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A001318 Generalized pentagonal numbers: n*(3*n-1)/2, n=0, +- 1, +- 2, +- 3,....
(Formerly M1336 N0511)
170
0, 1, 2, 5, 7, 12, 15, 22, 26, 35, 40, 51, 57, 70, 77, 92, 100, 117, 126, 145, 155, 176, 187, 210, 222, 247, 260, 287, 301, 330, 345, 376, 392, 425, 442, 477, 495, 532, 551, 590, 610, 651, 672, 715, 737, 782, 805, 852, 876, 925, 950, 1001, 1027, 1080, 1107, 1162, 1190, 1247, 1276, 1335 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Partial sums of A026741. - Jud McCranie; corrected by Omar E. Pol, Jul 05 2012

From R. K. Guy, Dec 28 2005: (Start)

"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."

(End)

The sequence 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 A090864; A000009(a(n)) = A051044(n). - Reinhard Zumkeller, Apr 22 2006

a(n) is the number of 3 X 3 matrices (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 Deléham, Apr 11 2007

Also numbers a(n) such that 24*a(n)+1 = (6*n-1)^2 are odd squares: 1, 25, 49, 121, 169, 289, 361,..., n=0, +-1, +-2,.... - Zak Seidov, Mar 08 2008

From Matthew Vandermast, 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*(3*n-1)/2 belongs to A000332, see A145919. (End)

Starting with offset 1 = row sums of triangle A168258. - Gary W. Adamson, Nov 21 2009

Starting with offset 1 = Triangle A101688 * [1, 2, 3,...]. - Gary W. Adamson, Nov 27 2009

Starting with offset 1 can be considered the first in an infinite set generated from A026741. Refer to the array in A175005. - Gary W. Adamson, Apr 03 2010

Vertex number of a square spiral whose edges have length A026741. The two axes of the spiral forming an "X" are A000326 and A005449. The four semi-axes forming an "X" are A049452, A049453, A033570 and the numbers >= 2 of A033568. - Omar E. Pol, Sep 08 2011

A general formula for the generalized k-gonal numbers is given by n*((k-2)*n-k+4)/2, n=0, +- 1, +- 2,..., k>=5. - Omar E. Pol, Sep 15 2011

A001318(n) is the number of 3-tuples (w,x,y) having all terms in {0,...,n} and 2*w = 2*x+y. - Clark Kimberling, Jun 04 2012

Generalized k-gonal numbers are second k-gonal numbers and positive terms of k-gonal numbers interleaved, k >= 5. - Omar E. Pol, Aug 04 2012

a(n) is the sum of the largest parts of the partitions of n+1 into exactly 2 parts. - Wesley Ivan Hurt, Jan 26 2013

Conway's relation mentioned by R. K. Guy is a relation between triangular numbers and generalized pentagonal numbers, two sequences from different families, but as triangular numbers are also generalized hexagonal numbers in this case we have a relation between two sequences from the same family. - Omar E. Pol, Feb 01 2013

REFERENCES

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, Contrib. Discr. Math. 3 (2) (2008), 76-114

E. Haga, A strange sequence and a brilliant discovery, chapter 5 of Exploring prime numbers on your PC and the Internet, first revised ed., 2007 (and earlier ed.), pp. 53-70.

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.

S. C. Schlicker, Numbers Simultaneously Polygonal and Centered Polygonal, Mathematics Magazine,  Vol. 84, No. 5, December 2011

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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

LINKS

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

G. E. Andrews, J. A. Sellers, Congruences for the Fishburn Numbers, arXiv preprint arXiv:1401.5345, 2014

P. Barry, On sequences with {-1, 0, 1} Hankel transforms, Arxiv preprint arXiv:1205.2565, 2012. - From N. J. A. Sloane, Oct 18 2012

S. Cooper and M. D. Hirschhorn, Results of Hurwitz type for three squares. Discrete Math. 274 (2004), no. 1-3, 9-24. See P(q).

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.

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

Alfred Hoehn, Illustration of initial terms

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

Johannes W. Meijer, Euler's Ship on the Pentagonal Sea, pdf and jpg.

J. W. Meijer and M. Nepveu, Euler's ship on the Pentagonal Sea, Acta Nova, Volume 4, No. 1, December 2008. pp. 176-187.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Eric Weisstein's World of Mathematics, Pentagonal numbers, Partition Function P.

Eric Weisstein's World of Mathematics, Pentagonal Number Theorem

M. Wohlgemuth, Pentagon, Kartenhaus und Summenzerlegung

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

FORMULA

Euler: prod(n>=1, 1-x^n ) = sum(n = -inf..inf, (-1)^n*x^(n*(3*n-1)/2) ).

