Search: id:A001107
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%I A001107 M4690
%S A001107 0,1,10,27,52,85,126,175,232,297,370,451,540,637,742,855,976,1105,1242,
%T A001107 1387,1540,1701,1870,2047,2232,2425,2626,2835,3052,3277,3510,3751,4000,
%U A001107 4257,4522,4795,5076,5365,5662,5967,6280,6601,6930,7267,7612,7965,8326
%N A001107 10-gonal (or decagonal) numbers: 4n^2 - 3n.
%C A001107 Write 0,1,2,... in clockwise spiral; sequence gives numbers on negative 
               y axis.
%C A001107 Number of divisors of 48^n - J. Lowell (jhbubby(AT)mindspring.com), Aug 
               30 2008
%D A001107 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001107 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, 
               p. 189.
%D A001107 Bruce C. Berndt, Ramanujan's Notebooks, Part II, Springer; see p. 23.
%D A001107 S. M. Ellerstein, The square spiral, J. Recreational Mathematics 29 (#3, 
               1998) 188; 30 (#4, 1999-2000), 246-250.
%D A001107 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, 
               Reading, MA, 2nd ed., 1994, p. 99.
%H A001107 T. D. Noe, <a href="b001107.txt">Table of n, a(n) for n=0..1000</a>
%H A001107 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A001107 Emilio Apricena, <a href="a035608.png">A version of the Ulam spiral</
               a>
%H A001107 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=344">
               Encyclopedia of Combinatorial Structures 344</a>
%H A001107 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 A001107 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 A001107 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               DecagonalNumber.html">Link to a section of The World of Mathematics.</
               a>
%F A001107 G.f.: A(x) = x(1+7x)/(1-x)^3.
%F A001107 Partial sums of odd numbers 1 mod 8, i.e. 1, 1+9, 1+9+17, ... - Jon Perry 
               (perry(AT)globalnet.co.uk), Dec 18 2004
%F A001107 a(n)=n+8*A000217(n-1) - Floor van Lamoen (fvlamoen(AT)hotmail.com), Oct 
               14 2005
%F A001107 1^3 + 3^3*(n-1)/(n+1) + 5^3*[(n-1)(n-2)]/[(n+1)(n+2)] + 7^3*[(n-1)(n-2)(n-3)]/
               [(n+1)(n+2)(n+3)] + ... = n(4n-3) [Ramanujan]. - Neven Juric, Apr 
               15 2008
%F A001107 Starting (1, 10, 27, 52,...) = binomial transform of [1, 9, 8, 0, 0, 
               0,...] - Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 30 2008
%F A001107 a(n)=3a(n-1)-3a(n-2)+a(n-3), a(0)=0, a(1)=1, a(2)=10 [From Jaume Oliver 
               Lafont (joliverlafont(AT)gmail.com), Dec 02 2008]
%F A001107 Also, let Dec(n)=decagonal numbers, T(n)=triangular numbers, then Dec(n)=T(n)+7*T(n-1) 
               [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 28 2009]
%F A001107 a(n)=8*n+a(n-1)-15 (with a(1)=0) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Nov 12 2009]
%e A001107 Part of the spiral:
%e A001107 16 17 18 19 ...
%e A001107 15 4 5 6 ...
%e A001107 14 3 0 7 ...
%e A001107 13 2 1 8 ...
%e A001107 For n=2, a(2)=8*2+0-15=1; n=3, a(3)=8*3+1-15=10; n=4, a(4)=8*4+10-15=27 
               [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 12 2009]
%p A001107 A001107:=-(1+7*z)/(z-1)**3; [S. Plouffe in his 1992 dissertation.]
%p A001107 a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=2*a[n-1]-a[n-2]+8 od: seq(a[n], 
               n=0..46); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 18 
               2008
%t A001107 lst={};Do[AppendTo[lst, 4*n^2-3*n], {n, 0, 5!}];lst...and/or... s=0;lst={s};
               Do[s+=n+1;AppendTo[lst, s], {n, 0, 6!, 8}];lst [From Vladimir Orlovsky 
               (4vladimir(AT)gmail.com), Oct 25 2008]
%o A001107 (PARI) a(n)=4*n^2-3*n
%o A001107 (Other) sage: [crt(0,n,4,3 )*crt(1, n, 4,3 )/4 for n in xrange(0, 47)]# 
               [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 30 2009]
%Y A001107 Cf. A000217, A001106, A051682.
%Y A001107 Cf. A007585. a(n)=A033954(-n)=A074377(2n-1).
%Y A001107 Cf. A093565 ((8, 1) Pascal, column m=2). Partial sums of A017077.
%Y A001107 Sequences from spirals: A001107, A002939, A007742, A033951, A033952, 
               A033953, A033954, A033989, A033990, A033991, A002943, A033996, A033988.
%Y A001107 Sequence in context: A045177 A043887 A161450 this_sequence A103135 A008468 
               A119548
%Y A001107 Adjacent sequences: A001104 A001105 A001106 this_sequence A001108 A001109 
               A001110
%K A001107 nonn,easy,nice,new
%O A001107 0,3
%A A001107 N. J. A. Sloane (njas(AT)research.att.com).

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