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%I #179 Feb 21 2022 00:59:24
%S 0,1,10,45,136,325,666,1225,2080,3321,5050,7381,10440,14365,19306,
%T 25425,32896,41905,52650,65341,80200,97461,117370,140185,166176,
%U 195625,228826,266085,307720,354061,405450,462241,524800,593505,668746,750925,840456,937765
%N a(n) = n^2*(n^2 + 1)/2.
%C Sum of first n^2 positive integers.
%C Start from xanthene and attach amino acids according to the reaction scheme that describes the reaction between the active sites. See the hyperlink below on chemistry. - _Robert G. Wilson v_, Aug 02 2002; _Amarnath Murthy_, Aug 01 2002
%C Sum of the next n multiples of n. - _Amarnath Murthy_, Aug 01 2002
%C The sum of the terms in an n X n spiral. These are also triangular numbers. - _William A. Tedeschi_, Feb 27 2008
%C Hypotenuse of Pythagorean triangles with smallest side a cube: A000578(n)^2 + A083374(n)^2 = a(n)^2. - _Martin Renner_, Nov 12 2011
%C For n>1, triangular numbers that can be represented as a sum of a square and a triangular number. For example, a(2)=10=4+6=9+1. - _Ivan N. Ianakiev_, Apr 24 2012
%C A037270 can be constructed in the following manner: Take A000217 and for every n not in A000290 delete the corresponding A000217(n). - _Ivan N. Ianakiev_, Apr 26 2012
%C Starting at a(1)=1 simply take 1*1=1, a(2)= 2*(2+3)=10, a(3)= 3*(4+5+6)=45, a(4)=4*(7+8+9+10) and so on. - _J. M. Bergot_, May 01 2015
%C Observation: The digital roots of the terms repeat in the sequence 1, 1, 9; e.g., the digital roots of 1, 10, 45, 136, 325, and 666 are 1, 1, 9, 1, 1, and 9. Verified for the first 10000 terms. - _Rob Barton_, Mar 28 2018
%C The above observation is easily explained and proved given that the digital root of a positive number equals the number modulo 9, and a(n + 9k) == a(n) (mod 9). - _M. F. Hasler_, Apr 05 2018
%C Number of unoriented rows of length 4 using up to n colors. For a(0)=0, there are no rows using no colors. For a(1)=1, there is one row using that one color for all positions. For a(2)=10, there are 4 achiral (AAAA, ABBA, BAAB, BBBB) and 6 chiral pairs (AAAB-BAAA, AABA-ABAA, AABB-BBAA, ABAB-BABA, ABBB-BBBA, BABB-BBAB). - _Robert A. Russell_, Nov 14 2018
%C For n > 0, a(2n+1) is the number of non-isomorphic 6C_m-snakes, where m = 2n+1 or m = 2n (for n>=2). A kC_n-snake is a connected graph in which the k>=2 blocks are isomorphic to the cycle C_n and the block-cutpoint graph is a path. - _Christian Barrientos_, May 15 2019
%C Number of achiral colorings of the edges of a tetrahedron with n available colors. - _Robert A. Russell_, Sep 07 2019
%D C. Alsina and R. B. Nelson, Charming Proofs: A Journey into Elegant Mathematics, MAA, 2010. See p. 5.
%D C. Barrientos, Graceful labelings of cyclic snakes, Ars Combin., 60(2001), 85-96.
%D Albert H. Beiler, Recreations in the theory of numbers, New York: Dover, (2nd ed.) 1966, p. 106, table 55.
%D T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.
%D T. A. Gulliver, Sequences from Cubes of Integers, Int. Math. Journal, 4 (2003), 439-445.
%D R. A. Wilson, Cosmic Trigger, epilogue of S.-P. Sirag.
%H T. D. Noe, <a href="/A037270/b037270.txt">Table of n, a(n) for n = 0..1000</a>
%H J. D. Bell, <a href="https://arxiv.org/abs/math/0408230">A translation of Leonhard Euler's "De Quadratis Magicis", E795</a>, arXiv:math/0408230 [math.CO], 2004-2005.
%H N. G de Bruijn, <a href="https://research.tue.nl/en/publications/some-classes-of-integer-valued-functions">Some classes of integer-valued functions</a>, Nederl. Akad. Wetensch. Proc, Ser. A, 58 (1955), 363-367. See page 363.
%H Th. Gruner, A. Kerber, R. Laue, and M. Meringer, <a href="https://doi.org/10.1007/978-3-642-60185-9_7">Mathematics for Combinatorial Chemistry</a>, In: F. Keil, W. Mackens, H. Voß and J. Wether, Scientific Computing in Chemical Engineering II, Springer, 1999, 74-81.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F a(n) = a(n-1) + n^3 + (n-1)^3.
