Search: id:A005448 Results 1-1 of 1 results found. %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, Table of n, a(n) for n=1..1000 %H A005448 Index entries for sequences related to linear recurrences with constant coefficients %H A005448 Milan Janjic, Two Enumerative Functions %H 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. %H A005448 S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992. %H A005448 Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004. %H A005448 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics. %H A005448 Index entries for sequences related to centered polygonal numbers %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 Search completed in 0.002 seconds