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

0,3

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

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).

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)

Example

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A273269 A273307 A093025 * A223417 A224699 A078575

Adjacent sequences: A104581 A104582 A104583 * A104585 A104586 A104587

Keyword

nonn,easy

Author

Gary W. Adamson, Mar 17 2005

Extensions

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

Status

approved