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%I #362 Mar 30 2024 02:45:11
%S 1,9,25,49,81,121,169,225,289,361,441,529,625,729,841,961,1089,1225,
%T 1369,1521,1681,1849,2025,2209,2401,2601,2809,3025,3249,3481,3721,
%U 3969,4225,4489,4761,5041,5329,5625,5929,6241,6561,6889,7225,7569
%N Odd squares: a(n) = (2n+1)^2. Also centered octagonal numbers.
%C 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_, Jan 26 2008
%C Numbers n such that tau(n) is odd where tau(x) denotes the Ramanujan tau function (A000594). - _Benoit Cloitre_, May 01 2003
%C 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
%C Binomial transform of [1, 8, 8, 0, 0, 0, ...]; Narayana transform (A001263) of [1, 8, 0, 0, 0, ...]. - _Gary W. Adamson_, Dec 29 2007
%C 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
%C 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
%C First quadrisection of A061038: A061038(4n). - _Paul Curtz_, Oct 26 2008
%C Equals the triangular numbers convolved with [1, 6, 1, 0, 0, 0, ...]. - _Gary W. Adamson_ & _Alexander R. Povolotsky_, May 29 2009
%C First differences: A008590(n) = a(n) - a(n-1) for n>0. - _Reinhard Zumkeller_, Nov 08 2009
%C Central terms of the triangle in A176271; cf. A000466, A053755. - _Reinhard Zumkeller_, Apr 13 2010
%C 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
%C Appear as numerators in the non-simple continued fraction expansion of Pi-3: Pi-3 = K_{k>=1} (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
%C Ulam's spiral (SE spoke). - _Robert G. Wilson v_, Oct 31 2011
%C 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
%C 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
%C Also: Odd numbers which have an odd sum of divisors (= sigma = A000203). - _M. F. Hasler_, Feb 23 2013
%C Consider primitive Pythagorean triangles (a^2 + b^2 = c^2, gcd(a, b) = 1) with hypotenuse c (A020882) and respective even leg b (A231100); sequence gives values c-b, sorted with duplicates removed. - _K. G. Stier_, Nov 04 2013
%C For n>1 a(n) is twice the area of the irregular quadrilateral created by the points ((n-2)*(n-1),(n-1)*n/2), ((n-1)*n/2,n*(n+1)/2), ((n+1)*(n+2)/2,n*(n+1)/2), and ((n+2)*(n+3)/2,(n+1)*(n+2)/2). - _J. M. Bergot_, May 27 2014
%C Number of pairs (x, y) of Z^2, such that max(abs(x), abs(y)) <= n. - _Michel Marcus_, Nov 28 2014
%C Except for a(1)=4, the number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 737", based on the 5-celled von Neumann neighborhood. - _Robert Price_, May 23 2016
%C a(n) is the sum of 2n+1 consecutive numbers, the first of which is n+1. - _Ivan N. Ianakiev_, Dec 21 2016
%C a(n) is the number of 2 X 2 matrices with all elements in {0..n} with determinant = 2*permanent. - _Indranil Ghosh_, Dec 25 2016
%C Engel expansion of Pi*StruveL_0(1)/2 where StruveL_0(1) is A197037. - _Benedict W. J. Irwin_, Jun 21 2018
%C Consider all Pythagorean triples (X,Y,Z=Y+1) ordered by increasing Z; the segments on the hypotenuse {p = a(n)/A001844(n), q = A060300(n)/A001844(n) = A001844(n) - p} and their ratio p/q = a(n)/A060300(n) are irreducible fractions in Q\Z. X values are A005408, Y values are A046092, Z values are A001844. - _Ralf Steiner_, Feb 25 2020
%C a(n) is the number of large or small squares that are used to tile primitive squares of type 2 (A344332). - _Bernard Schott_, Jun 03 2021
%C Also, positive odd integers with an odd number of odd divisors (for similar sequence with 'even', see A348005). - _Bernard Schott_, Nov 21 2021
%C a(n) is the least odd number k = x + y, with 0 < x < y, such that there are n distinct pairs (x,y) for which x*y/k is an integer; for example, a(2) = 25 and the two corresponding pairs are (5,20) and (10,15). The similar sequence with 'even' is A016742 (see Comment of Jan 26 2018). - _Bernard Schott_, Feb 24 2023
%D L. Lorentzen and H. Waadeland, Continued Fractions with Applications, North-Holland 1992, p. 586.
