0,3

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 (reinhard.zumkeller(AT)gmail.com), 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 (deutsch(AT)duke.poly.edu), May 21 2003

a(n)=number of bits in binary numbers from 1 to 111...1 (n bits). Partial sums of A001787. - Emeric Deutsch (deutsch(AT)duke.poly.edu), 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 (perry(AT)globalnet.co.uk), 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 (callan(AT)stat.wisc.edu), 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 (jvospost3(AT)gmail.com), 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 (ctibor.zizka(AT)seznam.cz), 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 (qntmpkt(AT)yahoo.com), 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.]. [From Jonathan Vos Post (jvospost3(AT)gmail.com), Jul 25 2009]

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

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.

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.

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 (jvospost3(AT)gmail.com), Jul 25 2009]

T. D. Noe, Table of n, a(n) for n=0..300

Ulrich Brehm and Egon Schulte, Polyhedral Maps. [From Jonathan Vos Post (jvospost3(AT)gmail.com), Jul 25 2009]

R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, J. Integer Seqs., Vol. 3 (2000), #00.1.6

T. Mansour, Restricted permutations by patterns of type 2-1.

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

Len Smiley, Hardy's algorithm

Eric Weisstein's World of Mathematics, Graph Genus

Index entries for sequences related to linear recurrences with constant coefficients

Binomial transform of A008574. - Paul Barry (pbarry(AT)wit.ie), Jul 21 2003

G.f.: x/((1-x)(1-2x)^2). 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 (pbarry(AT)wit.ie), Aug 19 2005

a(n) = A119258(n+1,n-1) for n>0. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 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 (g_m(AT)mcraefamily.com), Jul 12 2006

Sum(k*2^(k-1),k=1..n). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 19 2006

a(n)=a(n-1)+n*2^(n-1) [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Sep 20 2008]

A000337:=-1/(z-1)/(-1+2*z)**2; [Conjectured (correctly) by S. Plouffe in his 1992 dissertation.]

a:=n->sum (2^n-2^j, j=2..n): seq(a(n)/4, n=2..31); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 14 2008]

Table[Sum[(-1)^(n - k) k (-1)^(n - k) Binomial[n + 1, k + 1], {k, 0, n}], {n, 0, 28}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 08 2009]

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

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

Cf. A001787, A066185, A077436, A023758.

Cf. A014915.

Cf. A023758.

Sequence in context: A115981 A083091 A082753 this_sequence A147260 A146045 A086866

Adjacent sequences: A000334 A000335 A000336 this_sequence A000338 A000339 A000340

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).

Hardy reference from Len Smiley (smiley(AT)math.uaa.alaska.edu)

More terms from Graeme McRae (g_m(AT)mcraefamily.com), Jul 12 2006