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A049451 Twice second pentagonal numbers. 30

%I

%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

%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 Kival Ngaokrajang, <a href="/A049451/a049451.pdf">Illustration of 3 points circle center spiral</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

%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)

%p a:=n->sum(2*(n+j), j=1..n): seq(a(n), n=0..55); # _Zerinvary Lajos_, May 26 2008

%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)

%o 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

%Y Cf. A000567, A005449, A033580, A049450.

%Y Similar sequences are listed in A316466.

%K nonn,easy

%O 0,2

%A Joe Keane (jgk(AT)jgk.org)

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Last modified February 27 10:15 EST 2020. Contains 332304 sequences. (Running on oeis4.)