A317314 Multiples of 14 and odd numbers interleaved.

0, 1, 14, 3, 28, 5, 42, 7, 56, 9, 70, 11, 84, 13, 98, 15, 112, 17, 126, 19, 140, 21, 154, 23, 168, 25, 182, 27, 196, 29, 210, 31, 224, 33, 238, 35, 252, 37, 266, 39, 280, 41, 294, 43, 308, 45, 322, 47, 336, 49, 350, 51, 364, 53, 378, 55, 392, 57, 406, 59, 420, 61, 434, 63, 448, 65, 462, 67, 476, 69

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

Cf. A008596 and A005408 interleaved.

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.

Sequence in context: A088576 A204159 A040190 * A163647 A060782 A182431

Adjacent sequences: A317311 A317312 A317313 * A317315 A317316 A317317

Keyword

nonn,easy,mult

Author

Omar E. Pol, Jul 25 2018

Status

approved