OFFSET
0,3
COMMENTS
Partial sums give the generalized 18-gonal numbers (A274979).
a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 18-gonal numbers.
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).
FORMULA
a(2n) = 14*n, a(2n+1) = 2*n + 1.
From Colin Barker, Jul 29 2018: (Start)
G.f.: x*(1 + 14*x + x^2) / ((1 - x)^2*(1 + x)^2).
a(n) = 2*a(n-2) - a(n-4) for n>3. (End)
a(n) = 4*n + 3*n*(-1)^n. - Wesley Ivan Hurt, Nov 25 2021
Multiplicative with a(2^e) = 7*2^e, and a(p^e) = p^e for an odd prime p. - Amiram Eldar, Oct 14 2023
Dirichlet g.f.: zeta(s-1) * (1 + 3*2^(2-s)). - Amiram Eldar, Oct 25 2023
MATHEMATICA
Table[4 n + 3 n (-1)^n, {n, 0, 80}] (* Wesley Ivan Hurt, Nov 25 2021 *)
PROG
(PARI) a(n) = if(n%2==0, return(14*n/2), return(n)) \\ Felix Fröhlich, Jul 26 2018
(PARI) concat(0, Vec(x*(1 + 14*x + x^2) / ((1 - x)^2*(1 + x)^2) + O(x^60))) \\ Colin Barker, Jul 29 2018
CROSSREFS
Column 14 of A195151.
Sequences whose partial sums give the generalized k-gonal numbers: A026741 (k=5), A001477 (k=6), zero together with A080512 (k=7), A022998 (k=8), A195140 (k=9), zero together with A165998 (k=10), A195159 (k=11), A195161 (k=12), A195312 k=13), A195817 (k=14), A317311 (k=15), A317312 (k=16), A317313 (k=17).
Cf. A274979.
KEYWORD
nonn,easy,mult
AUTHOR
Omar E. Pol, Jul 25 2018
STATUS
approved