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A016754
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Odd squares: (2n+1)^2. Also centered octagonal numbers.
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86
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1, 9, 25, 49, 81, 121, 169, 225, 289, 361, 441, 529, 625, 729, 841, 961, 1089, 1225, 1369, 1521, 1681, 1849, 2025, 2209, 2401, 2601, 2809, 3025, 3249, 3481, 3721, 3969, 4225, 4489, 4761, 5041, 5329, 5625, 5929, 6241, 6561, 6889, 7225, 7569
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,2
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COMMENTS
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The brown rat (rattus norwegicus) breeds very quickly. It can give birth to other rats 7 times a year, starting at the age of three months. The average number of pups is 8. The present sequence gives the total number of rats, when the intervals are 12/7 of a year and a young rat starts having offspring at 24/7 of a year. - Hans Isdahl (hansi(AT)nordtroms.net), Jan 26 2008
Numbers n such that tau(n) is odd where tau(x) denotes the Ramanujan tau function (A000594). - Benoit Cloitre, May 01 2003
If Y is a fixed 2-subset of a (2n+1)-set X then a(n-1) is the number of 3-subsets of X intersecting Y. - Milan Janjic, Oct 21 2007
Binomial transform of [1, 8, 8, 0, 0, 0,...]; Narayana transform (A001263) of [1, 8, 0, 0, 0,...]. - Gary W. Adamson, Dec 29 2007
All terms of this sequence are of the form 8k+1. For numbers 8k+1 which aren't squares see A138393. Numbers 8k+1 are squares iff k is a triangular number from A000217. And squares have form 4n(n+1)+1. - Artur Jasinski, Mar 27 2008
Sequence arises from reading the line from 1, in the direction 1, 25,... and the line from 9, in the direction 9, 49,..., in the square spiral whose vertices are the squares A000290. - Omar E. Pol, May 24 2008
First quadrisection of A061038: A061038(4n). [Paul Curtz, Oct 26 2008]
Sum_{n>=0} 1/a(n) = Pi^2/8. [Jaume Oliver Lafont, Mar 07 2009]
Equals the triangular numbers convolved with [1, 6, 1, 0, 0, 0,...]. [Gary W. Adamson & Alexander R. Povolotsky, May 29 2009]
First differences: A008590(n) = a(n) - a(n-1) for n>0. [Reinhard Zumkeller, Nov 08 2009]
Central terms of the triangle in A176271; cf. A000466, A053755. [Reinhard Zumkeller, Apr 13 2010]
Odd numbers with odd abundance. Odd numbers with even abundance are in A088828. Even numbers with odd abundance are in A088827. Even numbers with even abundance are in A088829. - Jaroslav Krizek, May 07 2011.
Appear as numerators in the non-simple continued fraction expansion of Pi-3: Pi-3 = K_(k=1)^infinity (1-2*k)^2/6 = 1/(6+9/(6+25/(6+49/(6+...)))), see also the comment in A007509. - Alexander R. Povolotsky, Oct 12 2011
Ulam's spiral (SE spoke). - Robert G. Wilson v, Oct 31 2011
All terms end in 1, 5 or 9. Modulo 100, all terms are among { 1, 9, 21, 25, 29, 41, 49, 61, 69, 81, 89 }. - M. F. Hasler, Mar 19 2012
Right edge of both triangles A214604 and A214661: a(n) = A214604(n+1,n+1) = A214661(n+1,n+1). - Reinhard Zumkeller, Jul 25 2012
Also: Odd numbers which have an odd sum of divisors (= sigma = A000203). - M. F. Hasler, Feb 23 2013
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..1000
Index entries for sequences related to linear recurrences with constant coefficients
Milan Janjic, Two Enumerative Functions
B. C. Berndt & K. Ono, Ramanujan's unpublished manuscript on the partition and tau functions with proofs and commentary
Eric Weisstein's World of Mathematics, Moore Neighborhood
Index entries for sequences related to centered polygonal numbers
Index to sequences with linear recurrences with constant coefficients, signature (3,-3,1)
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FORMULA
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a(n) = 1 + sum(8*i, i=1..n) = 1+8*A000217(n). - Xavier Acloque, Jan 21 2003; Zak Seidov, May 07 2006; Robert G. Wilson v, Dec 29 2010
O.g.f.: (1+6*x+x^2)/(1-x)^3 . - R. J. Mathar, Jan 11 2008
a(n) = 4*n*(n + 1) + 1 = 4*n^2 + 4*n + 1. - Artur Jasinski, Mar 27 2008
a(n) = A000290(A005408(n)). - Reinhard Zumkeller, Nov 08 2009
a(n) = 8*n+a(n-1) with n>0, a(0)=1. - Vincenzo Librandi, Aug 01 2010
a(n) = A033951(n) + n. - Reinhard Zumkeller, May 17 2009
a(n) = A033996(n) + 1. - Omar E. Pol, Oct 03 2011
a(n) = (A005408(n))^2. - Moshe Levin, Nov 29 2011
Contribution from George F. Johnson, Sept 5 2012: (Start)
a(n+1) = a(n) + 4 + 4*sqrt(a(n)); a(n-1) = a(n) + 4 - 4*sqrt(a(n)).
a(n+1) = 2*a(n) - a(n-1) + 8; a(n+1) = 3*a(n) - 3*a(n-1) + a(n-2).
(a(n+1) - a(n-1))/8 = sqrt(a(n)); a(n+1)*a(n-1) = (a(n)-4)^2.
a(n) = 2*A046092(n) + 1 = 2*A001844(n) - 1 = A046092(n) + A001844(n).
Limit as n -> infinity of a(n)/a(n-1) = 1.
(End)
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MATHEMATICA
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Table[4n*(n + 1) + 1, {n, 0, 500}] - Artur Jasinski, Mar 27 2008
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PROG
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(PARI) (n+n+1)^2 \\ Charles R Greathouse IV, Jun 16 2011
(Haskell)
a016754 n = a016754_list !! n
a016754_list = scanl (+) 1 $ tail a008590_list
-- Reinhard Zumkeller, Apr 02 2012
(Maxima) A016754(n):=(n+n+1)^2$
makelist(A016754(n), n, 0, 20); /* Martin Ettl, Nov 12 2012 */
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CROSSREFS
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Cf. A005408, A033996, A001263, A138393, A000290, A001539, A016742, A016802, A016814, A016826, A016838.
Cf. A167661, A167700.
Cf. A000447 (partial sums).
Sequence in context: A075026 A113659 A113745 * A110487 A030156 A192775
Adjacent sequences: A016751 A016752 A016753 * A016755 A016756 A016757
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KEYWORD
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nonn,easy,changed
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Additional description from Terry Trotter, Apr 06 2002.
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STATUS
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approved
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