OFFSET
1,1
COMMENTS
a(1) = 2; then add 0 to the first number, then 1, 2, 3, 4, ... and so on.
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E. R. Berlekamp, A contribution to mathematical psychometrics, Unpublished Bell Labs Memorandum, Feb 08 1968 [Annotated scanned copy]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
KEYWORD
nonn,easy,changed
AUTHOR
Vladimir Joseph Stephan Orlovsky, Dec 15 2008
STATUS
approved