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A001082
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a(n) = n(3n-4)/4 if n even, (n-1)(3n+1)/4 if n odd.
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57
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0, 1, 5, 8, 16, 21, 33, 40, 56, 65, 85, 96, 120, 133, 161, 176, 208, 225, 261, 280, 320, 341, 385, 408, 456, 481, 533, 560, 616, 645, 705, 736, 800, 833, 901, 936, 1008, 1045, 1121, 1160, 1240, 1281, 1365, 1408, 1496, 1541, 1633, 1680, 1776, 1825, 1925, 1976
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| Numbers of the form 3n^2+2n, n an integer. Could also be called generalized octagonal numbers. Cf. A001318, generalized pentagonal numbers.
3*a(n)+1 is a perfect square.
Sequence allows us to find X values of the equation: 3*X^3 + X^2 = Y^2. - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Nov 06 2007
a(n) mod 10 belongs to a periodic sequence: 0, 1, 5, 8, 6, 1, 3, 0, 6, 5, 5, 6, 0, 3, 1, 6, 8, 5, 1, 0. [From Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Sep 04 2009]
Exponents of powers of q in one form of the quintuple product identity. (-x^-2 + 1) * q^0 + (x^-3 - x) * q^1 + (-x^-5 + x^3) * q^5 + (x^-6 - x^4) * q^8 + ... = Sum_n q^(3*n^2 + 2*n) * (x^(3*n) - x^(-3*n - 2)) = Product_{k>0} (1 - x * q^(2*k - 1)) * (1 - x^-1 * q^(2*k - 1)) * (1 - q^(2*k)) * (1 - x^2 * q^(4*k)) * (1 - x^-2 * q^(4*k - 4)). - Michael Somos, Dec 21 2011
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LINKS
| T. D. Noe, Table of n, a(n) for n = 1..1000
R. Stephan, On the solutions to 'px+1 is square'
Eric Weisstein's World of Mathematics, Quintuple Product Identity
Index to sequences with linear recurrences with constant coefficients, signature (1,2,-2,-1,1).
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FORMULA
| a(n) = n^2 - n - floor(n/2)^2.
G.f.: sum_{n=0..inf} (-1)^n*[x^(a(2n+1)) + x^(a(2n+2))] = 1/1 - (x-x^2)/1 - (x^2-x^4)/1 - (x^3-x^6)/1 -...- (x^k - x^(2k))/1 -... (continued fraction where k=1..inf). - Paul D. Hanna (pauldhanna(AT)juno.com), Aug 16 2002
a(n+1) = ceil(n/2)^2+A046092([n/2]).
a(2n)=n(3n-2)=A000567(n), a(2n+1)=n(3n+2)=A045944(n). - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Nov 06 2007
O.g.f.: -x^2*(x^2+4*x+1)/((x-1)^3*(1+x)^2). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 15 2008
a(n) = n^2+n-ceiling(n/2)^2 with offset 0 and a(0)=0 [From Gary Detlefs (gdetlefs(AT)aol.com), Feb 23 2010]
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MAPLE
| seq(n^2+n-ceil(n/2)^2, n=0..51); [From Gary Detlefs (gdetlefs(AT)aol.com), Feb 23 2010]
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MATHEMATICA
| Table[If[EvenQ[n], n*(3*n-4)/4, (n-1) (3*n+1)/4], {n, 100}]
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PROG
| (PARI) {a(n) = if( n%2, (n-1) * (3*n + 1) / 4, n * (3*n - 4) / 4)}
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CROSSREFS
| Partial sums of A022998.
A089801 is the characteristic function. - R. J. Mathar, 07 Oct 2011.
Cf. A005563, A046092.
Sequence in context: A141536 A065905 A126695 * A030006 A088586 A073136
Adjacent sequences: A001079 A001080 A001081 * A001083 A001084 A001085
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KEYWORD
| nonn,easy,changed
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com) and Tom Duff
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EXTENSIONS
| More terms from James A. Sellers (sellersj(AT)math.psu.edu), Sep 19 2000
More terms from Ralf Stephan (ralf(AT)ark.in-berlin.de), Jul 25 2003
Editorial changes by N. J. A. Sloane, Feb 03 2012
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