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A006002
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n*(n+1)^2/2.
(Formerly M1920)
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23
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0, 2, 9, 24, 50, 90, 147, 224, 324, 450, 605, 792, 1014, 1274, 1575, 1920, 2312, 2754, 3249, 3800, 4410, 5082, 5819, 6624, 7500, 8450, 9477, 10584, 11774, 13050, 14415, 15872, 17424, 19074, 20825, 22680
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Sum of nontriangular numbers between successive triangular numbers. 1, (2), 3, (4, 5), 6, (7, 8, 9), 10, (11, 12, 13, 14), 15, ... Sum of the terms in brackets. Or sum of n consecutive integers beginning with T(n) +1. T(n) = n(n+1)/2. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Aug 27 2005
Row sums of triangle A159797. [From Omar E. Pol (info(AT)polprimos.com), Jul 24 2009]
The same results occur when one plots the points (1,3), (3,6), (6,10), (10,15) and so on for all the triangle numbers and finds the area beneath. Take three consecutive triangle numbers and label them a, b, c; the area created is simply (b-a)*(b+c)/2. Thus for 6,10,15 the area beneath the line defined by the points (6,10) and (10,15) is (10-6)*(10+15)/2=50 [From J. M. Bergot Jun 28 2011]
Let P = ab where a and b are nonequal prime numbers > 1. Let Q=product of all divisors of P^n. Q can be expressed as P^k, where k=n*(n+1)^2/2. This follows from the fact that all divisors are of the form a^i*b^j, i,j=0 to n. Example given below. In the more general case where P = product of m nonequal prime numbers, k= n*(n+1)^m/2. When m=3, the sequence is the same as A092364. [James and Douglas Raymond, Dec 04 2011]
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REFERENCES
| S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..1000
Index entries for two-way infinite sequences
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FORMULA
| a(n) is the largest number which is not the sum of distinct numbers of form kn+1, k >= 0 (David W. Wilson).
G.f.: x(x+2)/(1-x)^4. - Michael Somos, Jan 30 2004
For n>0: C(1+n, 1)*C(1+n, 2). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 10 2006
(Apparently) this is also the splittance (as defined by Hammer & Simeone, 1977) of the Kneser graphs of the form K(n+3,2). [From Felix Goldberg (felixg(AT)tx.technion.ac.il), Jul 13 2009]
a(n) = 2*C(n+1,2) + 3*C(n+1,3) [From Gary Detlefs (gdetlefs(AT)aol.com), Jun 06 2010]
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EXAMPLE
| Let P^n=6^2. product of divisors of 36=10077796 = 6^9. i.e, for n=2, k=9. - James and Douglas Raymond, Dec 04 2011
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MAPLE
| seq(binomial(n, 2)*n, n=1..36); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 25 2007
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MATHEMATICA
| Table[(n^3 -n^2 )/2, {n, 1, 41}] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 21 2007
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PROG
| (PARI) a(n)=n*(n+1)^2/2
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CROSSREFS
| A002411(n)=-a(-1-n).
Cf. A159797, A163274. [From Omar E. Pol (info(AT)polprimos.com), Jul 24 2009]
Equals A035006(n+1) divided by 4. [From Johannes W. Meijer (meijgia(AT)hotmail.com), Feb 04 2010.]
Sequence in context: A101583 A204556 A185669 * A023662 A131357 A079997
Adjacent sequences: A005999 A006000 A006001 * A006003 A006004 A006005
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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