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A005448
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Centered triangular numbers: a(n) = 3n(n-1)/2 + 1.
(Formerly M3378)
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136
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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
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OFFSET
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1,2
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COMMENTS
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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
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
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 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
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REFERENCES
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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).
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LINKS
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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]
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FORMULA
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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
Euler transform of length 3 sequence [4, 0, -1]. - 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
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) = 2*a(n-1) - a(n-2) + 3. - Ant King, Jun 12 2012
Sum_{n>=1} a(n)/n! = 5*e/2 - 1.
Sum_{n>=1} (-1)^n * a(n)/n! = 5/(2*e) - 1. (End)
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EXAMPLE
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a(1) = 1:
*
/ \
/ \
/ \
*-------*
.................................................
a(2) = 4:
*
/ \
/ \
/ \
*---*---*
/ \
* / \ *
/ \ / \ / \
/ *-------* \
/ \ / \
*-------* *-------*
.................................................
a(3) = 10:
*
/ \
/ \
/ \
*---*---*
/ \
* / \ *
/ \ / \ / \
/ *---*---* \
/ \ / \ / \
*---*---* *---*---*
/ \ / \ / \
* / *---*---* \ *
/ \ / \ / \ / \ / \
/ *-------* *-------* \
/ \ / \ / \
*-------* *-------* *-------*
.................................................
a(4) = 19:
*
/ \
/ \
/ \
*---*---*
/ \
* / \ *
/ \ / \ / \
/ *---*---* \
/ \ / \ / \
*---*---* *---*---*
/ \ / \ / \
* / \---*---* \ *
/ \ / \ / \ / \ / \
/ *---*---* *---*---* \
/ \ / \ / \ / \ / \
*---*---* *---*---* *---*---*
/ \ / \ / \ / \ / \
* / *---*---* *---*---* \ *
/ \ / \ / \ / \ / \ / \ / \
/ *-------* *-------* *-------* \
/ \ / \ / \ / \
*-------* *-------* *-------* *-------*
(End)
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MAPLE
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MATHEMATICA
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Join[{1, 4}, Total/@Partition[Accumulate[Range[50]], 3, 1]] (* Harvey P. Dale, Aug 17 2012 *)
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 *)
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PROG
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(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 ..]
(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
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CROSSREFS
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Cf. A000217, A000292, A001263, A001844, A002061, A006003 = partial sums, A008486, A008585 = first differences, A045943, A134482, A226903, A242357, A255437.
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KEYWORD
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nonn,easy,nice
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AUTHOR
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STATUS
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approved
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