A317313 Multiples of 13 and odd numbers interleaved.

0, 1, 13, 3, 26, 5, 39, 7, 52, 9, 65, 11, 78, 13, 91, 15, 104, 17, 117, 19, 130, 21, 143, 23, 156, 25, 169, 27, 182, 29, 195, 31, 208, 33, 221, 35, 234, 37, 247, 39, 260, 41, 273, 43, 286, 45, 299, 47, 312, 49, 325, 51, 338, 53, 351, 55, 364, 57, 377, 59, 390, 61, 403, 63, 416, 65, 429, 67, 442, 69

Offset

0,3

Comments

Partial sums give the generalized 17-gonal numbers (A303305).

More generally, the partial sums of the sequence formed by the multiples of m and the odd numbers interleaved, give the generalized k-gonal numbers, with m >= 1 and k = m + 4.

a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 17-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) = 13*n, a(2n+1) = 2*n + 1.

From Colin Barker, Jul 29 2018: (Start)

G.f.: x*(1 + 13*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) = 13*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 + 11/2^s). - Amiram Eldar, Oct 25 2023

Mathematica

Table[{13n, 2n + 1}, {n, 0, 35}] // Flatten (* or *)
CoefficientList[Series[(x^3 + 13 x^2 + x)/(x^2 - 1)^2, {x, 0, 69}], x] (* or *)
LinearRecurrence[{0, 2, 0, -1}, {0, 1, 13, 3}, 70] (* Robert G. Wilson v, Jul 26 2018 *)

Prog

(PARI) a(n) = if(n%2==0, return((n/2)*13), return(n)) \\ Felix Fröhlich, Jul 26 2018
(PARI) concat(0, Vec(x*(1 + 13*x + x^2) / ((1 - x)^2*(1 + x)^2) + O(x^60))) \\ Colin Barker, Jul 29 2018

Crossrefs

Cf. A008595 and A005408 interleaved.

Column 13 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).

Cf. A303305.

Sequence in context: A264971 A107806 A138965 * A170922 A005602 A272175

Adjacent sequences: A317310 A317311 A317312 * A317314 A317315 A317316

Keyword

nonn,easy,mult

Author

Omar E. Pol, Jul 25 2018

Status

approved