A084858 -id:A084858 - cp's OEIS frontend

A001651 Numbers not divisible by 3.

1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26, 28, 29, 31, 32, 34, 35, 37, 38, 40, 41, 43, 44, 46, 47, 49, 50, 52, 53, 55, 56, 58, 59, 61, 62, 64, 65, 67, 68, 70, 71, 73, 74, 76, 77, 79, 80, 82, 83, 85, 86, 88, 89, 91, 92, 94, 95, 97, 98, 100, 101, 103, 104

Offset: 1

N. J. A. Sloane

Inverse binomial transform of A084858. - Benoit Cloitre, Jun 12 2003
Earliest monotonic sequence starting with (1,2) and satisfying the condition: "a(n)+a(n-1) is not in the sequence." - Benoit Cloitre, Mar 25 2004. [The numbers of the form a(n)+a(n-1) form precisely the complement with respect to the positive integers. - David W. Wilson, Feb 18 2012]
a(1) = 1; a(n) is least number which is relatively prime to the sum of all the previous terms. - Amarnath Murthy, Jun 18 2001
For n > 3, numbers having 3 as an anti-divisor. - Alexandre Wajnberg, Oct 02 2005
Also numbers n such that (n+1)*(n+2)/6 = A000292(n)/n is an integer. - Ctibor O. Zizka, Oct 15 2010
Notice the property described by Gary Detlefs in A113801: more generally, these numbers are of the form (2*h*n + (h-4)*(-1)^n-h)/4 (h, n natural numbers), therefore ((2*h*n + (h-4)*(-1)^n - h)/4)^2 - 1 == 0 (mod h); in this case, a(n)^2 - 1 == 0 (mod 3). - Bruno Berselli, Nov 17 2010
A001651 mod 9 gives A141425. - Paul Curtz, Dec 31 2010. (Correct for the modified offset 1. - M. F. Hasler, Apr 07 2015)
The set of natural numbers (1, 2, 3, ...), sequence A000027; represents the numbers of ordered compositions of n using terms in the signed set: (1, 2, -4, -5, 7, 8, -10, -11, 13, 14, ...). This follows from (1, 2, 3, ...) being the INVERT transform of A011655, signed and beginning: (1, 1, 0, -1, -1, 0, 1, 1, 0, ...). - Gary W. Adamson, Apr 28 2013
Union of A047239 and A047257. - Wesley Ivan Hurt, Dec 19 2013
Numbers whose sum of digits (and digital root) is != 0 (mod 3). - Joerg Arndt, Aug 29 2014
The number of partitions of 3*(n-1) into at most 2 parts. - Colin Barker, Apr 22 2015
a(n) is the number of partitions of 3*n into two distinct parts. - L. Edson Jeffery, Jan 14 2017
Conjectured (and like even easily proved) to be the graph bandwidth of the complete bipartite graph K_{n,n}. - Eric W. Weisstein, Apr 24 2017
Numbers k such that Fibonacci(k) mod 4 = 1 or 3. Equivalently, sequence lists the indices of the odd Fibonacci numbers (see A014437). - Bruno Berselli, Oct 17 2017
Minimum value of n_3 such that the "rectangular spiral pattern" is the optimal solution for Ripà's n_1 X n_2 x n_3 Dots Problem, for any n_1 = n_2. For example, if n_1 = n_2 = 5, n_3 = floor((3/2)*(n_1 - 1)) + 1 = a(5). - Marco Ripà, Jul 23 2018
For n >= 54, a(n) = sat(n, P_n), the minimum number of edges in a P_n-saturated graph on n vertices, where P_n is the n-vertex path (see Dudek, Katona, and Wojda, 2003; Frick and Singleton, 2005). - Danny Rorabaugh, Nov 07 2017
From Roger Ford, May 09 2021: (Start)
a(n) is the smallest sum of arch lengths for the top arches of a semi-meander with n arches. An arch length is the number of arches covered + 1.
      /\   The top arch has a length of 3. /\   The top arch has a length of 3.
     /  \  Both bottom arches have a      //\\  The middle arch has a length of 2.
    //\/\\ length of 1.                  ///\\\ The bottom arch has a length of 1.
  Example: a(6) = 8     /\      /\
                       //\\ /\ //\\ /\   2 + 1 + 1 + 2 + 1 + 1 = 8. (End)
This is the lexicographically earliest increasing sequence of positive integers such that no polynomial of degree d can be fitted to d+2 consecutive terms (equivalently, such that no iterated difference is zero). - Pontus von Brömssen, Dec 26 2021

			G.f.: x + 2*x^2 + 4*x^3 + 5*x^4 + 7*x^5 + 8*x^6 + 10*x^7 + 11*x^8 + 13*x^9 + ...

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).

