login
This site is supported by donations to The OEIS Foundation.

 

Logo

The OEIS is looking to hire part-time people to help edit core sequences, upload scanned documents, process citations, fix broken links, etc. - Neil Sloane, njasloane@gmail.com

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A000337 a(n) = (n-1)*2^n + 1.
(Formerly M3874 N1587)
60
0, 1, 5, 17, 49, 129, 321, 769, 1793, 4097, 9217, 20481, 45057, 98305, 212993, 458753, 983041, 2097153, 4456449, 9437185, 19922945, 41943041, 88080385, 184549377, 385875969, 805306369, 1677721601, 3489660929, 7247757313 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(n) also gives number of zeros in binary numbers 1 to 111..1 (n+1 bits). - Stephen G. Penrice (spenrice(AT)ets.org), Oct 01 2000

Numerator of m(n)=(m(n-1)+n)/2, m(0)=0. Denominator is A000072. - Reinhard Zumkeller, Feb 23 2002

a(n) = number of directed column-convex polyominoes of area n+2 having along the lower contour exactly one vertical step that is followed by a horizontal step (a reentrant corner). - Emeric Deutsch, May 21 2003

a(n) = number of bits in binary numbers from 1 to 111...1 (n bits). Partial sums of A001787. - Emeric Deutsch, May 24 2003

Genus of graph of n-cube = a(n-3) = 1+(n-4)*2^(n-3), n>1.

Sum of ordered partitions of n where each element is summed via T(e-1). See A066185 for more information. - Jon Perry, Dec 12 2003

a(n-2) = number of Dyck n-paths with exactly one peak at height >= 3. Example. There are 5 such paths with n=4: UUUUDDDD, UUDUUDDD, UUUDDUDD, UDUUUDDD, UUUDDDUD. - David Callan, Mar 23 2004

Permutations in S_{n+2} avoiding 12-3 that contain the pattern 13-2 exactly once.

a(n) is prime for n = 2, 3, 7, 27, 51, 55, 81. a(n) is semiprime for n = 4, 5, 6, 8, 9, 10, 11, 13, 15, 19, 28, 32, 39, 57, 63, 66, 75, 97. - Jonathan Vos Post, Jul 18 2005

A member of the family of sequences defined by a(n) = Sum_{i=1..n} i*[c(1)*...*c(r)]^(i-1). This sequence has c(1)=2, A014915 has c(1)=3. - Ctibor O. Zizka, Feb 23 2008

Starting with 1 = row sums of A023758 as a triangle by rows: (1; 2,3; 4,6,7; 8,12,14,15; ...). - Gary W. Adamson, Jul 18 2008

Equivalent formula given in Brehm: for each q >= 3 there exists a polyhedral map M_q of type {4, q} with [number of vertices] f_0 = 2^q and [genus] g = (2^(q-3))*(q-4) + 1 such that M_q and its dual have polyhedral embeddings in R^3 [McMullen et al.]. - Jonathan Vos Post, Jul 25 2009

Sums of rows of the triangle in A173787. - Reinhard Zumkeller, Feb 28 2010

This sequence is related to A000079 by a(n) = n*A000079(n)-sum(A000079(i), i=0..n-1). - Bruno Berselli, Mar 06 2012

(1+ 5x + 17x^2 + 49x^3 + ...) = (1 + 2x + 4x^2 + 8x^3 + ...) * (1 + 3x + 7x^2 + 15x^3) + ...). - Gary W. Adamson, Mar 14 2012

The first barycentrics coordinate of the centroid of Pascal triangles, assumming that numbers are weights, is A000295(n+1)/A000337(n), no matter what are the triangle sides. See attached figure. - César Eliud Lozada, Nov 14 2014

REFERENCES

F. Harary, Topological concepts in graph theory, pp. 13-17 of F. Harary and L. Beineke, editors, A seminar on Graph Theory, Holt, Rinehart and Winston, New York, 1967.

V. G. Gutierrez, S. L. de Medrano, Surfaces as complete intersections, in Riemann and Klein Surfaces, Automorphisms, Symmetries and Moduli Spaces, edited by Milagros Izquierdo, S. Allen Broughton, Antonio F. Costa, Contemp. Math. vol. 629, 2014, pp. 171-.

F. Harary, Graph Theory. Addison-Wesley, Reading, MA, 1969, p. 119.

G. H. Hardy, A Theorem Concerning the Infinite Cardinal Numbers, Quart. J. Math., 35 (1904), p. 90 = Collected Papers of G. H. Hardy, Vol. VII, p. 430.

