A317319 Multiples of 19 and odd numbers interleaved.

0, 1, 19, 3, 38, 5, 57, 7, 76, 9, 95, 11, 114, 13, 133, 15, 152, 17, 171, 19, 190, 21, 209, 23, 228, 25, 247, 27, 266, 29, 285, 31, 304, 33, 323, 35, 342, 37, 361, 39, 380, 41, 399, 43, 418, 45, 437, 47, 456, 49, 475, 51, 494, 53, 513, 55, 532, 57, 551, 59, 570, 61, 589, 63, 608, 65, 627, 67, 646, 69

Offset

0,3

Comments

Partial sums give the generalized 23-gonal numbers (A303303).

a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 23-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) = 19*n, a(2n+1) = 2*n + 1.

From Colin Barker, Jul 29 2018: (Start)

G.f.: x*(1 + 19*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) = 19*2^(e-1), and a(p^e) = p^e for an odd prime p. - Amiram Eldar, Oct 14 2023

Dirichlet g.f.: zeta(s-1) * (1 + 17/2^s). - Amiram Eldar, Oct 26 2023

Mathematica

a[n_] := If[OddQ[n], n, 19*n/2]; Array[a, 70, 0] (* Amiram Eldar, Oct 14 2023 *)

Prog

(PARI) concat(0, Vec(x*(1 + 19*x + x^2) / ((1 - x)^2*(1 + x)^2) + O(x^60))) \\ Colin Barker, Jul 29 2018

Crossrefs

Cf. A008601 and A005408 interleaved.

Column 19 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. A303303.

Sequence in context: A349406 A040353 A128160 * A002206 A040349 A274249

Adjacent sequences: A317316 A317317 A317318 * A317320 A317321 A317322

Keyword

nonn,easy,mult

Author

Omar E. Pol, Jul 25 2018

Status

approved