ceiling[nn/2]], Array[PolygonalNumber[pp, -#] &, Ceiling[nn/2]]]] (* Michael De Vlieger, Jun 06 2018 *)
Table[(30 n (n + 1) + 11 (2 n + 1) (-1)^n - 11)/16, {n, 0, 60}] (* Bruno Berselli, Jun 08 2018 *)
CoefficientList[ Series[-x (x^2 + 13x + 1)/((x - 1)^3 (x + 1)^2), {x, 0, 50}], x] (* or *)
LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 14, 17, 43}, 51] (* Robert G. Wilson v, Jul 28 2018 *)
PROG
(PARI) concat(0, Vec(x*(1 + 13*x + x^2)/((1 + x)^2*(1 - x)^3) + O(x^40))) \\ Colin Barker, Jun 12 2018
CROSSREFS
Sequences of generalized k-gonal numbers: A001318 (k=5), A000217 (k=6), A085787 (k=7), A001082 (k=8), A118277 (k=9), A074377 (k=10), A195160 (k=11), A195162 (k=12), A195313 (k=13), A195818 (k=14), A277082 (k=15), A274978 (k=16), this sequence (k=17), A274979 (k=18), A303813 (k=19), A218864 (k=20), A303298 (k=21), A303299 (k=22), A303303 (k=23), A303814 (k=24), A303304 (k=25), A316724 (k=26), A316725 (k=27), A303812 (k=28), A303815 (k=29), A316729 (k=30).
KEYWORD
nonn,easy
AUTHOR
Omar E. Pol, Jun 06 2018
STATUS
approved