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 A000302 Powers of 4: a(n) = 4^n. (Formerly M3518 N1428) 493
 1, 4, 16, 64, 256, 1024, 4096, 16384, 65536, 262144, 1048576, 4194304, 16777216, 67108864, 268435456, 1073741824, 4294967296, 17179869184, 68719476736, 274877906944, 1099511627776, 4398046511104, 17592186044416, 70368744177664, 281474976710656 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Same as Pisot sequences E(1, 4), L(1, 4), P(1, 4), T(1, 4). See A008776 for definitions of Pisot sequences. The convolution square root of this sequence is A000984, the central binomial coefficients: C(2n,n). - T. D. Noe, Jun 11 2002 With P(n) being the number of integer partitions of n, p(i) as the number of parts of the i-th partition of n, d(i) as the number of different parts of the i-th partition of n, m(i, j) the multiplicity of the j-th part of the i-th partition of n, one has a(n) = Sum_{i = 1..P(n)} p(i)!/(Product_{j = 1..d(i)} m(i, j)!) * 2^(n-1). - Thomas Wieder, May 18 2005 Sums of rows of the triangle in A122366. - Reinhard Zumkeller, Aug 30 2006 A000005(a(n)) = A005408(n+1). - Reinhard Zumkeller, Mar 04 2007 Hankel transform of A076035. - Philippe Deléham, Feb 28 2009 a(n) = A159991(n)/A001024(n) = A047653(n) + A181765(n). A160700(a(n)) = A010685(n). - Reinhard Zumkeller, May 02 2009 Equals the Catalan sequence: (1, 1, 2, 5, 14, ...), convolved with A032443: (1, 3, 11, 42, ...). - Gary W. Adamson, May 15 2009 a(n) = A188915(A006127(n)). - Reinhard Zumkeller, Apr 14 2011 Sum of coefficients of expansion of (1 + x + x^2 + x^3)^n. a(n) is number of compositions of natural numbers into n parts less than 4. For example, a(2) = 16 since there are 16 compositions of natural numbers into 2 parts less than 4. The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n >= 1, a(n) equals the number of 4-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011 Squares in A002984. - Reinhard Zumkeller, Dec 28 2011 a(n) is the minimum number whose arithmetic derivative is n times the number itself: 1' = 0 = 0 * 1; 4' = 4 = 1 * 4; 16' = 32 = 2 * 16; 64' = 192 = 3 * 64, etc. - Paolo P. Lava, Feb 21 2012 Row sums of Pascal's triangle using the rule that going left increases the value by a factor of k = 3. For example, the first three rows are {1}, {3, 1}, and {9, 6, 1}. Using this rule gives row sums as (k+1)^n. - Jon Perry, Oct 11 2012 Sum_{k = 0..n} binomial(2*k + l, k)*binomial(2*(n - k) - l, n - k) for every real number l. - Rui Duarte and António Guedes de Oliveira, Feb 16 2013 First differences of A002450. - Omar E. Pol, Feb 20 2013 Sum of all peak heights in Dyck paths of semilength n+1. - David Scambler, Apr 22 2013 Powers of 4 exceed powers of 2 by A020522 which is the m-th oblong number A002378(m), m being the n-th Mersenne number A000225(n); hence, we may write, a(n) = A000079(n) + A002378(A000225(n)). - Lekraj Beedassy, Jan 17 2014 a(n) is equal to 1 plus the sum for 0 < k < 2^n of the numerators and denominators of the reduced fractions k/2^n. - J. M. Bergot, Jul 13 2015 Binomial transform of A000244. - Tony Foster III, Oct 01 2016 From Ilya Gutkovskiy, Oct 01 2016: (Start) Number of nodes at level n regular 4-ary tree. Partial sums of