%I A001318 M1336 N0511
%S A001318 0,1,2,5,7,12,15,22,26,35,40,51,57,70,77,92,100,117,126,145,155,176,187,
               210,
%T A001318 222,247,260,287,301,330,345,376,392,425,442,477,495,532,551,590,610,651,
               672,
%U A001318 715,737,782,805,852,876,925,950,1001,1027,1080,1107,1162,1190,1247,1276,
               1335
%N A001318 Generalized pentagonal numbers: n(3n-1)/2, n=0, +- 1, +- 2,....
%C A001318 Comment from R. K. Guy, Dec 28 2005:
%C A001318 "Conway's relation twixt the triangular and pentagonal numbers: Divide 
               the triangular numbers by 3 (when you can exactly):
%C A001318 0 1 3 6 10 15 21 28 36 45 55 66 78 91 105 120 136 153 ...
%C A001318 0 - 1 2 .- .5 .7 .- 12 15 .- 22 26 .- .35 .40 .- ..51 ...
%C A001318 .....-.-.....+..+.....-..-.....+..+......-...-.......+....
%C A001318 "and you get the pentagonal numbers in pairs, one of positive rank and 
               the other negative.
%C A001318 "Append signs according as the pair have the same (+) or opposite (-) 
               parity.
%C A001318 "Then Euler's pentagonal number theorem is easy to remember:
%C A001318 "p(n-0)-p(n-1)-p(n-2)+p(n-5)+p(n-7)-p(n-12)-p(n-15)++-- =0^n
%C A001318 where p(n) is the partition function, the left side terminates before 
               the argument becomes negative and 0^n = 1 if n = 0 and = 0 if n > 
               0.
%C A001318 "E.g. p(0)=1, p(7)=p(7-1)+p(7-2)-p(7-5)-p(7-7)+0^7=11+7-2-1+0=15."
%C A001318 Sequence that may be used in order to compute sigma(n), as described 
               in Euler's article. - Thomas Baruchel (baruchel(AT)users.sourceforge.net), 
               Nov 19 2003
%C A001318 Number of levels in the partitions of n+1 with parts in {1,2}.
%C A001318 A080995(a(n))=1: complement of A118300; A000009(a(n))=A051044(n). - Reinhard 
               Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 22 2006
%C A001318 a(n) is the number of 3 X 3 matrix(symmetrical about each diagonal)M=[a,
               b,c;b,d,b;c,b,a] such that a+b+c=b+d+b=n+2, a,b,c,d natural numbers; 
               example : a(3)=5 because (a,b,c,d)=(2,2,1,1), (1,2,2,1), (1,1,3,3), 
               (3,1,1,3), (2,1,2,3). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Apr 11 2007
%C A001318 Also numbers a(n) such that 24a(n)+1=(6n-1)^2 are odd squares: 1, 25, 
               49, 121, 169, 289, 361,..., n=0, +-1, +-2,.... - Zak Seidov (zakseidov(AT)yahoo.com), 
               Mar 08 2008
%C A001318 Contribution from Matthew Vandermast (ghodges14(AT)comcast.net), Oct 
               28 2008: (Start)
%C A001318 Numbers n for which A000326(n) is a member of A000332. Cf. A145920.
%C A001318 This sequence contains all members of A000332 and all nonnegative members 
               of A145919. For values of n such that n(3n-1)/2 belongs to A000332, 
               see A145919. (End)
%C A001318 Also numbers a(n) such that a(n)=(n^2+n)/6, with n>1 and n=/=1 mod (3) 
               [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Dec 31 2008]
%C A001318 Starting with offset 1 = row sums of triangle A168258 [From Gary W. Adamson 
               (qntmpkt(AT)yahoo.com), Nov 21 2009]
%D A001318 L. Euler, Decouverte d'une loi tout extraordinaire des nombres par rapport 
               a la somme de leurs diviseurs, Opera Omnia, I, 2, pp. 241-253.
%D A001318 A. Fink, R. K. Guy and M. Krusemeyer, Partitions with parts occurring 
               at most thrice, in preparation.
%D A001318 E. Haga, A strange sequence and a brilliant discovery, chapter 5 of Exploring 
               prime numbers on your PC and the Internet, 1st revised ed., 2007 
               (and earlier ed.), pp. 53-70.
%D A001318 R. Honsberger, Ingenuity in Math., Random House, 1970, p. 117.
%D A001318 D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial 
               Algorithms, (to appear), section 7.2.1.4, equation (18).
%D A001318 I. Niven, Formal power series, Amer. Math. Monthly, 76 (1969), 871-889.
%D A001318 I. Niven and H. S. Zuckerman, An Introduction to the Theory of Numbers. 
               2nd ed., Wiley, NY, 1966, p. 231.
%D A001318 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001318 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001318 A. Weil, Two lectures on number theory, past and present, L'Enseign. 
               Math., XX (1974), 87-110; Oeuvres III, 279-302.
%H A001318 T. D. Noe, <a href="b001318.txt">Table of n, a(n) for n = 0..1000</a>
%H A001318 L. Euler, <a href="http://arXiv.org/abs/math.HO/0505373">On the remarkable 
               properties of the pentagonal numbers</a>
%H A001318 L. Euler, <a href="http://math.dartmouth.edu/~euler/pages/E542.html">
               De mirabilibus proprietatibus numerorum pentagonalium</a>, par. 2
%H A001318 L. Euler, <a href="http://math.dartmouth.edu/~euler/pages/E243.html">
               Observatio de summis divisorum</a> p. 8.
