A005448 Centered triangular numbers: a(n) = 3n(n-1)/2 + 1. (Formerly M3378)

1, 4, 10, 19, 31, 46, 64, 85, 109, 136, 166, 199, 235, 274, 316, 361, 409, 460, 514, 571, 631, 694, 760, 829, 901, 976, 1054, 1135, 1219, 1306, 1396, 1489, 1585, 1684, 1786, 1891, 1999, 2110, 2224, 2341, 2461, 2584, 2710, 2839, 2971, 3106, 3244, 3385, 3529

Offset

1,2

Comments

These are Hogben's central polygonal numbers

2

.P

3 n

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

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

Binomial transform of (1,3,3,0,0,0,...). - Paul Barry, Jul 01 2003

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

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

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

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

Equals the triangular numbers convolved with [1, 1, 1, 0, 0, 0, ...]. - Gary W. Adamson and Alexander R. Povolotsky, May 29 2009

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

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

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

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

The equation A000578(x) - A000578(x-1) = A000217(y) - A000217(y-2) is satisfied by y=a(x). - Bruno Berselli, Feb 19 2014

A242357(a(n)) = n. - Reinhard Zumkeller, May 11 2014

A255437(a(n)) = 1. - Reinhard Zumkeller, Mar 23 2015

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

References

R. Reed, The Lemming Simulation Problem, Mathematics in School, 3 (#6, Nov. 1974), front cover and pp. 5-6.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.

D. Bevan, D. Levin, P. Nugent, J. Pantone, and L. Pudwell, Pattern avoidance in forests of binary shrubs, arXiv:1510.08036 [math.CO], 2015.

Jarosław Grytczuk, Bartłomiej Pawlik, and Mariusz Pleszczyński, Variations on shuffle squares, arXiv:2308.13882 [math.CO], 2023. See p. 11.

F. Javier de Vega, On the parabolic partitions of a number, J. Alg., Num. Theor., and Appl. (2023) Vol. 61, No. 2, 135-169.

Guo-Niu Han, Enumeration of Standard Puzzles [Cached copy]

L. Hogben, Choice and Chance by Cardpack and Chessboard, Vol. 1, Max Parrish and Co, London, 1950, p. 22.

Milan Janjic, Two Enumerative Functions

Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.

Kival Ngaokrajang, Illustration of triangles expansion

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

R. Reed, The Lemming Simulation Problem, 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]

B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.

Eric Weisstein's World of Mathematics, Centered Triangular Number

Index entries for sequences related to centered polygonal numbers

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

Expansion of x*(1-x^3)/(1-x)^4.

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

a(n) = 1 + Sum_{j=0..n-1} (3*j). - Xavier Acloque, Oct 25 2003

a(n) = A000217(n) + A000290(n-1) = (3*A016754(n) + 5)/8. - Lekraj Beedassy, Nov 05 2005

Euler transform of length 3 sequence [4, 0, -1]. - Michael Somos, Sep 23 2006

a(1-n) = a(n). - Michael Somos, Sep 23 2006

a(n) = binomial(n+1,n-1) + binomial(n,n-2) + binomial(n-1,n-3). - Zerinvary Lajos, Sep 03 2006

Row sums of triangle A134482. - Gary W. Adamson, Oct 27 2007

Narayana transform (A001263) * [1, 3, 0, 0, 0, ...]. - Gary W. Adamson, Dec 29 2007

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

a(n) = A000217(n-1)*3 + 1 = A045943(n-1) + 1. - Omar E. Pol, Dec 27 2008

a(n) = a(n-1) + 3*n-3. - Vincenzo Librandi, Nov 18 2010

Sum_{n>=1} 1/a(n) = A306324. - Ant King, Jun 12 2012

a(n) = 2*a(n-1) - a(n-2) + 3. - Ant King, Jun 12 2012

a(n) = A101321(3,n-1). - R. J. Mathar, Jul 28 2016

E.g.f.: -1 + (2 + 3*x^2)*exp(x)/2. - Ilya Gutkovskiy, Jul 28 2016

a(n) = A002061(n) + A000217(n-1). - Bruce J. Nicholson, Apr 20 2017

From Amiram Eldar, Jun 20 2020: (Start)

Sum_{n>=1} a(n)/n! = 5*e/2 - 1.

Sum_{n>=1} (-1)^n * a(n)/n! = 5/(2*e) - 1. (End)

a(n) = A000326(n) - n + 1. - Charlie Marion, Nov 21 2020

Example

From Seiichi Manyama, Aug 12 2017: (Start)
a(1) = 1:
      *
     / \
    /   \
   /     \
  *-------*
.................................................
a(2) = 4:
            *
           / \
          /   \
         /     \
        *---*---*
           / \
      *   /   \   *
     / \ /     \ / \
    /   *-------*   \
   /     \     /     \
  *-------*   *-------*
.................................................
a(3) = 10:
                  *
                 / \
                /   \
               /     \
              *---*---*
                 / \
            *   /   \   *
           / \ /     \ / \
          /   *---*---*   \
         /     \ / \ /     \
        *---*---*   *---*---*
           / \ /     \ / \
      *   /   *---*---*   \   *
     / \ /     \ / \ /     \ / \
    /   *-------*   *-------*   \
   /     \     /     \     /     \
  *-------*   *-------*   *-------*
.................................................
a(4) = 19:
                        *
                       / \
                      /   \
                     /     \
                    *---*---*
                       / \
                  *   /   \   *
                 / \ /     \ / \
                /   *---*---*   \
               /     \ / \ /     \
              *---*---*   *---*---*
                 / \ /     \ / \
            *   /   \---*---*   \   *
           / \ /     \ / \ /     \ / \
          /   *---*---*   *---*---*   \
         /     \ / \ /     \ / \ /     \
        *---*---*   *---*---*   *---*---*
           / \ /     \ / \ /     \ / \
      *   /   *---*---*   *---*---*   \   *
     / \ /     \ / \ /     \ / \ /     \ / \
    /   *-------*   *-------*   *-------*   \
   /     \     /     \     /     \     /     \
  *-------*   *-------*   *-------*   *-------*
(End)

Maple

A005448 := n->(3*(n-1)^2+3*(n-1)+2)/2: seq(A005448(n), n=1..100);
A005448 := -(1+z+z**2)/(z-1)^3; # Simon Plouffe in his 1992 dissertation for offset 0

Mathematica

FoldList[#1 + #2 &, 1, 3 Range@ 50] (* Robert G. Wilson v, Feb 02 2011 *)
Join[{1, 4}, Total/@Partition[Accumulate[Range[50]], 3, 1]] (* Harvey P. Dale, Aug 17 2012 *)
LinearRecurrence[{3, -3, 1}, {1, 4, 10}, 50] (* Vincenzo Librandi, Sep 13 2015 *)
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 *)

Prog

(PARI) {a(n)=3*(n^2-n)/2+1} /* Michael Somos, Sep 23 2006 */
(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
(Haskell)
a005448 n = 3 * n * (n - 1) `div` 2 + 1
a005448_list = 1 : zipWith (+) a005448_list [3, 6 ..]
-- Reinhard Zumkeller, Jun 20 2013
(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

Crossrefs

Cf. A000217, A000292, A001263, A001844, A002061, A006003 = partial sums, A008486, A008585 = first differences, A045943, A134482, A226903, A242357, A255437.

Sequence in context: A162505 A025720 A022793 * A301247 A037040 A007077

Adjacent sequences: A005445 A005446 A005447 * A005449 A005450 A005451

Keyword

nonn,easy,nice

Author

N. J. A. Sloane, R. K. Guy, Dec 12 1974

Status

approved