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%I M3378 #255 Dec 21 2023 03:04:10
%S 1,4,10,19,31,46,64,85,109,136,166,199,235,274,316,361,409,460,514,
%T 571,631,694,760,829,901,976,1054,1135,1219,1306,1396,1489,1585,1684,
%U 1786,1891,1999,2110,2224,2341,2461,2584,2710,2839,2971,3106,3244,3385,3529
%N Centered triangular numbers: a(n) = 3*n*(n-1)/2 + 1.
%C These are Hogben's central polygonal numbers
%C 2
%C .P
%C 3 n
%C Also the sum of three consecutive triangular numbers (A000217); i.e., a(4) = 19 = T4 + T3 + T2 = 10 + 6 + 3. - _Robert G. Wilson v_, Apr 27 2001
%C For k>2, Sum_{n=1..k} a(n) gives the sum pertaining to the magic square of order k. E.g., Sum_{n=1..5} a(n) = 1 + 4 + 10 + 19 + 31 = 65. In general, Sum_{n=1..k} a(n) = k*(k^2 + 1)/2. - _Amarnath Murthy_, Dec 22 2001
%C Binomial transform of (1,3,3,0,0,0,...). - _Paul Barry_, Jul 01 2003
%C a(n) is the difference of two tetrahedral (or pyramidal) numbers: C(n+3,3) = (n+1)(n+2)(n+3)/6. a(n) = A000292(n) - A000292(n-3) = (n+1)(n+2)(n+3)/6 - (n-2)(n-1)(n)/6. - _Alexander Adamchuk_, May 20 2006
%C Partial sums are A006003(n) = n(n^2+1)/2. Finite differences are a(n+1) - a(n) = A008585(n) = 3n. - _Alexander Adamchuk_, Jun 03 2006
%C If X is an n-set and Y a fixed 3-subset of X then a(n-2) is equal to the number of 3-subsets of X intersecting Y. - _Milan Janjic_, Jul 30 2007
%C Equals (1, 2, 3, ...) convolved with (1, 2, 3, 3, 3, ...). a(4) = 19 = (1, 2, 3, 4) dot (3, 3, 2, 1) = (3 + 6 + 6 + 4). - _Gary W. Adamson_, May 01 2009
%C Equals the triangular numbers convolved with [1, 1, 1, 0, 0, 0, ...]. - _Gary W. Adamson_ and _Alexander R. Povolotsky_, May 29 2009
%C a(n) is the number of triples (w,x,y) having all terms in {0,...,n} and min(w+x,x+y,y+w) = max(w,x,y). - _Clark Kimberling_, Jun 14 2012
%C a(n) = number of atoms at graph distance <= n from an atom in the graphite or graphene network (cf. A008486). - _N. J. A. Sloane_, Jan 06 2013
%C In 1826, Shiraishi gave a solution to the Diophantine equation a^3 + b^3 + c^3 = d^3 with b = a(n) for n > 1; see A226903. - _Jonathan Sondow_, Jun 22 2013
%C For n > 1, a(n) is the remainder of n^2 * (n-1)^2 mod (n^2 + (n-1)^2). - _J. M. Bergot_, Jun 27 2013
%C The equation A000578(x) - A000578(x-1) = A000217(y) - A000217(y-2) is satisfied by y=a(x). - _Bruno Berselli_, Feb 19 2014
%C A242357(a(n)) = n. - _Reinhard Zumkeller_, May 11 2014
%C A255437(a(n)) = 1. - _Reinhard Zumkeller_, Mar 23 2015
%C The first differences give A008486. a(n) seems to give the total number of triangles in the n-th generation of the six patterns of triangle expansion shown in the link. - _Kival Ngaokrajang_, Sep 12 2015
%C Number of binary shuffle squares of length 2n which contains exactly two 1's. - _Bartlomiej Pawlik_, Sep 07 2023
%C The digital root has period 3 (1, 4, 1) (A146325), the same digital root as the centered 12-gonal numbers, or centered dodecagonal numbers A003154(n). - _Peter M. Chema_, Dec 20 2023
%D R. Reed, The Lemming Simulation Problem, Mathematics in School, 3 (#6, Nov. 1974), front cover and pp. 5-6.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Seiichi Manyama, <a href="/A005448/b005448.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from T. D. Noe)
%H Paul Barry, <a href="https://arxiv.org/abs/2104.01644">Centered polygon numbers, heptagons and nonagons, and the Robbins numbers</a>, arXiv:2104.01644 [math.CO], 2021.