%H Paolo Xausa, <a href="/A016754/b016754.txt">Table of n, a(n) for n = 0..9999</a> (terms 0..1000 from T. D. Noe)
%H Jeremiah Bartz, Bruce Dearden and Joel Iiams, <a href="https://arxiv.org/abs/1810.07895">Classes of Gap Balancing Numbers</a>, arXiv:1810.07895 [math.NT], 2018.
%H Bruce C. Berndt and Ken Ono, <a href="http://www.mat.univie.ac.at/~slc/wpapers/s42berndt.html">Ramanujan's unpublished manuscript on the partition and tau functions with proofs and commentary</a>, Séminaire Lotharingien de Combinatoire, B42c (1999), 63 pp.
%H John Elias, <a href="/A016754/a016754.png">Illustration: 8-fold Square Progression of Ulam's Spiral</a>
%H Milan Janjic, <a href="https://pmf.unibl.org/janjic/">Two Enumerative Functions</a>; also on <a href="https://www.semanticscholar.org/paper/Two-Enumerative-Functions-Janjic/801b6b226bfe1d6b002fb4946c3957d7052132bd?p2df">Semantic Scholar</a>.
%H Scientific American, <a href="/A244677/a244677.jpg">Cover of the March 1964 issue</a>.
%H Amelia Carolina Sparavigna, <a href="http://doi.org/10.5281/zenodo.3247003">Groupoids of OEIS A002378 and A016754 Numbers (oblong and odd square numbers)</a>, Politecnico di Torino (Italy, 2019).
%H Leo Tavares, <a href="/A016754/a016754.jpg">Illustration: Diamond Triangles</a>
%H Leo Tavares, <a href="/A016754/a016754_1.jpg">Illustration: Diamond Stars</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MooreNeighborhood.html">Moore Neighborhood</a>.
%H <a href="/index/Ce#CENTRALCUBE">Index entries for sequences related to centered polygonal numbers</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = 1 + Sum_{i=1..n} 8*i = 1 + 8*A000217(n). - Xavier Acloque, Jan 21 2003; _Zak Seidov_, May 07 2006; _Robert G. Wilson v_, Dec 29 2010
%F O.g.f.: (1+6*x+x^2)/(1-x)^3. - _R. J. Mathar_, Jan 11 2008
%F a(n) = 4*n*(n + 1) + 1 = 4*n^2 + 4*n + 1. - _Artur Jasinski_, Mar 27 2008
%F Sum_{n>=0} 1/a(n) = Pi^2/8. - _Jaume Oliver Lafont_, Mar 07 2009
%F a(n) = A000290(A005408(n)). - _Reinhard Zumkeller_, Nov 08 2009
%F a(n) = a(n-1) + 8*n with n>0, a(0)=1. - _Vincenzo Librandi_, Aug 01 2010
%F a(n) = A033951(n) + n. - _Reinhard Zumkeller_, May 17 2009
%F a(n) = A033996(n) + 1. - _Omar E. Pol_, Oct 03 2011
%F a(n) = (A005408(n))^2. - _Zak Seidov_, Nov 29 2011
%F From _George F. Johnson_, Sep 05 2012: (Start)
%F a(n+1) = a(n) + 4 + 4*sqrt(a(n)).
%F a(n-1) = a(n) + 4 - 4*sqrt(a(n)).
%F a(n+1) = 2*a(n) - a(n-1) + 8.
%F a(n+1) = 3*a(n) - 3*a(n-1) + a(n-2).
%F (a(n+1) - a(n-1))/8 = sqrt(a(n)).
%F a(n+1)*a(n-1) = (a(n)-4)^2.
%F a(n) = 2*A046092(n) + 1 = 2*A001844(n) - 1 = A046092(n) + A001844(n).