%H A001318 L. Euler, <a href="http://arXiv.org/abs/math.HO/0411587">An observation 
               on the sums of divisors</a> p. 8.
%H A001318 S. Heubach and T. Mansour, <a href="http://arXiv.org/abs/math.CO/0310197">
               Counting rises, levels and drops in compositions</a>
%H A001318 Alfred Hoehn, <a href="a001318.pdf">Illustration of initial terms</a>
%H A001318 B. H. Margolius, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               Permutations with inversions</a>, J. Integ. Seqs. Vol. 4 (2001), 
               #01.2.4.
%H A001318 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A001318 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A001318 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PentagonalNumber.html">Pentagonal numbers</a>
%H A001318 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PartitionFunctionP.html">Link to a section of The World of Mathematics.</
               a>
%H A001318 M. Wohlgemuth, <a href="http://matheplanet.com/default3.html?article=277">
               Pentagon, Kartenhaus und Summenzerlegung</a>
%H A001318 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PentagonalNumberTheorem.html">Pentagonal Number Theorem</a>
%H A001318 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%F A001318 Euler: Product_{n=1..inf} (1-x^n) = Sum_{n = -inf..inf} (-1)^n*x^(n(3n-1)/
               2).
%F A001318 G.f.: x*(1+x+x^2)/((1-x)*(1-x^2)^2).
%F A001318 a(n)=n(n+1)/6 when n runs through numbers == 0 or 2 mod 3 - Barry E. 
               Williams
%F A001318 a(n) = A008805(n-1) + A008805(n-2) + A008805(n-3), n>2. - Ralf Stephan, 
               Apr 26 2003
%F A001318 Sequence consists of the pentagonal numbers (A000326), followed by A000326(n)+n 
               and then the next pentagonal numbers. - Jon Perry (perry(AT)globalnet.co.uk), 
               Sep 11 2003
%F A001318 a(n)=(6n^2+6n+1)/16-(2n+1)(-1)^n/16; a(n+1)=b(n)-b(n-1) where b(n)=sum{k=0..floor((n+2)/
               2), ((n+2)/(n+2-k))(-1)^k*C(n+2-k, k)C(n-2k+2, 2)C(n-2k, floor((n-2k)/
               2))}; - Paul Barry (pbarry(AT)wit.ie), May 13 2005
%F A001318 a(n)=sum{k=1..floor((n+1)/2), n-k+1} - Paul Barry (pbarry(AT)wit.ie), 
               Sep 07 2005
%F A001318 A001318(n)=A000217(n)-A000217(int(n/2)). - Pierre CAMI (pierrecami(AT)tele2.fr), 
               Dec 09 2007
%F A001318 a(0)=0, a(1)=1; then if n even a(n)=a(n-1)+n/2 and if n odd a(n)=a(n-1)+n. 
               - Pierre CAMI (pierrecami(AT)tele2.fr), Dec 09 2007
%F A001318 Numbers of the form n*(3*n-+1)/2. [From Vladimir Orlovsky (4vladimir(AT)gmail.com), 
               Jan 06 2009]
%F A001318 a(n) = A000217(n) - A000217(floor(n/2)) = n*(n+1)/2 - floor(n/2)*(floor(n/
               2)+1)/2 [From Carl R. White (oeisfan(AT)phodd.net), Aug 10 2009]
%p A001318 A001318:=-(1+z+z**2)/(z+1)**2/(z-1)**3; [S. Plouffe in his 1992 dissertation. 
               Gives sequence without initial zero.]
%t A001318 #1. lst={};s=0;Do[s+=n/3;If[Floor[s]==s,AppendTo[lst,s]],{n,0,7!}];lst 
               #2. lst={};Do[AppendTo[lst,n*(3*n-1)/2];AppendTo[lst,n*(3*n+1)/2],
               {n,6!}];lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Jan 
               06 2009]
%Y A001318 Cf. A000326 (pentagonal numbers), A000217 (triangular numbers), A010815, 
               A034828, A000326, A005449.
%Y A001318 Indices of nonzero terms of A010815 [ David W. Wilson (davidwwilson(AT)comcast.net) 
               ], i.e. the (zero-based) indices of 1-bits of the infinite binary 
               word to which the terms of A068052 converge.
%Y A001318 First differences give A026741 (Jud McCranie, j.mccranie(AT)comcast.net).
%Y A001318 Cf. A000217.
%Y A001318 Cf. A153384.
%Y A001318 Cf. A074378, A057569, A057570 [From Vladimir Orlovsky (4vladimir(AT)gmail.com), 
               Jan 06 2009]
%Y A001318 Sequence in context: A129232 A088822 A080182 this_sequence A024702 A161664 
               A080547
%Y A001318 Cf. A168258 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 21 2009]
%Y A001318 Adjacent sequences: A001315 A001316 A001317 this_sequence A001319 A001320 
               A001321
%K A001318 nonn,easy,nice,new
%O A001318 0,3
%A A001318 N. J. A. Sloane (njas(AT)research.att.com).
%E A001318 More terms from David W. Wilson (davidwwilson(AT)comcast.net)
%E A001318 Removed attribute "conjectured" from Plouffe g.f R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Mar 11 2009