%H D. Bevan, D. Levin, P. Nugent, J. Pantone, and L. Pudwell, <a href="http://arxiv.org/abs/1510.08036">Pattern avoidance in forests of binary shrubs</a>, arXiv:1510.08036 [math.CO], 2015.
%H Jarosław Grytczuk, Bartłomiej Pawlik, and Mariusz Pleszczyński, <a href="https://arxiv.org/abs/2308.13882">Variations on shuffle squares</a>, arXiv:2308.13882 [math.CO], 2023. See p. 11.
%H F. Javier de Vega, <a href="https://doi.org/10.17654/0972555523015">On the parabolic partitions of a number</a>, J. Alg., Num. Theor., and Appl. (2023) Vol. 61, No. 2, 135-169.
%H Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a> [Cached copy]
%H L. Hogben, <a href="https://archive.org/details/chanceandchoiceb029729mbp/page/n25">Choice and Chance by Cardpack and Chessboard</a>, Vol. 1, Max Parrish and Co, London, 1950, p. 22.
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>
%H Clark Kimberling and John E. Brown, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Kimberling/kimber67.html">Partial Complements and Transposable Dispersions</a>, J. Integer Seqs., Vol. 7, 2004.
%H Kival Ngaokrajang, <a href="/A005448/a005448.pdf">Illustration of triangles expansion</a>
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H R. Reed, <a href="/A005448/a005448_1.pdf">The Lemming Simulation Problem</a>, Mathematics in School, 3 (#6, Nov. 1974), front cover and pp. 5-6. [Scanned photocopy of pages 5, 6 only, with annotations by R. K. Guy and N. J. A. Sloane]
%H Leo Tavares, <a href="/A005448/a005448.jpg">Illustration: Triple Triangles</a>
%H B. K. Teo and N. J. A. Sloane, <a href="http://dx.doi.org/10.1021/ic00220a025">Magic numbers in polygonal and polyhedral clusters</a>, Inorgan. Chem. 24 (1985), 4545-4558.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CenteredTriangularNumber.html">Centered Triangular Number</a>
%H <a href="/index/Ce#CENTRALCUBE">Index entries for sequences related to centered polygonal numbers</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F Expansion of x*(1-x^3)/(1-x)^4.
%F a(n) = C(n+3, 3)-C(n, 3) = C(n, 0)+3*C(n, 1)+3*C(n, 2). - _Paul Barry_, Jul 01 2003
%F a(n) = 1 + Sum_{j=0..n-1} (3*j). - Xavier Acloque, Oct 25 2003
%F a(n) = A000217(n) + A000290(n-1) = (3*A016754(n) + 5)/8. - _Lekraj Beedassy_, Nov 05 2005
%F Euler transform of length 3 sequence [4, 0, -1]. - _Michael Somos_, Sep 23 2006
%F a(1-n) = a(n). - _Michael Somos_, Sep 23 2006
%F a(n) = binomial(n+1,n-1) + binomial(n,n-2) + binomial(n-1,n-3). - _Zerinvary Lajos_, Sep 03 2006
%F Row sums of triangle A134482. - _Gary W. Adamson_, Oct 27 2007
%F Narayana transform (A001263) * [1, 3, 0, 0, 0, ...]. - _Gary W. Adamson_, Dec 29 2007
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), a(1)=1, a(2)=4, a(3)=10. - _Jaume Oliver Lafont_, Dec 02 2008
%F a(n) = A000217(n-1)*3 + 1 = A045943(n-1) + 1. - _Omar E. Pol_, Dec 27 2008
%F a(n) = a(n-1) + 3*n-3. - _Vincenzo Librandi_, Nov 18 2010
%F Sum_{n>=1} 1/a(n) = A306324. - _Ant King_, Jun 12 2012
%F a(n) = 2*a(n-1) - a(n-2) + 3. - _Ant King_, Jun 12 2012
%F a(n) = A101321(3,n-1). - _R. J. Mathar_, Jul 28 2016
%F E.g.f.: -1 + (2 + 3*x^2)*exp(x)/2. - _Ilya Gutkovskiy_, Jul 28 2016
%F a(n) = A002061(n) + A000217(n-1). - _Bruce J. Nicholson_, Apr 20 2017
%F From _Amiram Eldar_, Jun 20 2020: (Start)
%F Sum_{n>=1} a(n)/n! = 5*e/2 - 1.
