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A005448
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Centered triangular numbers: 3n(n-1)/2 + 1.
(Formerly M3378)
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54
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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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 (rgwv(AT)rgwv.com), 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 (amarnath_murthy(AT)yahoo.com), Dec 22 2001
Binomial transform of (1,3,3,0,0,0,.....). - Paul Barry (pbarry(AT)wit.ie), 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 (alex(AT)kolmogorov.com), 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 (Smith(AT)xxx.yyy.com), 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 R. Janjic (agnus(AT)blic.net), 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). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 01 2009]
Equals the triangular numbers convolved with [1, 1, 1, 0, 0, 0,...]. [From Gary W. Adamson & Alexander Povolotsky (qntmpkt(AT)yahoo.com), May 29 2009]
The subsequence of primes is A125602; the subsequence of semiprimes is A184481 [Jonathan Vos Post, Feb 12, 2011].
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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).
B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..1000
Guo-Niu Han, Enumeration of Standard Puzzles
Milan Janjic, Two Enumerative Functions
Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
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).
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FORMULA
| Expansion of x*(1-x^3 )/(1-x)^4.
a(n)=C(n+3, 3)-C(n, 3)=C(n, 0)+3C(n, 1)+3C(n, 2). - Paul Barry (pbarry(AT)wit.ie), Jul 01 2003
a(n) = 1+ sum_{j=0..n-1} (3*j). - Xavier Acloque Oct 25 2003
a(n) = T(n) + S(n-1) = A000217(n) + A000290(n-1) = (3*A016754(n) + 5)/8. - Lekraj Beedassy (blekraj(AT)yahoo.com), 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
binomial(n+1,n-1)+binomial(n,n-2)+binomial(n-1,n-3). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Sep 03 2006
Row sums of triangle A134482. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 27 2007
Narayana transform (A001263) * [1, 3, 0, 0, 0,...] - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 29 2007
a(n)=3a(n-1)-3a(n-2)+a(n-3), a(1)=1, a(2)=4, a(3)=10 [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Dec 02 2008]
a(n) = A000217(n-1)*3 + 1 = A045943(n-1) + 1. [From Omar E. Pol (info(AT)polprimos.com), Dec 27 2008]
a(n) = 3n(n+1)/2 + 1, if the offset were 0. [From Omar E. Pol (info(AT)polprimos.com), Nov 08 2009]
a(n)=a(n-1)+3*n-3. [From Vincenzo Librandi, Nov 18 2010]
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MAPLE
| A005448 := n->(3*n^2+3*n+2)/2;
A005448:=-(1+z+z**2)/(z-1)**3; [S. Plouffe in his 1992 dissertation for offset 0.]
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MATHEMATICA
| s=1; lst={s}; Do[s+=n; AppendTo[lst, s], {n, 3, 7!, 3}]; lst (* From Vladimir Orlovsky (4vladimir(AT)gmail.com), Dec 13 2008 *)
FoldList[#1 + #2 &, 1, 3 Range@ 50] (* Robert G. Wilson v, Feb 02 2011 *)
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PROG
| (PARI) {a(n)=3*(n^2-n)/2+1} /* Michael Somos Sep 23 2006 */
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CROSSREFS
| Cf. A045943, A001844, A000292, A006003 - partial sums, A008585 - finite differences, A134482, A001263.
Cf. A000217. [From Omar E. Pol (info(AT)polprimos.com), Dec 27 2008]
Sequence in context: A162505 A025720 A022793 * A037040 A007077 A009895
Adjacent sequences: A005445 A005446 A005447 * A005449 A005450 A005451
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), R. K. Guy, Dec 12, 1974
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EXTENSIONS
| More terms from Milan R. Janjic (agnus(AT)blic.net), Jul 30 2007
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