%F Limit_{n -> oo} a(n)/a(n-1) = 1. (End)
%F a(n) = binomial(2*n+2,2) + binomial(2*n+1,2). - _John Molokach_, Jul 12 2013
%F E.g.f.: (1 + 8*x + 4*x^2)*exp(x). - _Ilya Gutkovskiy_, May 23 2016
%F a(n) = A101321(8,n). - _R. J. Mathar_, Jul 28 2016
%F Product_{n>=1} A033996(n)/a(n) = Pi/4. - _Daniel Suteu_, Dec 25 2016
%F a(n) = A014105(n) + A000384(n+1). - _Bruce J. Nicholson_, Nov 11 2017
%F a(n) = A003215(n) + A002378(n). - _Klaus Purath_, Jun 09 2020
%F From _Amiram Eldar_, Jun 20 2020: (Start)
%F Sum_{n>=0} a(n)/n! = 13*e.
%F Sum_{n>=0} (-1)^(n+1)*a(n)/n! = 3/e. (End)
%F Sum_{n>=0} (-1)^n/a(n) = A006752. - _Amiram Eldar_, Oct 10 2020
%F From _Amiram Eldar_, Jan 28 2021: (Start)
%F Product_{n>=0} (1 + 1/a(n)) = cosh(Pi/2).
%F Product_{n>=1} (1 - 1/a(n)) = Pi/4 (A003881). (End)
%F From _Leo Tavares_, Nov 24 2021: (Start)
%F a(n) = A014634(n) - A002943(n). See Diamond Triangles illustration.
%F a(n) = A003154(n+1) - A046092(n). See Diamond Stars illustration. (End)
%F From _Peter Bala_, Mar 11 2024: (Start)
%F Sum_{k = 1..n+1} 1/(k*a(k)*a(k-1)) = 1/(9 - 3/(17 - 60/(33 - 315/(57 - ... - n^2*(4*n^2 - 1)/((2*n + 1)^2 + 2*2^2 ))))).
%F 3/2 - 2*log(2) = Sum_{k >= 1} 1/(k*a(k)*a(k-1)) = 1/(9 - 3/(17 - 60/(33 - 315/(57 - ... - n^2*(4*n^2 - 1)/((2*n + 1)^2 + 2*2^2 - ... ))))).
%F Row 2 of A142992. (End)
%F From _Peter Bala_, Mar 26 2024: (Start)
%F 8*a(n) = (2*n + 1)*(a(n+1) - a(n-1)).
%F Sum_{n >= 0} (-1)^n/(a(n)*a(n+1)) = 1/2 - Pi/8 = 1/(9 + (1*3)/(8 + (3*5)/(8 + ... + (4*n^2 - 1)/(8 + ... )))). For the continued fraction use Lorentzen and Waadeland, p. 586, equation 4.7.9 with n = 1. Cf. A057813. (End)
%t A016754[nmax_]:=Range[1,2nmax+1,2]^2;A016754[100] (* _Paolo Xausa_, Mar 05 2023 *)
%o (PARI) A016754(n)=(n<<1+1)^2 \\ _Charles R Greathouse IV_, Jun 16 2011, corrected and edited by _M. F. Hasler_, Apr 11 2023
%o (Haskell)
%o a016754 n = a016754_list !! n
%o a016754_list = scanl (+) 1 $ tail a008590_list
%o -- _Reinhard Zumkeller_, Apr 02 2012
%o (Maxima) A016754(n):=(n+n+1)^2$
%o makelist(A016754(n),n,0,20); /* _Martin Ettl_, Nov 12 2012 */
%o (Magma) [n^2: n in [1..100 by 2]]; // _Vincenzo Librandi_, Jan 03 2017
%o (Python)
%o def A016754(n): return ((n<<1)|1)**2 # _Chai Wah Wu_, Jul 06 2023
%Y Cf. A000290, A000384, A001263, A001539, A001844, A003881, A005408, A006752, A014105, A016742, A016802, A016814, A016826, A016838, A033996, A046092, A060300, A138393, A167661, A167700.
%Y Cf. A000447 (partial sums).
%Y Cf. A005917, A344330, A344332.
%Y Cf. A348005.
%Y Partial sums of A022144.
%Y Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
%Y Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.
%Y Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.
%Y Cf. A014634, A003154.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
%E Additional description from _Terrel Trotter, Jr._, Apr 06 2002
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