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A036499 Numbers of the form n*(n+1)/6 for n = 2 or 3 modulo 6. 6
1, 2, 12, 15, 35, 40, 70, 77, 117, 126, 176, 187, 247, 260, 330, 345, 425, 442, 532, 551, 651, 672, 782, 805, 925, 950, 1080, 1107, 1247, 1276, 1426, 1457, 1617, 1650, 1820, 1855, 2035, 2072, 2262, 2301, 2501, 2542, 2752, 2795, 3015, 3060, 3290, 3337, 3577, 3626 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Numbers with an odd number of partitions with an extra odd partition; coefficient of z^p in Product_{n >= 1}(1-z^n) has coefficient (-1).

n such that the number of partitions of n into distinct parts with an odd number of parts exceed by 1 the number of partitions of n into distinct parts with an even number of parts. [Euler's 1754/55 pentagonal number Theorem, see, e.g. the Freitag-Busam reference (in German). This reference from - Wolfdieter Lang, Jan 18 2016]

In formal power series, A010815=(product(1-x^k),k>0), ranks of coefficients -1. (A001318=ranks of nonzero (1 or -1) in A010815=ranks of odds terms in A000009).

Quasipolynomial of order 2. [Charles R Greathouse IV, Dec 08 2011]

Union of A033568 and A033570. - Ray Chandler, Dec 09 2011

REFERENCES

E. Freitag and R. Busam, Funktionentheorie 1, Springer, Vierte Auflage, 2006, p. 410.

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

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

FORMULA

a(n) = (3*n*n-5*n+2)/2 + (2*n-1)*(n mod 2). - Frank Ellermann, Mar 16 2002

G.f.: (1+x+8*x^2+x^3+x^4)/((1-x)^3*(1+x)^2). - Ray Chandler, Dec 09 2011

Bisection: a(2*k+1) = A001318(1+4*k) = (2*k+1)*(3*k+1) = A033570(k), a(2*(k+1)) = A001318(2+4*k) = (2*k+1)*(3*k+2) = A033568(k+1), k >= 0. - Wolfdieter Lang, Jan 18 2016

a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5) for n>5. - Wesley Ivan Hurt, Jan 18 2016

MAPLE

seq(seq((6*k+i)*(6*k+i+1)/6, i=2..3), k=0..50); # Robert Israel, Jan 18 2016

MATHEMATICA

Table[ 1/8*(3 + (-1)^k - 6*k)*(1 + (-1)^k - 2*k), {k, 64} ]

LinearRecurrence[{1, 2, -2, -1, 1}, {1, 2, 12, 15, 35}, 50] (* or *)

CoefficientList[Series[(1+x+8x^2+x^3+x^4)/((1-x)^3(1+x)^2), {x, 0, 100}], x] (* or *)

Table[(2n+1)(3n+{1, 2}), {n, 0, 24}]//Flatten (* Ray Chandler, Dec 09 2011 *)

PROG

(PARI) a(n)=n*(3*n-5)/2+1+n%2*(2*n-1) \\ Charles R Greathouse IV, Dec 08 2011

(MAGMA) [(3*n*n-5*n+2)/2+(2*n-1)*(n mod 2): n in [1..50]]; // Vincenzo Librandi, Jan 19 2016

CROSSREFS

Cf. A000009, A001318, A010815, A033568, A033570, A036498.

Sequence in context: A200183 A118516 A077103 * A107607 A102975 A120300

Adjacent sequences:  A036496 A036497 A036498 * A036500 A036501 A036502

KEYWORD

nonn,easy

AUTHOR

Wouter Meeussen

EXTENSIONS

Better description from Claude Lenormand (claude.lenormand(AT)free.fr), Feb 12 2001

Edited by Ray Chandler, Dec 09 2011

STATUS

approved

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Last modified August 12 21:10 EDT 2020. Contains 336440 sequences. (Running on oeis4.)