A002001. (End) Satisfies Benford's law [Berger-Hill, 2011]. - N. J. A. Sloane, Feb 08 2017 Also the number of connected dominating sets in the (n+1)-barbell graph. - Eric W. Weisstein, Jun 29 2017 Side length of the cells at level n in a pyramid scheme where a square grid is decomposed into overlapping 2 X 2 blocks (cf. Kropatsch, 1985). - Felix Fröhlich, Jul 04 2019 a(n-1) is the number of 3-compositions of n; see Hopkins & Ouvry reference. - Brian Hopkins, Aug 15 2020 REFERENCES R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd. ed., 1994, eq. (5.39), p. 187. D. Phulara and L. W. Shapiro, Descendants in ordered trees with a marked vertex, Congressus Numerantium, 205 (2011), 121-128. 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. 55. LINKS T. D. Noe, Table of n, a(n) for n = 0..100 Arno Berger and Theodore P. Hill, Benford's law strikes back: no simple explanation in sight for mathematical gem, The Mathematical Intelligencer 33.1 (2011): 85-91. Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão, and Graça Tomaz, Combinatorial Identities Associated with a Multidimensional Polynomial Sequence, J. Int. Seq., Vol. 21 (2018), Article 18.7.4. P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5. R. Duarte and A. G. de Oliveira, Short note on the convolution of binomial coefficients, arXiv preprint arXiv:1302.2100 [math.CO], 2013 and J. Int. Seq. 16 (2013) #13.7.6. Roudy El Haddad, Recurrent Sums and Partition Identities, arXiv:2101.09089 [math.NT], 2021. Roudy El Haddad, A generalization of multiple zeta value. Part 1: Recurrent sums. Notes on Number Theory and Discrete Mathematics, 28(2), 2022, 167-199, DOI: 10.7546/nntdm.2022.28.2.167-199. R. K. Guy, Letter to N. J. A. Sloane Brian Hopkins and Stéphane Ouvry, Combinatorics of Multicompositions, arXiv:2008.04937 [math.CO], 2020. INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 8 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 269 Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2. Tanya Khovanova, Recursive Sequences Craig Knecht, Number of tilings for a 6 sphinx tile repetitive unit. Walter G. Kropatsch, A pyramid that grows by powers of 2, Pattern Recognition Letters, Vol. 3, No. 5 (1985), 315-322 [Subscription required]. Mircea Merca, A Note on Cosine Power Sums J. Integer Sequences, Vol. 15 (2012), Article 12.5.3. Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009. Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992 Robert Price, Comments on A000302 concerning Elementary Cellular Automata, Feb 26 2016. Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1. Robert Schneider, Partition zeta functions, Research in Number Theory, 2(1):9., 2016. Paul K. Stockmeyer, The Pascal Rhombus and the Stealth Configuration, arXiv preprint arXiv:1504.04404 [math.CO], 2015. Eric Weisstein's World of Mathematics, Barbell Graph Eric Weisstein's World of Mathematics, Cantor Dust Eric Weisstein's World of Mathematics, Connected Dominating Set Eric Weisstein's World of Mathematics, Elementary Cellular Automaton S. Wolfram, A New Kind of Science Index entries for linear recurrences with constant coefficients, signature (4). FORMULA a(n) = 4^n. a(0) = 1; a(n) = 4*a(n-1). G.f.: 