A317323 Multiples of 23 and odd numbers interleaved.

0, 1, 23, 3, 46, 5, 69, 7, 92, 9, 115, 11, 138, 13, 161, 15, 184, 17, 207, 19, 230, 21, 253, 23, 276, 25, 299, 27, 322, 29, 345, 31, 368, 33, 391, 35, 414, 37, 437, 39, 460, 41, 483, 43, 506, 45, 529, 47, 552, 49, 575, 51, 598, 53, 621, 55, 644, 57, 667, 59, 690, 61, 713, 63, 736, 65, 759, 67, 782, 69

Offset: 0

Author

Omar E. Pol, Jul 25 2018

Keywords

• nonn
• easy
• mult

Comments

Partial sums give the generalized 27-gonal numbers (A316725).

a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 27-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).

Crossrefs

Cf. A008605 and A005408 interleaved.

Column 23 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. A316725.

Programs

Mathematica

With[{nn=40},Riffle[23*Range[0,nn],Range[1,2*nn,2]]] (* or *) LinearRecurrence[{0,2,0,-1},{0,1,23,3},80] (* Harvey P. Dale, May 05 2019 *)

PARI

concat(0, Vec(x*(1 + 23*x + x^2) / ((1 - x)^2*(1 + x)^2) + O(x^60))) \\ Colin Barker, Jul 29 2018

Formula

a(2n) = 23*n, a(2n+1) = 2*n + 1.

From Colin Barker, Jul 29 2018: (Start)

G.f.: x*(1 + 23*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) = 23*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 + 21/2^s). - Amiram Eldar, Oct 26 2023