%F Sum_{n>=1} (-1)^n * a(n)/n! = 5/(2*e) - 1. (End)
%F a(n) = A000326(n) - n + 1. - _Charlie Marion_, Nov 21 2020
%e From _Seiichi Manyama_, Aug 12 2017: (Start)
%e a(1) = 1:
%e *
%e / \
%e / \
%e / \
%e *-------*
%e .................................................
%e a(2) = 4:
%e *
%e / \
%e / \
%e / \
%e *---*---*
%e / \
%e * / \ *
%e / \ / \ / \
%e / *-------* \
%e / \ / \
%e *-------* *-------*
%e .................................................
%e a(3) = 10:
%e *
%e / \
%e / \
%e / \
%e *---*---*
%e / \
%e * / \ *
%e / \ / \ / \
%e / *---*---* \
%e / \ / \ / \
%e *---*---* *---*---*
%e / \ / \ / \
%e * / *---*---* \ *
%e / \ / \ / \ / \ / \
%e / *-------* *-------* \
%e / \ / \ / \
%e *-------* *-------* *-------*
%e .................................................
%e a(4) = 19:
%e *
%e / \
%e / \
%e / \
%e *---*---*
%e / \
%e * / \ *
%e / \ / \ / \
%e / *---*---* \
%e / \ / \ / \
%e *---*---* *---*---*
%e / \ / \ / \
%e * / \---*---* \ *
%e / \ / \ / \ / \ / \
%e / *---*---* *---*---* \
%e / \ / \ / \ / \ / \
%e *---*---* *---*---* *---*---*
%e / \ / \ / \ / \ / \
%e * / *---*---* *---*---* \ *
%e / \ / \ / \ / \ / \ / \ / \
%e / *-------* *-------* *-------* \
%e / \ / \ / \ / \
%e *-------* *-------* *-------* *-------*
%e (End)
%p A005448 := n->(3*(n-1)^2+3*(n-1)+2)/2: seq(A005448(n), n=1..100);
%p A005448 := -(1+z+z**2)/(z-1)^3; # _Simon Plouffe_ in his 1992 dissertation for offset 0
%t FoldList[#1 + #2 &, 1, 3 Range@ 50] (* _Robert G. Wilson v_, Feb 02 2011 *)
%t Join[{1,4},Total/@Partition[Accumulate[Range[50]],3,1]] (* _Harvey P. Dale_, Aug 17 2012 *)
%t LinearRecurrence[{3, -3, 1}, {1, 4, 10}, 50] (* _Vincenzo Librandi_, Sep 13 2015 *)
%t Table[ j! Coefficient[Series[Exp[x]*(1 + 3 x^2/2)-1, {x, 0, 20}], x, j], {j, 0, 20}] (* _Nikolaos Pantelidis_, Feb 07 2023 *)
%o (PARI) {a(n)=3*(n^2-n)/2+1} /* _Michael Somos_, Sep 23 2006 */
%o (PARI) isok(n) = my(k=(2*n-2)/3, m); (n==1) || ((denominator(k)==1) && (m=sqrtint(k)) && (m*(m+1)==k)); \\ _Michel Marcus_, May 20 2020
%o (Haskell)
%o a005448 n = 3 * n * (n - 1) `div` 2 + 1
%o a005448_list = 1 : zipWith (+) a005448_list [3, 6 ..]
%o -- _Reinhard Zumkeller_, Jun 20 2013
%o (Magma) I:=[1,4,10]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..60]]; // _Vincenzo Librandi_, Sep 13 2015
%Y Cf. A000217, A000292, A001263, A001844, A002061, A003154, A006003 (partial sums), A008486, A008585 = first differences, A045943, A134482, A146325, A226903, A242357, A255437.
%K nonn,easy,nice
%O 1,2
%A _N. J. A. Sloane_, _R. K. Guy_, Dec 12 1974
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