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 A002024 n appears n times; a(n) = floor(sqrt(2n) + 1/2). (Formerly M0250 N0089) 224
 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Integer inverse function of the triangular numbers A000217. The function trinv(n) = floor((1+sqrt(1+8n))/2), n>=0, gives the values 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, ..., that is, the same sequence with offset 0. - N. J. A. Sloane, Jun 21 2009 Array T(k,n) = n+k-1 read by antidiagonals. Eigensequence of the triangle = A001563. - Gary W. Adamson, Dec 29 2008 a(A169581(n)) = A038567(n). - Reinhard Zumkeller, Dec 02 2009 Can apparently also be defined via a(n+1)=b(n) for n>=2 where b(0)=b(1)=1 and b(n) = b(n-b(n-2))+1. Tested to be correct at least up to n<=150000. - José María Grau Ribas, Jun 10 2011 For any n >= 0, a(n+1) is the least integer m such that A000217(m)=m(m+1)/2 is larger than n. This is useful when enumerating representations of n as difference of triangular numbers; see also A234813. - M. F. Hasler, Apr 19 2014 Number of binary digits of A023758, i.e., a(n) = ceiling(log_2(A023758(n+2))). - Andres Cicuttin, Apr 29 2016 a(n) and A002260(n) give respectively the x(n) and y(n) coordinates of the sorted sequence of points in the integer lattice such that x(n) > 0, 0 < y(n) <= x(n), and min(x(n), y(n)) < max(x(n+1), y(n+1)) for n > 0. - Andres Cicuttin, Dec 25 2016 Partial sums (A060432) are given by S(n) = (-a(n)^3 + a(n)*(1+6n))/6. - Daniel Cieslinski, Oct 23 2017 REFERENCES E. S. Barbeau et al., Five Hundred Mathematical Challenges, Prob. 441, pp. 41, 194. MAA 1995. R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 97. K. Hardy & K. S. Williams, The Green Book of Mathematical Problems, p. 59, Solution to Prob. 14, Dover NY, 1985 R. Honsberger, Mathematical Morsels, pp. 133-4, MAA 1978. J. F. Hurley, Litton's Problematical Recreations, pp. 152; 313-4 Prob. 22, VNR Co. NY 1971 D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 1, p. 43. Raphael Schumacher, The self-counting identity, Fib. Quart., 55 (No. 2 2017), 157-167. N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 129. LINKS Franklin T. Adams-Watters, Table of n, a(n) for n = 1..5050 Jaegug Bae, Sungjin Choi, A generalization of a subset-sum-distinct sequence, J. Korean Math. Soc. 40 (2003), no. 5, 757--768. MR1996839 (2004d:05198). See b(n). H. T. Freitag and H. W. Gould, Solution to Problem 571, Math. Mag., 38 (1965), 185-187. H. T. Freitag (Proposer) and H. W. Gould (Solver), Problem 571: An Ordering of the Rationals, Math. Mag., 38 (1965), 185-187 [Annotated scanned copy] S. W. Golomb, Discrete chaos: sequences satisfying "strange" recursions, unpublished manuscript, circa 1990 [cached copy, with permission (annotated)] Henry W. Gould, Letters to N. J. A. Sloane, Oct 1973 and Jan 1974. Abraham Isgur, Vitaly Kuznetsov, and Stephen Tanny, A combinatorial approach for solving certain nested recursions with non-slow solutions, arXiv:1202.0276 [math.CO], 2012. Stanley Wu-Wei Liu, Closed form expressions for A002024(n) M. A. Nyblom, Some curious sequences involving floor and ceiling functions, Am. Math. Monthly 109 (#6, 200), 559-564. Boris Putievskiy, Transformations [Of] Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012. Raphael Schumacher, Extension of Summation Formulas involving Stirling series, arXiv:1605.09204 [math.NT], 2016. N. J. A. Sloane, Handwritten notes on Self-Generating Sequences, 1970 (note that A1148 has now become A005282) Michael Somos, Sequences used for indexing triangular or square arrays L. J. Upton, Letter to N. J. A. Sloane, May 22 1991 Eric Weisstein's World of Mathematics, Self-Counting Sequence FORMULA a(n) = floor( 1/2 + sqrt(2n) ). Also a(n) = ceiling((sqrt(1+8n)-1)/2). [See the Liu link for a large collection of explicit formulas. - N. J. A. Sloane, Oct 30 2019] a((k - 1 ) * k / 2 + i) = k for k > 0 and 0 < i <= k. - Reinhard Zumkeller, Aug 30 2001 a(n) = a(n - a(n-1)) + 1, with a(1)=1. - Ian M. Levitt (ilevitt(AT)duke.poly.edu), Aug 18 2002 a(n) = round(sqrt(2n)). