

A152948


a(n) = (n^2  3*n + 6)/2.


20



2, 2, 3, 5, 8, 12, 17, 23, 30, 38, 47, 57, 68, 80, 93, 107, 122, 138, 155, 173, 192, 212, 233, 255, 278, 302, 327, 353, 380, 408, 437, 467, 498, 530, 563, 597, 632, 668, 705, 743, 782, 822, 863, 905, 948, 992, 1037, 1083, 1130, 1178, 1227, 1277, 1328, 1380
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OFFSET

1,1


COMMENTS

a(1) = 2; then add 0 to the first number, then 1, 2, 3, 4, ... and so on.
Essentially the same as A022856, A089071 and A133263.  R. J. Mathar, Dec 19 2008
First differences are A001477.
If we ignore the zero polygonal numbers, then for n >= 3, a(n) is the minimal k such that the kth ngonal number is a sum of two ngonal numbers (see formula and example).  Vladimir Shevelev, Jan 20 2014
Numbers m such that 8m  15 is a square.  Bruce J. Nicholson, Jul 24 2017


LINKS

Michael De Vlieger, Table of n, a(n) for n = 1..10000
E. R. Berlekamp, A contribution to mathematical psychometrics, Unpublished Bell Labs Memorandum, Feb 08 1968 [Annotated scanned copy]
KyuHwan Lee, Sejin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.
Ângela Mestre, José Agapito, Square Matrices Generated by Sequences of Riordan Arrays, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
Index entries for linear recurrences with constant coefficients, signature (3,3,1).


FORMULA

a(n) = a(n1) + n2 (with a(1)=2).  Vincenzo Librandi, Nov 26 2010
a(n) = 3*a(n1)  3*a(n2) + a(n3).
G.f.: x*(2  4*x + 3*x^2) / (x1)^3.  R. J. Mathar, Oct 30 2011
If the zero polygonal numbers are ignored, then for n >= 4, the a(n)th ngonal number is a sum of the (a(n)1)th ngonal number and the (n1)th ngonal number.  Vladimir Shevelev, Jan 20 2014


EXAMPLE

a(7)=17. This means that the 17th (positive) heptagonal number 697 (cf. A000566) is the smallest heptagonal number which is a sum of two (positive) heptagonal numbers. We have 697 = 616 + 81 with indices 17, 16, 6 in A000566.  Vladimir Shevelev, Jan 20 2014


MATHEMATICA

Array[(#^2  3 # + 6)/2 &, 54] (* or *) Rest@ CoefficientList[Series[x (2  4 x + 3 x^2)/(x  1)^3, {x, 0, 54}], x] (* Michael De Vlieger, Mar 25 2020 *)


PROG

(Sage) [2+binomial(n, 2) for n in range(0, 54)] # Zerinvary Lajos, Mar 12 2009
(MAGMA) [ (n^23*n+6)/2: n in [1..60] ];
(PARI) a(n)=(n^23*n+6)/2 \\ Charles R Greathouse IV, Sep 28 2015


CROSSREFS

Cf. A000124, A000217, A152947.
Sequence in context: A179523 A087729 A039890 * A018136 A243853 A293419
Adjacent sequences: A152945 A152946 A152947 * A152949 A152950 A152951


KEYWORD

nonn,easy,changed


AUTHOR

Vladimir Joseph Stephan Orlovsky, Dec 15 2008


STATUS

approved