%F a(n) = A000537(n)+A000537(n-1), i.e., square of sum of first n integers plus square of sum of first n-1 integers. - _Henry Bottomley_, Oct 15 2001
%F a(n) = Sum_{k=0..n^2} k. - _William A. Tedeschi_, Feb 27 2008
%F a(n) = (1/8)*sinh(2*arcsinh(n)). - _Artur Jasinski_, Feb 10 2010
%F G.f.: x*(1+x)*(1+4*x+x^2)/(1-x)^5. - _Colin Barker_, Mar 22 2012
%F a(n) = a(n-1) + A005898(n-1). - _Ivan N. Ianakiev_, May 13 2012
%F a(n) = 2 * A000217(n-1) * A000217(n) + A000290(n). - _Ivan N. Ianakiev_, May 26 2012
%F a(n) = A000217(n^2). - _J. M. Bergot_, Jun 07 2012
%F a(n) = 5*a(n-1) -10*a(n-2) +10*a(n-3) -5*a(n-4) +a(n-5) n>4, a(0)=0, a(1)=1, a(2)=10, a(3)=45, a(4)=136. - _Yosu Yurramendi_, Sep 02 2013
%F For n>0, a(n) = A000217(n)^2 + A000217(n-1)^2. - _Richard R. Forberg_, Dec 25 2013
%F a(n) = T(T(n)) + T(T(n-1)) + T(T(n)-1) + T(T(n-1)-1), where T(n) = A000217(n). - _Charlie Marion_, Sep 10 2016
%F a(n) = t(n-3)*t(n)+t(n-1)*t(n+2), with t(n)=A000217(n). - _J. M. Bergot_, Apr 07 2018
%F From _Robert A. Russell_, Nov 14 2018: (Start)
%F a(n) = (A000583(n) + A000290(n)) / 2 = (n^4 + n^2) / 2.
%F a(n) = A000583(n) - A083374(n) = A083374(n) + A000290(n).
%F G.f.: (Sum_{j=1..4} S2(4,j)*j!*x^j/(1-x)^(j+1) + Sum_{j=1..2} S2(2,j)*j!*x^j/(1-x)^(j+1)) / 2, where S2 is the Stirling subset number A008277.
%F G.f.: Sum_{k=1..4} A145882(4,k) * x^k / (1-x)^5.
%F E.g.f.: (Sum_{k=1..4} S2(4,k)*x^k + Sum_{k=1..2} S2(2,k)*x^k) * exp(x) / 2, where S2 is the Stirling subset number A008277.
%F For n>4, a(n) = Sum_{j=1..5} -binomial(j-6,j) * a(n-j). (End)
%F a(n) = n*A006003(n). - _Kritsada Moomuang_, Dec 16 2018
%F For n > 0, a(n) = Sum_{k=1..n} A317617(n,k). - _Stefano Spezia_, Jan 10 2019
%F Sum_{n>=1} 1/a(n) = 1 + Pi^2/3 - Pi*coth(Pi) = 1.13652003875929052467672874379... - _Vaclav Kotesovec_, Jan 21 2019
%F Sum_{n>=1} (-1)^(n+1)/a(n) = Pi*csch(Pi) + Pi^2/6 - 1. - _Amiram Eldar_, Nov 02 2021
%p seq(n^2*(n^2+1)/2,n=0..30); # _Muniru A Asiru_, Mar 28 2018
%t Table[ n^2*((n^2 + 1)/2), {n, 0, 30} ]
%t Table[(1/8) Round[N[Sinh[2 ArcSinh[n]]^2, 100]], {n, 0, 30}] (* _Artur Jasinski_, Feb 10 2010 *)
%t LinearRecurrence[{5,-10,10,-5,1},{0,1,10,45,136},30] (* _Harvey P. Dale_, Aug 03 2014 *)
%o (PARI) a(n)=binomial(n^2+1,2) \\ _Charles R Greathouse IV_, Apr 25 2012
%o (GAP) a:=List([0..30],n->n^2*(n^2+1)/2); # _Muniru A Asiru_, Mar 28 2018
%o (Python) for n in range(0,30): print(n**2*(n**2+1)/2, end=', ') # _Stefano Spezia_, Jan 10 2019
%o (Magma) [n^2*(n^2 + 1)/2: n in [0..30]] // _Stefano Spezia_, Jan 15 2019
%Y Cf. A000217, A236770 (see crossrefs).
%Y Row 4 of A277504.
%Y Cf. A000583 (oriented), A083374 (chiral), A000290 (achiral).
%Y Cf. A317617.
%Y Row 3 of A327086 (achiral simplex edge colorings).
%K nonn,easy,nice
%O 0,3
%A Aaron Gulliver (gulliver(AT)elec.canterbury.ac.nz)
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