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%I M4379 #119 Apr 29 2024 12:35:55
%S 1,7,23,63,159,383,895,2047,4607,10239,22527,49151,106495,229375,
%T 491519,1048575,2228223,4718591,9961471,20971519,44040191,92274687,
%U 192937983,402653183,838860799,1744830463,3623878655,7516192767
%N Woodall (or Riesel) numbers: n*2^n - 1.
%C For n>1, a(n) is base at which zero is reached for the function "write f(j) in base j, read as base j+1 and then subtract 1 to give f(j+1)" starting from f(n) = n^2 - 1. - _Henry Bottomley_, Aug 06 2000
%C Sequence corresponds also to the maximum chain length of the classic puzzle whereby, under agreed commercial terms, an asset of unringed golden chain, when judiciously fragmented into as few as n pieces and n-1 opened links (through n-1 cuts), might be used to settle debt sequentially, with a golden link covering for unit cost. Here beside the n-1 opened links, the n fragmented pieces have lengths n, 2*n, 4*n, ..., 2^(n-1)*n. For instance, the chain of original length a(5)=159, if segregated by 4 cuts into 5+1+10+1+20+1+40+1+80, may be used to pay sequentially, i.e., a link-cost at a time, for an equivalent cost up to 159 links, to the same creditor. - _Lekraj Beedassy_, Feb 06 2003
%D A. Brousseau, Number Theory Tables. Fibonacci Association, San Jose, CA, 1973, p. 159.
%D K. R. Bhutani and A. B. Levin, "The Problem of Sawing a Chain", Journal of Recreational Mathematics 2002-3 31(1) 32-35.
%D G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
%D M. Gardner, Martin Gardner's Sixth Book of Mathematical Diversions from Scientific American, "Gold Links", Problem 4, pp. 50-51; 57-58, University of Chicago Press, 1983.
%D O. O’Shea, Mathematical Brainteasers with Surprising Solutions, Problem 76, pp. 183-185, Prometheus Books, Guilford, Connecticut, 2020.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vladimir Pletser, <a href="/A003261/b003261.txt">Table of n, a(n) for n = 1..3000</a> (terms 1..300 from T. D. Noe).
%H Ray Ballinger, <a href="http://web.archive.org/web/20161028080439/http://www.prothsearch.net/woodall.html">Woodall Primes: Definition and Status</a>.
%H Attila Bérczes, István Pink, and Paul Thomas Young, <a href="https://doi.org/10.1016/j.jnt.2024.03.006">Cullen numbers and Woodall numbers in generalized Fibonacci sequences</a>, J. Num. Theor. (2024) Vol. 262, 86-102.
%H Alfred Brousseau, <a href="http://www.fq.math.ca/fibonacci-tables.html">Fibonacci and Related Number Theoretic Tables</a>, Fibonacci Association, San Jose, CA, 1972. See p. 159.
%H C. K. Caldwell, <a href="https://t5k.org/glossary/page.php?sort=WoodallNumber">Woodall Numbers</a>.
%H Orhan Eren and Yüksel Soykan, <a href="https://doi.org/10.9734/ACRI/2023/v23i8611">Gaussian Generalized Woodall Numbers</a>, Arch. Current Res. Int'l (2023) Vol. 23, Iss. 8, Art. No. ACRI.108618, 48-68. See p. 50.
%H Paul Leyland, <a href="http://www.leyland.vispa.com/numth/factorization/cullen_woodall/cw.htm">Factors of Cullen and Woodall numbers</a>.
%H Paul Leyland, <a href="http://www.leyland.vispa.com/numth/factorization/cullen_woodall/gcw.htm">Generalized Cullen and Woodall numbers</a>.
%H D. Marques, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Marques/marques5r2.html">On Generalized Cullen and Woodall Numbers That are Also Fibonacci Numbers</a>, Journal of Integer Sequences, 17 (2014), #14.9.4.
%H Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha122.htm">Factorizations of many number sequences: Riesel numbers, n=1..100</a>, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha123.htm">n=101..200</a>, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha124.htm">n=201..300</a>, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha125.htm">n=301..323</a>.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H T. Sillke, <a href="http://www.mathematik.uni-bielefeld.de/~sillke/PUZZLES/chain-link-pay">Using Chains Links To Pay For A Room</a>.
%H Amelia Carolina Sparavigna, <a href="https://doi.org/10.5281/zenodo.2634312">On the generalized sums of Mersenne, Fermat, Cullen and Woodall Numbers</a>, Politecnico di Torino (Italy, 2019).
%H Amelia Carolina Sparavigna, <a href="https://doi.org/10.18483/ijSci.2044">Composition Operations of Generalized Entropies Applied to the Study of Numbers</a>, International Journal of Sciences (2019) Vol. 8, No. 4, 87-92.
%H Amelia Carolina Sparavigna, <a href="https://doi.org/10.5281/zenodo.3471358">The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences</a>, Politecnico di Torino, Italy (2019), [math.NT].
%H Amelia Carolina Sparavigna, <a href="https://doi.org/10.18483/ijSci.2188">Some Groupoids and their Representations by Means of Integer Sequences</a>, International Journal of Sciences (2019) Vol. 8, No. 10.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/WoodallNumber.html">Woodall Number</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Woodall_number">Woodall number</a>.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,-8,4).
%F G.f.: x*(-1-2*x+4*x^2) / ( (x-1)*(-1+2*x)^2 ). - _Simon Plouffe_ in his 1992 dissertation
%F Binomial transform of A133653 and double binomial transform of [1, 5, -1, 1, -1, 1, ...]. - _Gary W. Adamson_, Sep 19 2007
%F a(n) = -(2)^n * A006127(-n) for all n in Z. - _Michael Somos_, Nov 04 2018
%e G.f. = x + 7*x^2 + 23*x^3 + 63*x^4 + 159*x^5 + 383*x^6 + 895*x^7 + ... - _Michael Somos_, Nov 04 2018
%p for n from 1 to 3000 do n, n*2^n -1; end do; # _Vladimir Pletser_, Dec 30 2022
%t Table[n*2^n-1,{n,3*4!}] (* _Vladimir Joseph Stephan Orlovsky_, Apr 25 2010 *)
%t LinearRecurrence[{5,-8,4},{1,7,23},30] (* _Harvey P. Dale_, Mar 13 2022 *)
%o (Haskell)
%o a003261 = (subtract 1) . a036289 -- _Reinhard Zumkeller_, Mar 05 2012
%o (PARI) A003261(n)=n*2^n-1 \\ _M. F. Hasler_, Oct 31 2012
%o (Magma) [n*2^n - 1: n in [1..30]]; // _G. C. Greubel_, Nov 04 2018
%o (Python) [n*2**n - 1 for n in range(1, 29)] # _Michael S. Branicky_, Jan 07 2021
%Y Cf. A002234, A002064, A005849, A050918, A006127.
%Y a(n) = A036289(n) - 1 = A002064(n) - 2.
%Y Cf. A133653.
%K nonn,easy,nice
%O 1,2
%A _N. J. A. Sloane_
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