A317318 Multiples of 18 and odd numbers interleaved.

0, 1, 18, 3, 36, 5, 54, 7, 72, 9, 90, 11, 108, 13, 126, 15, 144, 17, 162, 19, 180, 21, 198, 23, 216, 25, 234, 27, 252, 29, 270, 31, 288, 33, 306, 35, 324, 37, 342, 39, 360, 41, 378, 43, 396, 45, 414, 47, 432, 49, 450, 51, 468, 53, 486, 55, 504, 57, 522, 59, 540, 61, 558, 63, 576, 65, 594, 67, 612, 69

Offset

0,3

Comments

Partial sums give the generalized 22-gonal numbers (A303299).

a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 22-gonal numbers.

Continued fraction expansion of 3*A298241. - Nicolas Bělohoubek, Apr 11 2026

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) = 18*n, a(2n+1) = 2*n + 1.

From Colin Barker, Jul 29 2018: (Start)

G.f.: x*(1 + 18*x + x^2) / ((1 - x)^2*(1 + x)^2).

a(n) = 2*a(n-2) - a(n-4) for n>3. (End)

Multiplicative with a(2^e) = 9*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 + 2^(4-s)). - Amiram Eldar, Oct 25 2023

E.g.f.: x*(cosh(x) + 9*sinh(x)). - Stefano Spezia, Apr 11 2026

Mathematica

a[n_] := If[OddQ[n], n, 9*n]; Array[a, 70, 0] (* Amiram Eldar, Oct 14 2023 *)

Prog

(PARI) concat(0, Vec(x*(1 + 18*x + x^2) / ((1 - x)^2*(1 + x)^2) + O(x^60))) \\ Colin Barker, Jul 29 2018

Crossrefs

Cf. A008600 and A005408 interleaved.

Column 18 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).

Cf. A303299.

Cf. A298241.

Sequence in context: A135216 A135252 A040317 * A321263 A040313 A051522

Adjacent sequences: A317315 A317316 A317317 * A317319 A317320 A317321

Keyword

nonn,easy,mult

Author

Omar E. Pol, Jul 25 2018

Status

approved