A104584 a(n) = (1/2) * ( 3*n^2 + n*(-1)^n ).

0, 1, 7, 12, 26, 35, 57, 70, 100, 117, 155, 176, 222, 247, 301, 330, 392, 425, 495, 532, 610, 651, 737, 782, 876, 925, 1027, 1080, 1190, 1247, 1365, 1426, 1552, 1617, 1751, 1820, 1962, 2035, 2185, 2262, 2420

Offset: 0

Author

Gary W. Adamson, Mar 17 2005

Keywords

• nonn
• easy

Comments

Previous name was: Pentagonal wave sequence of the first kind.

Odd-indexed terms = A033570, pentagonal numbers with odd index (1, 12, 35, 70, ...). Even-indexed terms = A049453, 2nd pentagonal numbers with even index (0, 7, 26, 57, 100, ...).

Companion sequence A104585 (Pentagonal wave sequence of the second kind), switches odd with even applications and vice versa. The pentagonal wave sequence triangle A104586 has A104584 in odd columns and A104585 in even columns.

Exponents of q in the identity Sum_{n >= 0} ( q^n*Product_{k = 1..n} (1 - q^(4*k-3)) ) = 1 + q - q^7 - q^12 + q^26 + q^35 - - + + .... Compare with Euler's pentagonal number theorem: Product_{n >= 1} (1 - q^n) = 1 - Sum_{n >= 1} ( q^n*Product_{k = 1..n-1} (1 - q^k) ) = 1 - q - q^2 + q^5 + q^7 - q^12 - q^15 + + - - .... - Peter Bala, Dec 03 2020

Examples

			a(5) = 35 = A000326(5).
a(6) = 57 = A005449(6).
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Wikipedia, Pentagonal number theorem.
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

Cf. A000326, A005449, A049453, A033568, A104585, A104586.

Programs

Magma

I:=[0, 1, 7, 12, 26]; [n le 5 select I[n] else Self(n-1)+2*Self(n-2)-2*Self(n-3)-Self(n-4)+Self(n-5): n in [1..50]]; // Vincenzo Librandi, Apr 04 2013

Mathematica

Table[(1/2) (3 n^2 + n (-1)^n), {n, 0, 100}] (* Vincenzo Librandi, Apr 04 2013 *)
LinearRecurrence[{1,2,-2,-1,1},{0,1,7,12,26},50] (* Harvey P. Dale, Feb 14 2023 *)

Formula

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5). - Vincenzo Librandi, Apr 04 2013

a(n) = (1/2) * (3*n^2 + n*(-1)^n ). - Ralf Stephan, May 20 2007

G.f. -x*(1+6*x+3*x^2+2*x^3) / ( (1+x)^2*(x-1)^3 ). - R. J. Mathar, Jan 10 2011

From Amiram Eldar, Feb 22 2022: (Start)

Sum_{n>=1} 1/a(n) = 6 - Pi/sqrt(3) - 4*log(2).

Sum_{n>=1} (-1)^(n+1)/a(n) = 2*Pi/sqrt(3) + 3*log(3) - 6. (End)

Extensions

Better name, using formula from Ralf Stephan, Joerg Arndt, Sep 17 2013