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%I #122 Mar 18 2024 07:43:55
%S 0,4,14,30,52,80,114,154,200,252,310,374,444,520,602,690,784,884,990,
%T 1102,1220,1344,1474,1610,1752,1900,2054,2214,2380,2552,2730,2914,
%U 3104,3300,3502,3710,3924,4144,4370,4602,4840,5084,5334,5590,5852,6120,6394,6674,6960,7252,7550,7854
%N Twice second pentagonal numbers.
%C From _Floor van Lamoen_, Jul 21 2001: (Start)
%C Write 1,2,3,4,... in a hexagonal spiral around 0, then a(n) is the sequence found by reading the line from 0 in the direction 0,4,... . The spiral begins:
%C .
%C 52
%C . \
%C 33--32--31--30 51
%C / . \ \
%C 34 16--15--14 29 50
%C / / . \ \ \
%C 35 17 5---4 13 28 49
%C / / / . \ \ \ \
%C 36 18 6 0 3 12 27 48
%C / / / / / / / /
%C 37 19 7 1---2 11 26 47
%C \ \ \ / / /
%C 38 20 8---9--10 25 46
%C \ \ / /
%C 39 21--22--23--24 45
%C \ /
%C 40--41--42--43--44
%C (End)
%C Number of edges in the join of the complete bipartite graph of order 2n and the cycle graph of order n, K_n,n * C_n. - _Roberto E. Martinez II_, Jan 07 2002
%C The average of the first n elements starting from a(1) is equal to (n+1)^2. - Mario Catalani (mario.catalani(AT)unito.it), Apr 10 2003
%C If Y is a 4-subset of an n-set X then, for n >= 4, a(n-4) is the number of (n-4)-subsets of X having either one element or two elements in common with Y. - _Milan Janjic_, Dec 28 2007
%C With offset 1: the maximum possible sum of numbers in an N x N standard Minesweeper grid. - _Dmitry Kamenetsky_, Dec 14 2008
%C a(n) = A001399(6*n-2), number of partitions of 6*n-2 into parts < 4. For example a(2)=14 where the partitions of 6*2-2=10 into parts < 4 are [1,1,1,1,1,1,1,1,1,1], [1,1,1,1,1,1,1,1,2], [1,1,1,1,1,1,1,3], [1,1,1,1,1,1,2,2], [1,1,1,1,1,2,3], [1,1,1,1,2,2,2], [1,1,1,1,3,3], [1,1,1,2,2,3], [1,1,2,2,2,2], [1,1,2,3,3], [1,2,2,2,3], [2,2,2,2,2], [1,3,3,3], [2,2,3,3]. - _Adi Dani_, Jun 07 2011
%C A003056 is the following array A read by antidiagonals:
%C 0, 1, 2, 3, 4, 5, ...
%C 1, 2, 3, 4, 5, 6, ...
%C 2, 3, 4, 5, 6, 7, ...
%C 3, 4, 5, 6, 7, 8, ...
%C 4, 5, 6, 7, 8, 9, ...
%C 5, 6, 7, 8, 9, 10, ...
%C and a(n) is the hook sum Sum_{k=0..n} A(n,k) + Sum_{r=0..n-1} A(r,n). - _R. J. Mathar_, Jun 30 2013
%C a(n)*Pi is the total length of 3 points circle center spiral after n rotations. The spiral length at each rotation (L(n)) is A016957. The spiral length ratio rounded down [floor(L(n)/L(1))] is A001651. See illustration in links. - _Kival Ngaokrajang_, Dec 27 2013
%C Partial sums give A114364. - _Leo Tavares_, Feb 25 2022
%C For n >= 1, the continued fraction expansion of sqrt(27*a(n)) is [9n+1; {2, 2n-1, 1, 4, 1, 2n-1, 2, 18n+2}]. - _Magus K. Chu_, Oct 13 2022
%D L. B. W. Jolley, Summation of Series, Dover Publications, 1961, p. 12.
%H Ivan Panchenko, <a href="/A049451/b049451.txt">Table of n, a(n) for n = 0..1000</a>
%H Raghavendra N. Bhat, Cristian Cobeli, and Alexandru Zaharescu, <a href="https://arxiv.org/abs/2403.10500">A lozenge triangulation of the plane with integers</a>, arXiv:2403.10500 [math.NT], 2024.
