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A005448 Centered triangular numbers: 3n(n-1)/2 + 1.
(Formerly M3378)
64
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 (list; graph; refs; listen; history; text; internal format)
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 n>2 sigma(a(n)) gives the sum pertaining to the magic square of order n. E.g., for n = 5 we have sigma(a(n)) = 1+4+10+19+31= 65. In general sigma( a(n)) = n(n^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

The subsequence of primes is A125602; the subsequence of semiprimes is A184481. - Jonathan Vos Post, Feb 12 2011

The limiting value of the partial sums of the reciprocals of a(n) is 2Pi/sqrt(15)*tanh(Pi/2*sqrt(5/3)) = 1.567065131264... - Ant King, Jun 12 2012

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

REFERENCES

L. Hogben, Choice and Chance by Cardpack and Chessboard. Vol. 1, Chanticleer Press, NY, 1950, p. 22.

R. Reed, The Lemming Simulation Problem, Math. in School, 3 (#6, Nov. 1974), 5-6.

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

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

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

Milan Janjic, Two Enumerative Functions

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

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

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 sequences related to 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)=3a(n-1)-3a(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

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

MAPLE

A005448 := n->(3*n^2+3*n+2)/2;

A005448:=-(1+z+z**2)/(z-1)**3; # Simon Plouffe in his 1992 dissertation for offset 0

MATHEMATICA

s=1; lst={s}; Do[s+=n; AppendTo[lst, s], {n, 3, 7!, 3}]; lst (* Vladimir Joseph Stephan Orlovsky, Dec 13 2008 *)

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 *)

PROG

(PARI) {a(n)=3*(n^2-n)/2+1} /* Michael Somos, Sep 23 2006 */

(Haskell)

a005448 n = 3 * n * (n - 1) `div` 2 + 1

a005448_list = 1 : zipWith (+) a005448_list [3, 6 ..]

-- Reinhard Zumkeller, Jun 20 2013

CROSSREFS

Cf. A045943, A001844, A000292, A006003 = partial sums, A008585 = finite differences, A134482, A001263, A008486, A000217, A226903.

Sequence in context: A162505 A025720 A022793 * A037040 A007077 A009895

Adjacent sequences:  A005445 A005446 A005447 * A005449 A005450 A005451

KEYWORD

nonn,easy,nice

AUTHOR

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

EXTENSIONS

More terms from Milan Janjic, Jul 30 2007

STATUS

approved

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Last modified November 28 10:40 EST 2014. Contains 250311 sequences.