%I A005448 M3378
%S A005448 1,4,10,19,31,46,64,85,109,136,166,199,235,274,316,361,409,460,514,571,
%T A005448 631,694,760,829,901,976,1054,1135,1219,1306,1396,1489,1585,1684,1786,
%U A005448 1891,1999,2110,2224,2341,2461,2584,2710,2839,2971,3106,3244,3385,3529
%N A005448 Centered triangular numbers: 3n(n-1)/2 + 1.
%C A005448 These are Hogben's central polygonal numbers
%C A005448 2
%C A005448 .P
%C A005448 3 n
%C A005448 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
%C A005448 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
%C A005448 Binomial transform of (1,3,3,0,0,0,.....). - Paul Barry (pbarry(AT)wit.ie), 
               Jul 01 2003
%C A005448 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
%C A005448 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
%C A005448 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
%C A005448 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]
%C A005448 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]
%C A005448 Except for the first term, a(n)=3*n+a(n-1), (with a(1)=4) [From Vincenzo 
               Librandi (vincenzo.librandi(AT)tin.it), Oct 23 2009]
%D A005448 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A005448 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.
%D A005448 L. Hogben, Choice and Chance by Cardpack and Chessboard. Vol. 1, Chanticleer 
               Press, NY, 1950, p. 22.
%D A005448 R. Reed, The Lemming Simulation Problem, Math. in School, 3 (#6, Nov. 
               1974), 5-6.
%D A005448 B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral 
               clusters, Inorgan. Chem. 24 (1985), 4545-4558.
%H A005448 T. D. Noe, <a href="b005448.txt">Table of n, a(n) for n=1..1000</a>
%H A005448 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A005448 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative 
               Functions</a>
%H A005448 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A005448 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A005448 Clark Kimberling and John E. Brown, <a href="http://www.cs.uwaterloo.ca/
               journals/JIS/index.html">Partial Complements and Transposable Dispersions</
               a>, J. Integer Seqs., Vol. 7, 2004.
%H A005448 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               CenteredTriangularNumber.html">Link to a section of The World of 
               Mathematics.</a>
%H A005448 <a href="Sindx_Ce.html#CENTRALCUBE">Index entries for sequences related 
               to centered polygonal numbers</a>
%F A005448 Expansion of (1-x^3 )/(1-x)^4.
%F A005448 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
%F A005448 a(n) = 1+ sum (3*n) - Xavier Acloque Oct 25 2003
%F A005448 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
%F A005448 Euler transform of length 3 sequence [ 4, 0, -1]. - Michael Somos Sep 
               23 2006
%F A005448 a(1-n)=a(n). - Michael Somos Sep 23 2006
%F A005448 binomial(n+1,n-1)+binomial(n,n-2)+binomial(n-1,n-3). - Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), Sep 03 2006
%F A005448 Row sums of triangle A134482. - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Oct 27 2007
%F A005448 Narayana transform (A001263) * [1, 3, 0, 0, 0,...] - Gary W. Adamson 
               (qntmpkt(AT)yahoo.com), Dec 29 2007
%F A005448 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]
%F A005448 a(n) = A000217(n-1)*3 + 1 = A045943(n-1) + 1. [From Omar E. Pol (info(AT)polprimos.com), 
               Dec 27 2008]
%F A005448 a(n) = 3n(n+1)/2 + 1, with offset 0. [From Omar E. Pol (info(AT)polprimos.com), 
               Nov 08 2009]
%F A005448 a(n)=3*n+a(n-1)-3 (with a(1)=1) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Nov 08 2009]
%e A005448 For n=2, a(2)=3*2+1-3=4; n=3, a(3)=3*3+4-3=10; n=4, a(4)=3*4+10-3=19 
               [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 08 2009]
%p A005448 A005448 := n->(3*n^2+3*n+2)/2;
%p A005448 A005448:=-(1+z+z**2)/(z-1)**3; [Conjectured by S. Plouffe in his 1992 
               dissertation.]
%t A005448 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]
%o A005448 (PARI) {a(n)=3*(n^2-n)/2+1} /* Michael Somos Sep 23 2006 */
%Y A005448 Cf. A045943, A001844.
%Y A005448 Cf. A000292.
%Y A005448 Cf. A006003 - partial sums. Cf. A008585 - finite differences.
%Y A005448 Cf. A134482.
%Y A005448 Cf. A001263.
%Y A005448 Cf. A000217. [From Omar E. Pol (info(AT)polprimos.com), Dec 27 2008]
%Y A005448 Sequence in context: A152946 A025720 A022793 this_sequence A037040 A007077 
               A009895
%Y A005448 Adjacent sequences: A005445 A005446 A005447 this_sequence A005449 A005450 
               A005451
%K A005448 nonn,easy,nice,new
%O A005448 1,2
%A A005448 N. J. A. Sloane (njas(AT)research.att.com), R. K. Guy, Dec 12, 1974
%E A005448 More terms from Milan R. Janjic (agnus(AT)blic.net), Jul 30 2007