Demonstration of the
On-Line Encyclopedia of Integer Sequences
(Page 3)
Identifying a Sequence: Supplying a Formula
A secondary goal of the
On-Line Encyclopedia of Integer Sequences
is to provide a place where the general public has access to interesting
parts of mathematics.
Suppose someone rediscovers the sequence of tetrahedral
numbers, the number of balls in a triangular pyramid,
shown here:
The first few numbers are easy to calculate by hand:
1, 4, 10, 20, 35, 56, ...
This person might be a high-school student in Tokyo,
a medical doctor in Paris,
or a retired mountain climber in South Dakota.
He or she would like to know if
there is a formula for these numbers, what they are called,
and a reference where they can find out more about them.
As long as they have access to the Internet or to electronic mail,
they can consult the
On-Line Encyclopedia of Integer Sequences.
(If they don't have access to either the Internet or email,
even if they do not have electricity - like the correspondent in South Dakota -
they can still refer to the
book version,
published in 1995 by Academic Press. This is now somewhat out of date,
but it includes the majority of the most important
sequences.)
For the moment, let us suppose they can access the Internet.
(Consulting the database via email will be discussed in a later demonstration.)
They go to the
main web page,
where they see the following.
You replace the example by your sequence and click "Submit":
The reply shows several sequences that match these terms, but the top
entry is the sequence that is sought:
Search: 1,4,10,20,35,56
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| A000292 |
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Tetrahedral (or pyramidal) numbers: C(n+3,3) = (n+1)(n+2)(n+3)/6.
(Formerly M3382 N1363)
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+0
147
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| 0, 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, 455, 560, 680, 816, 969, 1140, 1330, 1540, 1771, 2024, 2300, 2600, 2925, 3276, 3654, 4060, 4495, 4960, 5456, 5984, 6545, 7140, 7770, 8436, 9139, 9880, 10660, 11480, 12341, 13244, 14190, 15180
(list)
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OFFSET
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-1,3
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COMMENT
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The number of balls in a triangular pyramid in which each edge contains n+1 balls. The sum of the first n triangular numbers (A000217).
Also (1/6)*(n^3+3*n^2+2*n) is the number of ways to color vertices of a triangle using <= n colors, allowing rotations and reflections. Group is the dihedral group D_6 with cycle index (x1^3+2*x3+3*x1*x2)/6.
Also the convolution of the natural numbers with themselves - Felix Goldberg (felixg(AT)tx.technion.ac.il), Feb 01 2001
Connected with the Eulerian numbers (1,4,1) via 1*a(x-2)+4*a(x-1)+1*a(x) = x^3. - Gottfried Helms (helms(AT)uni-kassel.de), Apr 15 2002
a(n) = sum |i-j| for all 1 <= i <= j <= n. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Aug 05 2002
a(n) = sum of the all possible products p*q where (p,q) are ordered pairs and p+q = n+1. a(5) = 5 + 8 + 9 + 8 + 5 = 35. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), May 29 2003
Number of labeled graphs on n+3 nodes that are triangles. - Jon Perry (perry(AT)globalnet.co.uk), Jun 14 2003
Number of permutations of n+3 which have exactly 1 descent and avoid the pattern 1324. - Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Nov 05 2004
Schaefli symbol for this polyhedron: {3,3}
Transform of n^2 under the Riordan array (1/(1-x^2),x). - Paul Barry (pbarry(AT)wit.ie), Apr 16 2005
a(n) = -A108299(n+5,6) = A108299(n+6,7). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jun 01 2005
a(n) = -A110555(n+4,3). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jul 27 2005
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REFERENCES
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 194.
J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 83.
H. S. M. Coxeter, Polyhedral numbers, pp. 25-35 of R. S. Cohen, J. J. Stachel and M. W. Wartofsky, eds., For Dirk Struik: Scientific, historical and political essays in honor of Dirk J. Struik, Reidel, Dordrecht, 1974.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 4.
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (1).
J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.
D. Wells, The Penguin Dictionary of Curious and interesting Numbers, pp 126-7 Penguin Books 1987.
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LINKS
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O. Aichholzer and H. Krasser, The point set order type data base: a collection of applications and results, pp. 17-20 in Abstracts 13-th Canadian Conference on Computational Geometry (CCCG '01), Waterloo, Aug. 13-15, 2001.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
R. Jovanovic, First 2500 Tetrahedral numbers
Hyun Kwang Kim, On Regular Polytope Numbers
Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7
N. J. A. Sloane, Illustration of initial terms
N. J. A. Sloane, Pyramid of 20 balls corresponding to a(3)=20.
G. Villemin's Almanach of Numbers, Nombres Tetraedriques
E. W. Weisstein, Link to a section of The World of Mathematics (1).
E. W. Weisstein, Link to a section of The World of Mathematics (2).
Index entries for "core" sequences
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FORMULA
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Partial sums of the triangular numbers (A000217).
G.f.: 1/(1-x)^4. a(-4-n)=-a(n).
a(n)=(n+3)/n*a(n-1) - Ralf Stephan, Apr 26 2003
Sums of three consecutive terms give A006003. - Ralf Stephan, Apr 26 2003
a(n)=C[1,2,]+C[2,2]+...+C[n-1,2]+C[n,2]; n=5: a(5)=0+1+3+6+10=20. - Labos E. (labos(AT)ana1.sote.hu), May 09 2003
a(n)=sum{k=0..n, k(n-k)} (offset 1). - Paul Barry (pbarry(AT)wit.ie), Jul 21 2003
Determinant of the n X n symmetric Pascal matrix M_(i,j)=C(i+j+2,i) - Benoit Cloitre (abmt(AT)wanadoo.fr), Aug 19 2003
The sum of a series constructed by the products of the index and the length of the series (n) minus the index (i): a(n) = sum[i(n-i)]. Also the sum of n terms of A000217. - Martin Steven McCormick (mathseq(AT)wazer.net), Apr 06 2005
a(n)=sum{k=0..floor((n-1)/2), (n-2k)^2} [offset 0]; a(n+1)=sum{k=0..n, k^2*(1-(-1)^(n+k-1))/2} [offset 0]; - Paul Barry (pbarry(AT)wit.ie), Apr 16 2005
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EXAMPLE
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a(2) = 3*4*5/6 = 10, the number of balls in a pyramid of 3 layers of balls, 6 in a triangle at the bottom, 3 in the middle layer and 1 on top.
Consider the square array
1 2 3 4 5 6...
2 4 6 8 10 12...
3 6 9 12 16 20...
4 8 12 16 20 24...
5 10 15 20 25 30...
...
then a(n) = sum of n-th antidiagonal. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Apr 06 2003
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MAPLE
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A000292 := n->binomial(n+3, 3);
Or, f:=n->(1/6)*(n^3+3*n^2+2*n);
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MATHEMATICA
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Rest[FoldList[Plus, 0, Rest[FoldList[Plus, 0, Range[50]]]]]
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PROGRAM
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(PARI) a(n)=(n+3)*(n+2)*(n+1)/6
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CROSSREFS
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Sums of 2 consecutive terms give A000330.
a(3n-3)=A006566(n). A000447(n)=a(2n-2). A002492(n)=a(2n+1).
First differences give triangular numbers.
Cf. A001044, A003991, A061552.
Column 0 of triangle A094415.
Sequence in context: A038406 A038409 A090579 this_sequence A101552 A038419 A057319
Adjacent sequences: A000289 A000290 A000291 this_sequence A000293 A000294 A000295
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KEYWORD
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nonn,core,easy,nice
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AUTHOR
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njas
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EXTENSIONS
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More terms from Michael Somos
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