A317321 Multiples of 21 and odd numbers interleaved.

0, 1, 21, 3, 42, 5, 63, 7, 84, 9, 105, 11, 126, 13, 147, 15, 168, 17, 189, 19, 210, 21, 231, 23, 252, 25, 273, 27, 294, 29, 315, 31, 336, 33, 357, 35, 378, 37, 399, 39, 420, 41, 441, 43, 462, 45, 483, 47, 504, 49, 525, 51, 546, 53, 567, 55, 588, 57, 609, 59, 630, 61, 651, 63, 672, 65, 693, 67, 714, 69

Offset: 0

Author

Omar E. Pol, Jul 25 2018

Keywords

• nonn
• easy
• mult

Comments

Partial sums give the generalized 25-gonal numbers (A303304).

a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 25-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. A008603 and A005408 interleaved.

Column 21 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. A303304.

Programs

Mathematica

a[n_] := If[OddQ[n], n, 21*n/2]; Array[a, 70, 0] (* Amiram Eldar, Oct 14 2023 *)

PARI

concat(0, Vec(x*(1 + 21*x + x^2) / ((1 - x)^2*(1 + x)^2) + O(x^60))) \\ Colin Barker, Jul 29 2018

Formula

a(2n) = 21*n, a(2n+1) = 2*n + 1.

Multiplicative with a(2^e) = 21*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 + 19/2^s). - Amiram Eldar, Oct 26 2023