%H Kival Ngaokrajang, <a href="/A049451/a049451.pdf">Illustration of 3 points circle center spiral</a>.
%H Leo Tavares, <a href="/A049451/a049451.jpg">Illustration: Double Hexagonal Trapezoids</a>.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = n*(3*n+1).
%F G.f.: 2*x*(2+x)/(1-x)^3.
%F Sum_{i=1..n} a(i) = A045991(n+1). - _Gary W. Adamson_, Dec 20 2006
%F a(n) = 2*A005449(n). - _Omar E. Pol_, Dec 18 2008
%F a(n) = a(n-1) + 6*n -2, n > 0. - _Vincenzo Librandi_, Aug 06 2010
%F a(n) = A100104(n+1) - A100104(n). - _Reinhard Zumkeller_, Jul 07 2012
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) with a(0) = 0, a(1) = 4, a(2) = 14. - _Philippe Deléham_, Mar 26 2013
%F a(n) = A174709(6*n+3). - _Philippe Deléham_, Mar 26 2013
%F a(n) = (24/(n+2)!)*Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*j^(n+2). - _Bruno Berselli_, Jun 04 2013 - after the similar formula of _Vladimir Kruchinin_ in A002411
%F a(n) = A002061(n+1) + A056220(n). - _Bruce J. Nicholson_, Sep 21 2017
%F a(n) = Sum_{i = 2..5} P(i,n), where P(i,m) = m*((i-2)*m-(i-4))/2. - _Bruno Berselli_, Jul 04 2018
%F E.g.f.: x*(4 + 3*x)*exp(x). - _G. C. Greubel_, Sep 01 2019
%F a(n) = A003215(n) - A005408(n). - _Leo Tavares_, Feb 25 2022
%F From _Amiram Eldar_, Feb 27 2022: (Start)
%F Sum_{n>=1} 1/a(n) = 3 - Pi/(2*sqrt(3)) - 3*log(3)/2.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/sqrt(3) + 2*log(2) - 3. (End)
%F a(n) = A001105(n) + A002378(n). - _Torlach Rush_, Jul 11 2022
%e From _Dmitry Kamenetsky_, Dec 14 2008, with slight rewording by Raymond Martineau (mart0258(AT)yahoo.com), Dec 16 2008: (Start)
%e For an N x N Minesweeper grid the highest sum of numbers is (N-1)(3*N-2). This is achieved by filling every second row with mines (shown as 'X'). For example, when N=5 the best grids are:
%e .
%e X X X X X
%e 4 6 6 6 4
%e X X X X X
%e 4 6 6 6 4
%e X X X X X
%e .
%e and
%e .
%e 2 3 3 3 2
%e X X X X X
%e 4 6 6 6 4
%e X X X X X
%e 2 3 3 3 2
%e .
%e each giving a total of 52. (End)
%t Table[n(3n+1), {n,0,55}] (* or *) CoefficientList[Series[2x(2+x)/(1-x)^3, {x,0,55}], x] (* _Michael De Vlieger_, Apr 05 2017 *)
%o (Haskell) a049451 n = n * (3 * n + 1) -- _Reinhard Zumkeller_, Jul 07 2012
%o (PARI) a(n)=n*(3*n+1) \\ _Charles R Greathouse IV_, Sep 24 2015
%o (Magma) [n*(3*n+1): n in [0..55]]; // _G. C. Greubel_, Sep 01 2019
%o (Sage) [n*(3*n+1) for n in (0..55)] # _G. C. Greubel_, Sep 01 2019
%o (GAP) List([0..55], n-> n*(3*n+1)); # _G. C. Greubel_, Sep 01 2019
%o (Python) [n*(3*n+1) for n in range(60)] # _Gennady Eremin_, Feb 27 2022
%Y Cf. A000567, A001105, A002378, A005449, A033580, A049450.
%Y Similar sequences are listed in A316466.
%Y Cf. A003215, A005408, A114364.
%K nonn,easy
%O 0,2
%A Joe Keane (jgk(AT)jgk.org)
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