login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A037270 a(n) = n^2*(n^2 + 1)/2. 49
0, 1, 10, 45, 136, 325, 666, 1225, 2080, 3321, 5050, 7381, 10440, 14365, 19306, 25425, 32896, 41905, 52650, 65341, 80200, 97461, 117370, 140185, 166176, 195625, 228826, 266085, 307720, 354061, 405450, 462241, 524800, 593505, 668746, 750925, 840456, 937765 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Sum of first n^2 positive integers.

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

Sum of the next n multiples of n. - Amarnath Murthy, Aug 01 2002

The sum of the terms in an n X n spiral. These are also triangular numbers. - William A. Tedeschi, Feb 27 2008

Hypotenuse of Pythagorean triangles with smallest side a cube: A000578(n)^2 + A083374(n)^2 = a(n)^2. - Martin Renner, Nov 12 2011

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

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

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

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

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

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

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

Number of achiral colorings of the edges of a tetrahedron with n available colors. - Robert A. Russell, Sep 07 2019

REFERENCES

C. Alsina and R. B. Nelson, Charming Proofs: A Journey into Elegant Mathematics, MAA, 2010. See p. 5.

C. Barrientos, Graceful labelings of cyclic snakes, Ars Combin., 60(2001), 85-96.

Albert H. Beiler, Recreations in the theory of numbers, New York: Dover, (2nd ed.) 1966, p. 106, table 55.

T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.

T. A. Gulliver, Sequences from Cubes of Integers, Int. Math. Journal, 4 (2003), 439-445.

R. A. Wilson, Cosmic Trigger, epilogue of S.-P. Sirag.

LINKS

T. D. Noe, Table of n, a(n) for n = 0..1000

J. D. Bell, A translation of Leonhard Euler's "De Quadratis Magicis", E795, arXiv:math/0408230 [math.CO], 2004-2005.

N. G de Bruijn, Some classes of integer-valued functions, Nederl. Akad. Wetensch. Proc, Ser. A, 58 (1955), 363-367. See page 363.

Th. Gruner, A. Kerber, R. Laue, and M. Meringer, Mathematics for Combinatorial Chemistry, In: F. Keil, W. Mackens, H. Voß and J. Wether, Scientific Computing in Chemical Engineering II, Springer, 1999, 74-81.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

FORMULA

a(n) = a(n-1) + n^3 + (n-1)^3.

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

a(n) = Sum_{k=0..n^2} k. - William A. Tedeschi, Feb 27 2008

a(n) = (1/8)*sinh(2*arcsinh(n)). - Artur Jasinski, Feb 10 2010

G.f.: x*(1+x)*(1+4*x+x^2)/(1-x)^5. - Colin Barker, Mar 22 2012

a(n) = a(n-1) + A005898(n-1). - Ivan N. Ianakiev, May 13 2012

a(n) = 2 * A000217(n-1) * A000217(n) + A000290(n). - Ivan N. Ianakiev, May 26 2012

a(n) = A000217(n^2). - J. M. Bergot, Jun 07 2012

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

For n>0, a(n) = A000217(n)^2 + A000217(n-1)^2. - Richard R. Forberg, Dec 25 2013

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

a(n) = t(n-3)*t(n)+t(n-1)*t(n+2), with t(n)=A000217(n). - J. M. Bergot, Apr 07 2018

From Robert A. Russell, Nov 14 2018: (Start)

a(n) = (A000583(n) + A000290(n)) / 2 = (n^4 + n^2) / 2.

a(n) = A000583(n) - A083374(n) = A083374(n) + A000290(n).

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.

G.f.: Sum_{k=1..4} A145882(4,k) * x^k / (1-x)^5.

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.

For n>4, a(n) = Sum_{j=1..5} -binomial(j-6,j) * a(n-j). (End)

a(n) = n*A006003(n). - Kritsada Moomuang, Dec 16 2018

For n > 0, a(n) = Sum_{k=1..n} A317617(n,k). - Stefano Spezia, Jan 10 2019

Sum_{n>=1} 1/a(n) = 1 + Pi^2/3 - Pi*coth(Pi) = 1.13652003875929052467672874379... - Vaclav Kotesovec, Jan 21 2019

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi*csch(Pi) + Pi^2/6 - 1. - Amiram Eldar, Nov 02 2021

MAPLE

seq(n^2*(n^2+1)/2, n=0..30); # Muniru A Asiru, Mar 28 2018

MATHEMATICA

Table[ n^2*((n^2 + 1)/2), {n, 0, 30} ]

Table[(1/8) Round[N[Sinh[2 ArcSinh[n]]^2, 100]], {n, 0, 30}] (* Artur Jasinski, Feb 10 2010 *)

LinearRecurrence[{5, -10, 10, -5, 1}, {0, 1, 10, 45, 136}, 30] (* Harvey P. Dale, Aug 03 2014 *)

PROG

(PARI) a(n)=binomial(n^2+1, 2) \\ Charles R Greathouse IV, Apr 25 2012

(GAP) a:=List([0..30], n->n^2*(n^2+1)/2); # Muniru A Asiru, Mar 28 2018

(Python) for n in range(0, 30): print(n**2*(n**2+1)/2, end=', ') # Stefano Spezia, Jan 10 2019

(Magma) [n^2*(n^2 + 1)/2: n in [0..30]] // Stefano Spezia, Jan 15 2019

CROSSREFS

Cf. A000217, A236770 (see crossrefs).

Row 4 of A277504.

Cf. A000583 (oriented), A083374 (chiral), A000290 (achiral).

Cf. A317617.

Row 3 of A327086 (achiral simplex edge colorings).

Sequence in context: A211032 A179095 A213188 * A027800 A005714 A175705

Adjacent sequences: A037267 A037268 A037269 * A037271 A037272 A037273

KEYWORD

nonn,easy,nice

AUTHOR

Aaron Gulliver (gulliver(AT)elec.canterbury.ac.nz)

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 5 19:04 EST 2022. Contains 358588 sequences. (Running on oeis4.)