A317326 Multiples of 26 and odd numbers interleaved.

0, 1, 26, 3, 52, 5, 78, 7, 104, 9, 130, 11, 156, 13, 182, 15, 208, 17, 234, 19, 260, 21, 286, 23, 312, 25, 338, 27, 364, 29, 390, 31, 416, 33, 442, 35, 468, 37, 494, 39, 520, 41, 546, 43, 572, 45, 598, 47, 624, 49, 650, 51, 676, 53, 702, 55, 728, 57, 754, 59, 780, 61, 806, 63, 832, 65, 858, 67, 884, 69

Offset

0,3

Comments

a(n) is the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 30-gonal numbers (A316729).

Partial sums give the generalized 30-gonal numbers.

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.

From Bruno Berselli, Jul 27 2018: (Start)

Also, this type of sequence is characterized by:

O.g.f.: x*(1 + m*x + x^2)/(1 - x^2)^2;

E.g.f.: x*(2 - m + (2 + m)*exp(2*x))*exp(-x)/4;

a(n) = -a(-n) = (2 + m - (2 - m)*(-1)^n)*n/4;

a(n) = (m/2)^((1 + (-1)^n)/2)*n;

a(n) = 2*a(n-2) - a(n-4), with signature (0,2,0,-1). (End)

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(2*n) = 26*n, a(2*n+1) = 2*n + 1.

From Bruno Berselli, Jul 27 2018: (Start)

O.g.f.: x*(1 + 26*x + x^2)/(1 - x^2)^2.

E.g.f.: x*(-6 + 7*exp(2*x))*exp(-x).

a(n) = -a(-n) = (7 + 6*(-1)^n)*n.

a(n) = 13^((1 + (-1)^n)/2)*n.

a(n) = 2*a(n-2) - a(n-4). (End)

Multiplicative with a(2^e) = 13*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^(3-s)). - Amiram Eldar, Oct 26 2023

Mathematica

Table[(7 + 6 (-1)^n) n, {n, 0, 70}] (* Bruno Berselli, Jul 27 2018 *)

Prog

(Julia) [13^div(1+(-1)^n, 2)*n for n in 0:70] |> println # Bruno Berselli, Jul 28 2018

Crossrefs

Cf. A252994 and A005408 interleaved.

Column 26 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), A317314 (k=18), A317315 (k=19), A317316 (k=20), A317317 (k=21), A317318 (k=22), A317319 (k=23), A317320 (k=24), A317321 (k=25), A317322 (k=26), A317323 (k=27), A317324 (k=28), A317325 (k=29), this sequence (k=30).

Cf. A316729.

Sequence in context: A040666 A317471 A291151 * A040660 A040661 A140871

Adjacent sequences: A317323 A317324 A317325 * A317327 A317328 A317329

Keyword

nonn,mult,easy,changed

Author

Omar E. Pol, Jul 25 2018

Status

approved