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 A016754 Odd squares: a(n) = (2n+1)^2. Also centered octagonal numbers. 257
 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)
 OFFSET 0,2 COMMENTS 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 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 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 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 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 Number of pairs (x, y) of Z^2, such that max(abs(x), abs(y)) <= n. - Michel Marcus, Nov 28 2014 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 a(n) is the sum of 2n+1 consecutive numbers, the first of which is n+1. - Ivan N. Ianakiev, Dec 21 2016 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 Engel expansion of Pi*StruveL_0(1)/2 where StruveL_0(1) is A197037. - Benedict W. J. Irwin, Jun 21 2018 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 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 Also, positive odd integers with an odd number of odd divisors (for similar sequence with 'even', see A348005). - Bernard Schott, Nov 21 2021 LINKS T. D. Noe, Table of n, a(n) for n = 0..1000 Jeremiah Bartz, Bruce Dearden and Joel Iiams, Classes of Gap Balancing Numbers, arXiv:1810.07895 [math.NT], 2018. Bruce C. Berndt and Ken Ono, Ramanujan's unpublished manuscript on the partition and tau functions with proofs and commentary, Séminaire Lotharingien de Combinatoire, B42c (1999), 63 pp. Milan Janjic, Two Enumerative Functions; also on Semantic Scholar. Scientific American, Cover of the March 1964 issue. Amelia Carolina Sparavigna, Groupoids of OEIS A002378 and A016754 Numbers (oblong and odd square numbers), Politecnico di Torino (Italy, 2019). Leo Tavares, Illustration: Diamond Triangles Leo Tavares, Illustration: Diamond Stars Eric Weisstein's World of Mathematics, Moore Neighborhood. Index entries for linear recurrences with constant coefficients, signature (3,-3,1). FORMULA 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 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 Sum_{n>=0} 1/a(n) = Pi^2/8. - Jaume Oliver Lafont, Mar 07 2009 a(n) = A000290(A005408(n)). - Reinhard Zumkeller, Nov 08 2009 a(n) = a(n-1) + 8*n 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. - Zak Seidov, Nov 29 2011 From George F. Johnson, Sep 05 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_{n -> infinity} a(n)/a(n-1) = 1. (End) a(n) = binomial(2*n+2,2) + binomial(2*n+1,2). - John Molokach, Jul 12 2013 E.g.f.: (1 + 8*x + 4*x^2)*exp(x). - Ilya Gutkovskiy, May 23 2016 a(n) = A101321(8,n). - R. J. Mathar, Jul 28 2016 Product_{n>=1} A033996(n)/a(n) = Pi/4. - Daniel Suteu, Dec 25 2016 a(n) = A014105(n) + A000384(n+1). - Bruce J. Nicholson, Nov 11 2017 a(n) = A003215(n) + A002378(n). - Klaus Purath, Jun 09 2020 From Amiram Eldar, Jun 20 2020: (Start) Sum_{n>=0} a(n)/n! = 13*e. Sum_{n>=0} (-1)^(n+1)*a(n)/n! = 3/e. (End) Sum_{n>=0} (-1)^n/a(n) = A006752. - Amiram Eldar, Oct 10 2020 From Amiram Eldar, Jan 28 2021: (Start) Product_{n>=0} (1 + 1/a(n)) = cosh(Pi/2). Product_{n>=1} (1 - 1/a(n)) = Pi/4 (A003881). (End) From Leo Tavares, Nov 24 2021: (Start) a(n) = A014634(n) - A002943(n). See Diamond Triangles illustration. a(n) = A003154(n+1) - A046092(n). See Diamond Stars illustration. (End) MATHEMATICA nterms=100; Table[(2n+1)^2, {n, 0, nterms-1}] (* Paolo Xausa, Nov 24 2021 *) PROG (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 */ (Magma) [n^2: n in [1..100 by 2]]; // Vincenzo Librandi, Jan 03 2017 CROSSREFS Cf. A000290, A000384, A001263, A001539, A001844, A003881, A005408, A006752, A014105, A016742, A016802, A016814, A016826, A016838, A033996, A046092, A060300, A138393, A167661, A167700. Cf. A000447 (partial sums). Cf. A005917, A344330, A344332. Cf. A348005. Partial sums of A022144. Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951. 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. 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. Cf. A014634, A003154. Sequence in context: A325701 A113745 A348742 * A110487 A259417 A030156 Adjacent sequences:  A016751 A016752 A016753 * A016755 A016756 A016757 KEYWORD nonn,easy AUTHOR EXTENSIONS Additional description from Terrel Trotter, Jr., Apr 06 2002 STATUS approved

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Last modified October 2 06:43 EDT 2022. Contains 357191 sequences. (Running on oeis4.)