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).

LINKS

T. D. Noe and Indranil Ghosh, Table of n, a(n) for n = 0..1000(First 301 terms from T. D. Noe)

L. W. Beineke and F. Harary, The genus of the n-cube, Canad. J. Math., 17 (1965), 494-496.

Ulrich Brehm and Egon Schulte, Polyhedral Maps. [Jonathan Vos Post, Jul 25 2009]

Harry Crane, Left-right arrangements, set partitions, and pattern avoidance, Australasian Journal of Combinatorics, 61(1) (2015), 57-72.

R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6

S. Kitaev, J. Remmel and M. Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, arXiv preprint arXiv:1201.6243 [math.CO], 2012.

Sergey Kitaev, Jeffrey Remmel, Mark Tiefenbruck, Quadrant Marked Mesh Patterns in 132-Avoiding Permutations II, Electronic Journal of Combinatorial Number Theory, Volume 15 #A16. (arXiv:1302.2274 [math.CO])

César Eliud Lozada, Centroids of Pascal triangles

P. McMullen, Ch. Schulz and J.M. Wills, Polyhedral manifolds in E^3 with unusually large genus, Israel J. Math. 46:127-144, 1983. [From Jonathan Vos Post, Jul 25 2009]

T. Mansour, Restricted permutations by patterns of type (2,1), arXiv:math/0202219 [math.CO], 2002.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Len Smiley, Hardy's algorithm

Eric Weisstein's World of Mathematics, Graph Genus

A. F. Y. Zhao, Pattern Popularity in Multiply Restricted Permutations, Journal of Integer Sequences, 17 (2014), #14.10.3.

Index entries for linear recurrences with constant coefficients, signature (5,-8,4).

FORMULA

Binomial transform of A004273. Binomial transform of A008574 if the leading zero is dropped.

G.f.: x/((1-x)(1-2x)^2). - Simon Plouffe in his 1992 dissertation

E.g.f.: exp(x) - exp(2x)(1-2x). a(n) = 4*a(n-1) - 4*a(n-2)+1, n>0. Series reversion of g.f. A(x) is x*A034015(-x). - Michael Somos

Binomial transform of n/(n+1) is a(n)/(n+1). - Paul Barry, Aug 19 2005

a(n) = A119258(n+1,n-1) for n>0. - Reinhard Zumkeller, May 11 2006

Convolution of "Number of fixed points in all 231-avoiding involutions in S_n" (A059570) with "The odd numbers" (A005408), treating the result as if offset=0. - Graeme McRae, Jul 12 2006

a(n) = Sum_{k=1..n} k*2^(k-1), partial sums of A001787. - Zerinvary Lajos, Oct 19 2006

a(n) = 5*a(n-1) - 8*a(n-2) + 4*a(n-3), n>2. - Harvey P. Dale, Jun 21 2011

MAPLE

A000337 := proc(n) 1+(n-1)*2^n ; end proc: # R. J. Mathar, Oct 10 2011

MATHEMATICA

Table[Sum[(-1)^(n - k) k (-1)^(n - k) Binomial[n + 1, k + 1], {k, 0, n}], {n, 0, 28}] (* Zerinvary Lajos, Jul 08 2009 *)

Table[(n-1)2^n+1, {n, 0, 40}] (* or *) LinearRecurrence[{5, -8, 4}, {0, 1, 5}, 40] (* Harvey P. Dale, Jun 21 2011 *)

CoefficientList[Series[x / ((1 - x) (1 - 2 x)^2), {x, 0, 50}], x] (* Vincenzo Librandi, Nov 21 2014 *)

PROG

(PARI) a(n)=if(n<0, 0, (n-1)*2^n+1)

(MAGMA) [(n-1)*2^n + 1: n in [0..40]]; // Vincenzo Librandi, Nov 21 2014

(Python) a=lambda n:((n-1)<<(n))+1 # Indranil Ghosh, Jan 05 2017

CROSSREFS

a(n) = T(3, n), array T given by A048472. A036799/2.

Cf. A001787, A066185, A077436, A023758, A014915.

Cf. A000079, A209359.

Sequence in context: A083091 A176953 A082753 * A147260 A273688 A146045

Adjacent sequences:  A000334 A000335 A000336 * A000338 A000339 A000340

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

Hardy reference from Len Smiley

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified May 26 01:26 EDT 2017. Contains 287073 sequences.