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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 (list; graph; refs; listen; history; text; internal format)
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 k-th n-gonal number is a sum of two n-gonal 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E. R. Berlekamp, A contribution to mathematical psychometrics, Unpublished Bell Labs Memorandum, Feb 08 1968 [Annotated scanned copy]

Kyu-Hwan Lee, Se-jin 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(n-1) + n-2 (with a(1)=2). - Vincenzo Librandi, Nov 26 2010

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

G.f.: -x*(2 - 4*x + 3*x^2) /  (x-1)^3. - R. J. Mathar, Oct 30 2011

If the zero polygonal numbers are ignored, then for n >= 4, the a(n)-th n-gonal number is a sum of the (a(n)-1)-th n-gonal number and the (n-1)-th n-gonal 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^2-3*n+6)/2: n in [1..60] ];

(PARI) a(n)=(n^2-3*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

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Last modified March 28 17:51 EDT 2020. Contains 333103 sequences. (Running on oeis4.)