A195312 Multiples of 9 and odd numbers interleaved.

0, 1, 9, 3, 18, 5, 27, 7, 36, 9, 45, 11, 54, 13, 63, 15, 72, 17, 81, 19, 90, 21, 99, 23, 108, 25, 117, 27, 126, 29, 135, 31, 144, 33, 153, 35, 162, 37, 171, 39, 180, 41, 189, 43, 198, 45, 207, 47, 216, 49, 225, 51, 234, 53, 243, 55, 252, 57, 261, 59, 270, 61

Offset

0,3

Comments

Partial sums give the generalized 13-gonal (or tridecagonal) numbers A195313.

a(n) is also the length of the n-th line segment of a rectangular spiral on the infinite square grid. The vertices of the spiral are the generalized 13-gonal numbers. - Omar E. Pol, Jul 27 2018

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Formula

From Bruno Berselli, Sep 15 2011: (Start)

G.f.: x*(1+9*x+x^2)/((1-x)^2*(1+x)^2).

a(n) = (7*(-1)^n+11)*n/4.

a(n) + a(n-1) = A175885(n).

Sum_{i=0..n} a(i) = A195313(n). (End)

Multiplicative with a(2^e) = 9*2^(e-1), a(p^e) = p^e for odd prime p. - Andrew Howroyd, Jul 23 2018

Dirichlet g.f.: zeta(s-1) * (1 + 7/2^s). - Amiram Eldar, Oct 25 2023

Mathematica

With[{nn=30}, Riffle[9Range[0, nn], Range[1, 2nn+1, 2]]] (* Harvey P. Dale, Sep 24 2011 *)

Prog

(Magma) /* By definition */ &cat[[9*n, 2*n+1]: n in [0..33]]; // Bruno Berselli, Sep 16 2011
(PARI) a(n)=(7*(-1)^n+11)*n/4 \\ Charles R Greathouse IV, Oct 07 2015

Crossrefs

Column 9 of A195151.

Sequences whose partial sums give the generalized n-gonal numbers, if n>=5: A026741, A001477, zero together with A080512, A022998, A195140, zero together with A165998, A195159, A195161, this sequence.

Cf. A195313, A175885.

Sequence in context: A309791 A103935 A040077 * A248311 A367826 A302536

Adjacent sequences: A195309 A195310 A195311 * A195313 A195314 A195315

Keyword

nonn,easy,mult

Author

Omar E. Pol, Sep 14 2011

Status

approved