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
Examples
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
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- E. R. Berlekamp, A contribution to mathematical psychometrics, Unpublished Bell Labs Memorandum, Feb 08 1968 [Annotated scanned copy]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).
Programs
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Magma
[ (n^2-3*n+6)/2: n in [1..60] ];
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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 *) -
PARI
a(n)=(n^2-3*n+6)/2 \\ Charles R Greathouse IV, Sep 28 2015
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Sage
[2+binomial(n,2) for n in range(0, 54)] # Zerinvary Lajos, Mar 12 2009
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
Comments