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Nov 01 2002 T(n,k) = A003602(A118413(n,k)); = T(n,k) = A001511(A118416(n,k)). - Reinhard Zumkeller, Apr 27 2006 G.f.: x/(1-x)*Product_{k>0} (1-x^(2*k))/(1-x^(2*k-1)). - Vladeta Jovovic, Oct 06 2003 Equals A127899 * A004736. - Gary W. Adamson, Feb 09 2007 a(n) = sum{i=0..n-1, A010054(i)}. - Paolo P. Lava, Apr 02 2007 Sum_{i=1..n} Sum_{j=i..n+i-1} T(j,i) = A000578(n); Sum_{i=1..n} T(n,i) = A000290(n). - Reinhard Zumkeller, Jun 24 2007 a(n) + n = A014132(n). - Vincenzo Librandi, Jul 08 2010 a(n) = ceiling( -1/2 + sqrt(2n) ). - Branko Curgus, May 12 2009 We know that a(n)=round(sqrt(2n))=round(sqrt(2*n-1)); now exist exactly a and b greater than zero, that: 2n = 2+(a+b)^2 -(a+3*b), we have: a(n)=(a+b-1) in closed formula. - Fabio Civolani (civox(AT)tiscali.it), Feb 23 2010 A005318(n+1) = 2*A005318(n)-A205744(n), A205744(n) = A005318(A083920(n)), A083920(n) = n - a(n). - N. J. A. Sloane, Feb 11 2012 Expansion of psi(x) * x / (1 - x) in powers of x where psi() is a Ramanujan theta function. - Michael Somos, Mar 19 2014 G.f.: x/(1-x) * Product_{n>=1} (1 + x^n) * (1 - x^(2*n)). - Paul D. Hanna, Feb 27 2016 a(n) = 1 + Sum{i=1..n/2} ceiling(floor(2(n-1)/(i^2+i))/(2n)). - José de Jesús Camacho Medina, Jan 07 2017 a(n) = floor((sqrt(8*n-7)+1)/2). - Néstor Jofré, Apr 24 2017 a(n) = floor((A000196(8*n)+1)/2). - Pontus von Brömssen, Dec 10 2018 EXAMPLE From Clark Kimberling, Sep 16 2008: (Start) As a rectangular array, a northwest corner: 1 2 3 4 5 6 2 3 4 5 6 7 3 4 5 6 7 8 4 5 6 7 8 9 This is the weight array (cf. A144112) of A107985 (formatted as a rectangular array). (End) G.f. = x + 2*x^2 + 2*x^3 + 3*x^4 + 3*x^5 + 3*x^6 + 4*x^7 + 4*x^9 + 4*x^9 + 4*x^10+ ... MAPLE A002024 := n-> ceil((sqrt(1+8*n)-1)/2); seq(A002024(n), n=1..100); MATHEMATICA a = 1; a[n_] := a[n] = a[n - a[n - 1]] + 1 (* Branko Curgus, May 12 2009 *) Table[n, {n, 13}, {n}] // Flatten (* Robert G. Wilson v, May 11 2010 *) Table[PadRight[{}, n, n], {n, 15}]//Flatten (* Harvey P. Dale, Jan 13 2019 *) PROG /* The PARI functions t1, t2 can be used to read a triangular array T(n, k) (n >= 1, 1 <= k <= n) by rows from left to right: n -> T(t1(n), t2(n)). * The PARI functions t1, t3 can be used to read a triangular array T(n, k) (n >= 1, 1 <= k <= n) by rows from right to left: n -> T(t1(n), t3(n)). * The PARI functions t1, t4 can be used to read a triangular array T(n, k) (n >= 1, 0 <= k <= n-1) by rows from left to right: n -> T(t1(n), t4(n)). - Michael Somos, Aug 23, 2002 */ (PARI) t1(n)=floor(1/2+sqrt(2*n)) /* A002024 = this sequence */ (PARI) t2(n)=n-binomial(floor(1/2+sqrt(2*n)), 2) /* A002260(n-1) */ (PARI) t3(n)=binomial(floor(3/2+sqrt(2*n)), 2)-n+1 /* A004736 */ (PARI) t4(n)=n-1-binomial(floor(1/2+sqrt(2*n)), 2) /* A002260(n-1)-1 */ (PARI) A002024(n)=(sqrtint(n*8)+1)\2 \\ M. F. Hasler, Apr 19 2014 (PARI) a(n)=(sqrtint(8*n-7)+1)\2 (PARI) a(n)=my(k=1); while(binomial(k+1, 2)+1<=n, k++); k \\ R. J. Cano, Mar 17 2014 (Haskell) a002024 n k = a002024_tabl !! (n-1) !! (k-1) a002024_row n = a002024_tabl !! (n-1) a002024_tabl = iterate (\xs@(x:_) -> map (+ 1) (x : xs))  a002024_list = concat a002024_tabl a002024' = round . sqrt . (* 2) . fromIntegral -- Reinhard Zumkeller, Jul 05 2015, Feb 12 2012, Mar 18 2011 (Haskell) a002024_list = [1..] >>= \n -> replicate n n (Haskell) a002024 = (!!) \$ [1..] >>= \n -> replicate n n -- Sascha Mücke, May 10 2016 (MAGMA) [Floor(Sqrt(2*n) + 1/2): n in [1..80]]; // Vincenzo Librandi, Nov 19 2014 (Sage) [floor(sqrt(2*n) +1/2) for n in (1..80)] # G. C. Greubel, Dec 10 2018 CROSSREFS a(n+1) = 1+A003056(n), A022846(n)=a(n^2), a(n+1)=A002260(n)+A025581(n). Cf. A001462, A002262, A004736, A127899, A107985, A001563, A014132, A000194, A005145, A131507, A093995, A060432 (partial sums). A123578 is an essentially identical sequence. Sequence in context: A274093 A087847 A107436 * A123578 A087836 A087845 Adjacent sequences:  A002021 A002022 A002023 * A002025 A002026 A002027 KEYWORD nonn,easy,nice,tabl AUTHOR STATUS approved

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Last modified April 16 17:01 EDT 2021. Contains 343050 sequences. (Running on oeis4.)