Euler transform of length 3 sequence [ 2, 2, -1]. - Michael Somos, Mar 24 2011

a(-1 - n) = a(n). a(2*n) = A005449(n). a(2*n - 1) = A000326(n). - Michael Somos, Mar 24 2011

a(n) = 3+2*a(n-2)-a(n-4). - _Ant King, Aug 23 2011

Product_{k>0} (1 - x^k) = Sum_{k>=0} (-1)^k * x^a(k). - Michael Somos, Mar 24 2011

G.f.: x*(1+x+x^2)/ ( (1+x)^2*(1-x)^3 ).

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, Sep 11 2003

a(n) = (6*n^2+6*n+1)/16-(2*n+1)*(-1)^n/16; a(n) = A034828(n+1)-A034828(n). - Paul Barry, May 13 2005

a(n) = sum_{k=1..floor((n+1)/2)} (n-k+1). - Paul Barry, Sep 07 2005

a(n) = A000217(n)-A000217(floor(n/2)). - Pierre CAMI, Dec 09 2007

If n even a(n) = a(n-1)+n/2 and if n odd a(n) = a(n-1)+n, n>=2. - Pierre CAMI, Dec 09 2007

a(n)-a(n-1) = A026741(n) and it follows that the difference between consecutive terms is equal to n if n is odd and to n/2 if n is even. Hence this is a self-generating sequence that can be simply constructed from knowledge of the first term alone. - Ant King, Sep 26 2011

a(n) = 1/2*ceil(n/2)*ceil((3*n+1)/2). - Mircea Merca, Jul 13 2012

a(n) = floor [n^2/4] + ((floor[(n + 1)/2])^2 + (floor[(n + 1)/2]))/2. - Raphie Frank, Oct 05 2012

a(n) = (A008794(n+1) + A000217(n))/2 = A002378(n) - A085787(n). - Omar E. Pol, Jan 12 2013

a(n) = floor((n+1)/2)*((n+1) - (1/2)*floor((n+1)/2) - (1/2)). - Wesley Ivan Hurt, Jan 26 2013

From Oskar Wieland, Apr 10 2013: (Start)

a(n) = a(n+1) - A026741(n),

a(n) = a(n+2) - A001651(n),

a(n) = a(n+3) - A184418(n),

a(n) = a(n+4) - A007310(n),

a(n) = a(n+6) - A001651(n)*3 = a(n+6) - A016051(n),

a(n) = a(n+8) - A007310(n)*2 = a(n+8) - A091999(n),

a(n) = a(n+10)- A001651(n)*5 = a(n+10)- A072703(n),

a(n) = a(n+12)- A007310(n)*3,

a(n) = a(n+14)- A001651(n)*7. (End)

a(n) = (A007310^2 - 1)/24. - Richard R. Forberg, May 27 2013

a(n) = sum_{i = ceil((n+1)/2)..n} i. - Wesley Ivan Hurt, Jun 08 2013

G.f.: x*G(0), where G(k)= 1  + x*(3*k+4)/(3*k+2 - x*(3*k+2)*(3*k^2+11*k+10)/(x*(3*k^2+11*k+10) + (k+1)*(3*k+4)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 16 2013

MAPLE

A001318 := -(1+z+z**2)/(z+1)**2/(z-1)**3; # Simon Plouffe in his 1992 dissertation; gives sequence without initial zero

A001318 := proc(n) (6*n^2+6*n+1)/16-(2*n+1)*(-1)^n/16 ; end proc: # R. J. Mathar, Mar 27 2011

MATHEMATICA

Table[n*(n+1)/6, {n, Select[Range[0, 100], Mod[#, 3] != 1 &]}]

PROG

(PARI) {a(n) = (3*n^2 + 2*n + (n%2) * (2*n + 1)) / 8} /* Michael Somos, Mar 24 2011 */

(PARI) {a(n) = if( n<0, 0, polcoeff( x * (1 - x^3) / ((1 - x) * (1-x^2))^2 + x * O(x^n), n))} /* Michael Somos, Mar 24 2011 */

(Sage)

@CachedFunction

def A001318(n):

    if n == 0 : return 0

    inc = n//2 if is_even(n) else n

    return inc + A001318(n-1)

[A001318(n) for n in (0..59)] # Peter Luschny, Oct 13 2012

CROSSREFS

A080995 is the characteristic function; see also A010815.

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

Indices of nonzero terms of A010815, i.e., the (zero-based) indices of 1-bits of the infinite binary word to which the terms of A068052 converge.

Union of A036498 and A036499.

Cf. A153384, A168258, A101688, A174739, A175005.

Cf. A074378, A057569, A057570.

Generalized k-gonal numbers, k=5..14: this sequence, A000217, A085787, A001082, A118277, A074377, A195160, A195162, A195313, A195818.

Column 1 of A195152.

Squares in APs: A221671, A221672.

Sequence in context: A129232 A088822 A080182 * A024702 A226084 A161664

Adjacent sequences:  A001315 A001316 A001317 * A001319 A001320 A001321

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from David W. Wilson

STATUS

approved

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Last modified July 31 21:15 EDT 2014. Contains 245088 sequences.