%I A000566 M4358 N1826
%S A000566 0,1,7,18,34,55,81,112,148,189,235,286,342,403,469,540,616,697,783,874,
               970,
%T A000566 1071,1177,1288,1404,1525,1651,1782,1918,2059,2205,2356,2512,2673,2839,
%U A000566 3010,3186,3367,3553,3744,3940,4141,4347,4558,4774,4995,5221,5452,5688
%N A000566 Heptagonal numbers (or 7-gonal numbers) n(5n-3)/2.
%C A000566 Binomial transform of (0,1,5,0,0,0,...) Binomial transform is A084899. 
               - Paul Barry (pbarry(AT)wit.ie), Jun 10 2003
%C A000566 Also the partial sums of A016861, a zero added in front; therefore a(n) 
               = n (mod 5). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 19 
               2008
%C A000566 a(n+1) = A153126(n) + n mod 2; a(2*n+1)=A033571(n); a(2*(n+1))=A153127(n)+1. 
               [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Dec 20 
               2008]
%C A000566 Also, let Hep(n) = heptagonal numbers, T(n)= triangular numbers, then 
               Hep(n)= T(n)+4*T(n-1) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Jan 28 2009]
%C A000566 Comment from Ken Rosenbaum, Dec 02 2009: if you multiply the terms of 
               this sequence by 40 and add 9, you get A017354, which is the list 
               of squares of all whole numbers ending in 7 (this is easy to prove).
%D A000566 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, 
               p. 189.
%D A000566 L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 
               256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see 
               vol. 2, p. 2.
%D A000566 B. S. Rao, Heptagonal numbers in the Pell sequence and Diophantine equations 
               2x^2 = y^2(5y-3)^2 +- 2, Fib. Quarterly, 43 (2005), 194-201.
%D A000566 B. S. Rao, Heptagonal numbers in the associated Pell sequence ..., Fib. 
               Quarterly, 43 (2005), 302-306.
%D A000566 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000566 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A000566 T. D. Noe, <a href="b000566.txt">Table of n, a(n) for n=0..1000</a>
%H A000566 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A000566 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=341">
               Encyclopedia of Combinatorial Structures 341</a>
%H A000566 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 A000566 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 A000566 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               HeptagonalNumber.html">Link to a section of The World of Mathematics.</
               a>
%F A000566 G.f.: x(1+4x)/(1-x)^3; a(n)=C(n, 1)+5C(n, 2). - Paul Barry (pbarry(AT)wit.ie), 
               Jun 10 2003
%F A000566 a(n)=sum{k=1..n, 4n-3k}. - Paul Barry (pbarry(AT)wit.ie), Sep 06 2005
%F A000566 a(n)=n+5*A000217(n-1) - Floor van Lamoen (fvlamoen(AT)hotmail.com), Oct 
               14 2005
%F A000566 Row sums of triangle A131413 - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Jul 08 2007
%F A000566 Sequence starting (1, 7, 18, 34,...) = binomial transform of (1, 6, 5, 
               0, 0, 0,...). Also row sums of triangle A131896. - Gary W. Adamson 
               (qntmpkt(AT)yahoo.com), Jul 24 2007
%F A000566 a(n)=3a(n-1)-3a(n-2)+a(n-3), a(0)=0, a(1)=1, a(2)=7 [From Jaume Oliver 
               Lafont (joliverlafont(AT)gmail.com), Dec 02 2008]
%F A000566 a(n)=5*n+a(n-1)-9 (with a(1)=0) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Nov 12 2009]
%e A000566 For n=2, a(2)=5*2+0-9=1; n=3, a(3)=5*3+1-9=7; n=4, a(4)=5*4+7-9=18 [From 
               Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 12 2009]
%p A000566 A000566:=-(1+4*z)/(z-1)**3; [S. Plouffe in his 1992 dissertation.]
%p A000566 a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=2*a[n-1]-a[n-2]+5 od: seq(a[n], 
               n=0..48); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 18 
               2008
%t A000566 s=0;lst={s};Do[s+=n+1;AppendTo[lst, s], {n, 0, 6!, 5}];lst [From Vladimir 
               Orlovsky (4vladimir(AT)gmail.com), Oct 25 2008]
%Y A000566 Cf. A014637, A014640, A014773, A014792, A069099.
%Y A000566 a(n)= A093562(n+1, 2), (5, 1)-Pascal column.
%Y A000566 Cf. A131413.
%Y A000566 Cf. A131896.
%Y A000566 Cf. A134483.
%Y A000566 Cf. A000217, A000384, A000567.
%Y A000566 Sequence in context: A049532 A156619 A033537 this_sequence A133673 A023166 
               A002764
%Y A000566 Adjacent sequences: A000563 A000564 A000565 this_sequence A000567 A000568 
               A000569
%K A000566 nonn,easy,nice,new
%O A000566 0,3
%A A000566 N. J. A. Sloane (njas(AT)research.att.com).