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 A152948 a(n) = (n^2 - 3*n + 6)/2. 23

%I

%S 2,2,3,5,8,12,17,23,30,38,47,57,68,80,93,107,122,138,155,173,192,212,

%T 233,255,278,302,327,353,380,408,437,467,498,530,563,597,632,668,705,

%U 743,782,822,863,905,948,992,1037,1083,1130,1178,1227,1277,1328,1380

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

%C a(1) = 2; then add 0 to the first number, then 1, 2, 3, 4, ... and so on.

%C Essentially the same as A022856, A089071 and A133263. - _R. J. Mathar_, Dec 19 2008

%C First differences are A001477.

%C 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

%C Numbers m such that 8m - 15 is a square. - _Bruce J. Nicholson_, Jul 24 2017

%H Michael De Vlieger, <a href="/A152948/b152948.txt">Table of n, a(n) for n = 1..10000</a>

%H Marilena Barnabei, Flavio Bonetti, Niccolò Castronuovo, and Matteo Silimbani, <a href="https://doi.org/10.37236/9482">Permutations avoiding a simsun pattern</a>, The Electronic Journal of Combinatorics (2020) Vol. 27, Issue 3, P3.45.

%H E. R. Berlekamp, <a href="/A257113/a257113.pdf">A contribution to mathematical psychometrics</a>, Unpublished Bell Labs Memorandum, Feb 08 1968 [Annotated scanned copy]

%H Kyu-Hwan Lee and Se-jin Oh, <a href="http://arxiv.org/abs/1601.06685">Catalan triangle numbers and binomial coefficients</a>, arXiv:1601.06685 [math.CO], 2016.

%H Ângela Mestre and José Agapito, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL22/Mestre/mestre2.html">Square Matrices Generated by Sequences of Riordan Arrays</a>, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F a(n) = a(n-1) + n-2 (with a(1)=2). - _Vincenzo Librandi_, Nov 26 2010

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

%F G.f.: -x*(2 - 4*x + 3*x^2) / (x-1)^3. - _R. J. Mathar_, Oct 30 2011

%F 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

%e 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

%t 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 *)

%o (Sage) [2+binomial(n,2) for n in range(0, 54)] # _Zerinvary Lajos_, Mar 12 2009

%o (MAGMA) [ (n^2-3*n+6)/2: n in [1..60] ];

%o (PARI) a(n)=(n^2-3*n+6)/2 \\ _Charles R Greathouse IV_, Sep 28 2015

%Y Cf. A000124, A000217, A152947.

%K nonn,easy

%O 1,1

%A _Vladimir Joseph Stephan Orlovsky_, Dec 15 2008

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