1/(1-4*x). E.g.f.: exp(4*x). a(n) = Sum_{k = 0..n} binomial(2k, k) * binomial(2(n - k), n - k). - Benoit Cloitre, Jan 26 2003 [See Graham et al., eq. (5.39), p. 187. -Wolfdieter Lang, Aug 16 2019] 1 = Sum_{n >= 1} 3/a(n) = 3/4 + 3/16 + 3/64 + 3/256 + 3/1024, ...; with partial sums: 3/4, 15/16, 63/64, 255/256, 1023/1024, ... - Gary W. Adamson, Jun 16 2003 a(n) = A001045(2*n) + A001045(2*n+1). - Paul Barry, Apr 27 2004 a(n) = Sum_{j = 0..n} 2^(n - j)*binomial(n + j, j). - Peter C. Heinig (algorithms(AT)gmx.de), Apr 06 2007 Hankel transform of A115967. - Philippe Deléham, Jun 22 2007 a(n) = 6*StirlingS2(n+1, 4) + 6*StirlingS2(n+1, 3) + 3*StirlingS2(n+1, 2) + 1 = 2*StirlingS2(2^n, 2^n - 1) + StirlingS2(n+1, 2) + 1. - Ross La Haye, Jun 26 2008 ((2+sqrt(4))^n - (2-sqrt(4))^n)/4. Offset 1. a(3) = 16. - Al Hakanson (hawkuu(AT)gmail.com), Dec 31 2008 a(n) = Sum_{k = 0..n} binomial(2*n+1, k). - Mircea Merca, Jun 25 2011 Sum_{n >= 1} mobius(n)/a(n) = 0.1710822479183... - R. J. Mathar, Aug 12 2012 a(n) = 5*a(n - 1) - 4*a(n - 2). - Jean-Bernard François, Sep 12 2013 a(n) = (2*n+1) * binomial(2*n,n) * Sum_{j=0..n} (-1)^j/(2*j+1)*binomial(n,j). - Vaclav Kotesovec, Sep 15 2013 a(n) = A000217(2^n - 1) + A000217(2^n). - J. M. Bergot, Dec 28 2014 a(n) = (2^n)^2 = A000079(n)^2. - Doug Bell, Jun 23 2015 a(n) = A002063(n)/3 - A004171(n). - Zhandos Mambetaliyev, Nov 19 2016 a(n) = (1/2) * Product_{k = 0..n} (1 + (2*n + 1)/(2*k + 1)). - Peter Bala, Mar 06 2018 a(n) = A001045(n+1)*A001045(n+2) + A001045(n)^2. - Ezhilarasu Velayutham, Aug 30 2019 a(n) = denominator(zeta_star({2}_(n + 1))/zeta(2*n + 2)) where zeta_star is the multiple zeta star values and ({2}_n} represents (2, ..., 2) where the multiplicity of 2 is n. - Roudy El Haddad, Feb 22 2022 MAPLE A000302 := n->4^n; for n from 0 to 10 do sum(2^(n-j)*binomial(n+j, j), j=0..n); od; # Peter C. Heinig (algorithms(AT)gmx.de), Apr 06 2007 A000302:=-1/(-1+4*z); # Simon Plouffe in his 1992 dissertation. MATHEMATICA Table[4^n, {n, 0, 30}] (* Stefan Steinerberger, Apr 01 2006 *) CoefficientList[Series[1/(1 - 4 x), {x, 0, 50}], x]  (* Vincenzo Librandi, May 29 2014 *) NestList[4 # &, 1, 30] (* Harvey P. Dale, Mar 26 2015 *) 4^Range[0, 30] (* Eric W. Weisstein, Jun 29 2017 *) LinearRecurrence[{4}, {1}, 31] (* Robert A. Russell, Nov 08 2018 *) PROG (PARI) A000302(n)=4^n \\ Michael B. Porter, Nov 06 2009 (Haskell) a000302 = (4 ^) a000302_list = iterate (* 4) 1  -- Reinhard Zumkeller, Apr 04 2012 (Maxima) A000302(n):=4^n\$ makelist(A000302(n), n, 0, 30); /* Martin Ettl, Oct 24 2012 */ (Scala) (List.fill(20)(4: BigInt)).scanLeft(1: BigInt)(_ * _) // Alonso del Arte, Jun 22 2019 (Python) print([4**n for n in range(25)]) # Michael S. Branicky, Jan 04 2021 CROSSREFS Cf. A024036, A052539, A032443, A000351 (Binomial transform). Cf. A249307. Cf. A083420. Sequence in context: A206450 A294452 A270142 * A262710 A050734 A075614 Adjacent sequences:  A000299 A000300 A000301 * A000303 A000304 A000305 KEYWORD easy,nonn,nice,core AUTHOR EXTENSIONS Partially edited by Joerg Arndt, Mar 11 2010 STATUS approved

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Last modified September 26 08:00 EDT 2022. Contains 356987 sequences. (Running on oeis4.)