0,3

Comment from R. K. Guy, Dec 28 2005:

"Conway's relation twixt the triangular and pentagonal numbers: Divide the triangular numbers by 3 (when you can exactly):

0 1 3 6 10 15 21 28 36 45 55 66 78 91 105 120 136 153 ...

0 - 1 2 .- .5 .7 .- 12 15 .- 22 26 .- .35 .40 .- ..51 ...

.....-.-.....+..+.....-..-.....+..+......-...-.......+....

"and you get the pentagonal numbers in pairs, one of positive rank and the other negative.

"Append signs according as the pair have the same (+) or opposite (-) parity.

"Then Euler's pentagonal number theorem is easy to remember:

"p(n-0)-p(n-1)-p(n-2)+p(n-5)+p(n-7)-p(n-12)-p(n-15)++-- =0^n

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.

"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."

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

Number of levels in the partitions of n+1 with parts in {1,2}.

A080995(a(n))=1: complement of A118300; A000009(a(n))=A051044(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 22 2006

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

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

Contribution from Matthew Vandermast (ghodges14(AT)comcast.net), Oct 28 2008: (Start)

Numbers n for which A000326(n) is a member of A000332. Cf. A145920.

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)

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]

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

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.

A. Fink, R. K. Guy and M. Krusemeyer, Partitions with parts occurring at most thrice, in preparation.

R. Honsberger, Ingenuity in Math., Random House, 1970, p. 117.

D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, (to appear), section 7.2.1.4, equation (18).

I. Niven, Formal power series, Amer. Math. Monthly, 76 (1969), 871-889.

I. Niven and H. S. Zuckerman, An Introduction to the Theory of Numbers. 2nd ed., Wiley, NY, 1966, p. 231.

A. Weil, Two lectures on number theory, past and present, L'Enseign. Math., XX (1974), 87-110; Oeuvres III, 279-302.

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.

T. D. Noe, Table of n, a(n) for n = 0..1000

Index entries for sequences related to linear recurrences with constant coefficients

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

B. H. Margolius, Permutations with inversions, J. Integ. Seqs. Vol. 4 (2001), #01.2.4.

Eric Weisstein's World of Mathematics, Pentagonal numbers

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

M. Wohlgemuth, Pentagon, Kartenhaus und Summenzerlegung

S. Heubach and T. Mansour, Counting rises, levels and drops in compositions

L. Euler, On the remarkable properties of the pentagonal numbers

L. Euler, De mirabilibus proprietatibus numerorum pentagonalium, par. 2

L. Euler, Observatio de summis divisorum p. 8.

L. Euler, An observation on the sums of divisors p. 8.

Eric Weisstein's World of Mathematics, Pentagonal Number Theorem

Euler: Product_{n=1..inf} (1-x^n) = Sum_{n = -inf..inf} (-1)^n*x^(n(3n-1)/2).

G.f.: x*(1+x+x^2)/((1-x)*(1-x^2)^2).

a(n)=n(n+1)/6 when n runs through numbers == 0 or 2 mod 3 - Barry E. Williams

a(n) = A008805(n-1) + A008805(n-2) + A008805(n-3), n>2. - Ralf Stephan, Apr 26 2003

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

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

a(n)=sum{k=1..floor((n+1)/2), n-k+1} - Paul Barry (pbarry(AT)wit.ie), Sep 07 2005

A001318(n)=A000217(n)-A000217(int(n/2)). - Pierre CAMI (pierrecami(AT)tele2.fr), Dec 09 2007

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

Numbers of the form n*(3*n-+1)/2. [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Jan 06 2009]

A001318:=-(1+z+z**2)/(z+1)**2/(z-1)**3; [S. Plouffe in his 1992 dissertation. Gives sequence without initial zero.]

#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]

Cf. A000326 (pentagonal numbers), A000217 (triangular numbers), A010815, A034828, A000326, A005449.

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.

First differences give A026741 (Jud McCranie, j.mccranie(AT)comcast.net).

Cf. A000217.

Cf. A153384.

Cf. A074378, A057569, A057570 [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Jan 06 2009]

Adjacent sequences: A001315 A001316 A001317 this_sequence A001319 A001320 A001321

Sequence in context: A129232 A088822 A080182 this_sequence A024702 A161664 A080547

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).

More terms from David W. Wilson (davidwwilson(AT)comcast.net)

Removed attribute "conjectured" from Plouffe g.f R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 11 2009