A152948 a(n) = (n^2 - 3*n + 6)/2.

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

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.

From Vladimir Shevelev, Jan 20 2014: (Start)

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).

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. (End)

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

Marilena Barnabei, Flavio Bonetti, Niccolò Castronuovo, and Matteo Silimbani, Permutations avoiding a simsun pattern, The Electronic Journal of Combinatorics (2020) Vol. 27, Issue 3, P3.45.

E. R. Berlekamp, A contribution to mathematical psychometrics, Unpublished Bell Labs Memorandum, Feb 08 1968 [Annotated scanned copy]

Kyu-Hwan Lee and Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.

Ângela Mestre and 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

Sum_{n>=1} 1/a(n) = 1/2 + 2*Pi*tanh(sqrt(15)*Pi/2)/sqrt(15). - Amiram Eldar, Dec 13 2022

E.g.f.: exp(x)*(6 - 2*x + x^2)/2 - 3. - Stefano Spezia, Nov 14 2024

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, A000566, A001477, A022856, A089071, A133263, A152947.

Sequence in context: A087729 A039890 A347546 * A018136 A243853 A293419

Adjacent sequences: A152945 A152946 A152947 * A152949 A152950 A152951

Keyword

nonn,easy

Author

Vladimir Joseph Stephan Orlovsky, Dec 15 2008

Status

approved