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
L. Carlitz, R. Scoville and T. Vaughan, Some arithmetic functions related to Fibonacci numbers, Fib. Quart., Vol. 11, No. 4 (1973), pp. 337-386.
Aneta Dudek, Gyula Y. Katona, and A.Pawel Wojda, m_Path Cover Saturated Graphs, Electronic Notes in Discrete Math., Vol. 13 (April 2003), pp. 41-44.
Marietjie Frick and Joy Singleton, Lower Bound for the Size of Maximal Nontraceable Graphs, Electron. J. Combin., 12#R32 (2005), 9 pp.
Aviezri S. Fraenkel, New games related to old and new sequences, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 4, Paper G6, 2004. (See Table 5.)
Brian Hopkins, Euler's Enumerations, Enumerative Combinatorics and Applications, Vol. 1, No. 1 (2021), Article #S1H1.
G. Ledin, Jr., Is Eratosthenes out?, Fib. Quart., Vol. 6, No. 4 (1968), pp. 261-265.
Gerard P. Michon, Counting Polyhedra.
Melvyn B. Nathanson, On the fractional parts of roots of positive real numbers, Amer. Math. Monthly, Vol. 120, No. 5 (2013), pp. 409-429 [see p. 417].
M. A. Nyblom, Some curious sequences involving floor and ceiling functions, Am. Math. Monthly, Vol. 109, No. 6 (2002), pp. 559-564, Ex. 2.2.
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.
Marco Ripà, The rectangular spiral or the n1 X n2 X ... X nk Points Problem , Notes on Number Theory and Discrete Mathematics, Vol. 20, No. 1 (2014), pp. 59-71.
Eric Weisstein's World of Mathematics, Complete Bipartite Graph, Graph Bandwidth, and RATS Sequence.
Dominika Závacká, Cristina Dalfó, and Miquel Angel Fiol, Integer sequences from k-iterated line digraphs, CEUR: Proc. 24th Conf. Info. Tech. - Appl. and Theory (ITAT 2024) Vol 3792, 156-161. See p. 161, Table 2.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Differs from A059564 after 35 = a(24) = A059564(24).
Cf. A000726, A001082, A003105, A005408 (n=1 or 3 mod 4), A007494, A008585 (complement), A011655, A026386, A032766, A073010, A191967, A225227, A004526, A248897.
Cf. A000027, A000217, A000292, A000982, A001477, A008619, A014437, A040001, A047239, A047257, A077043, A084858, A113801, A141425, A215879.

Filtered([0..110],n->n mod 3<>0); # Muniru A Asiru, Jul 24 2018

a001651 = (`div` 2) . (subtract 1) . (* 3)
a001651_list = filter ((/= 0) . (`mod` 3)) [1..]
-- Reinhard Zumkeller, Jul 07 2012, Aug 23 2011

[3*(2*n-1)/4-(-1)^n/4: n in [1..80]]; // Vincenzo Librandi, Jun 07 2011

A001651 := n -> 3*floor(n/2) - (-1)^n; # Corrected by M. F. Hasler, Apr 07 2015
A001651:=(1+z+z**2)/(z+1)/(z-1)**2; # Simon Plouffe in his 1992 dissertation
a[1]:=1:a[2]:=2:for n from 3 to 100 do a[n]:=a[n-2]+3 od: seq(a[n], n=1..69); # Zerinvary Lajos, Mar 16 2008, offset corrected by M. F. Hasler, Apr 07 2015

Select[Table[n,{n,200}],Mod[#,3]!=0&] (* Vladimir Joseph Stephan Orlovsky, Feb 18 2011 *)
Drop[Range[200 + 1], {1, -1, 3}] - 1 (* József Konczer, May 24 2016 *)
Floor[(3 Range[70] - 1)/2] (* Eric W. Weisstein, Apr 24 2017 *)
CoefficientList[Series[(x^2 + x + 1)/((x - 1)^2 (x + 1)), {x, 0, 70}],
  x] (* or *)
LinearRecurrence[{1, 1, -1}, {1, 2, 4}, 70] (* Robert G. Wilson v, Jul 25 2018 *)

{a(n) = n + (n-1)\2}; /* Michael Somos, Jan 15 2011 */

x='x+O('x^100); Vec(x*(1+x+x^2)/((1-x)*(1-x^2))) \\ Altug Alkan, Oct 22 2015

print([k for k in range(1, 105) if k%3]) # Michael S. Branicky, Sep 06 2021

def A001651(n): return (n<<1)-(n>>1)-1 # Chai Wah Wu, Mar 05 2024

a(n) = 3 + a(n-2) for n > 2.
a(n) = a(n-1) + a(n-2) - a(n-3) for n > 3.
a(2*n+1) = 3*n+1, a(2*n) = 3*n-1.
G.f.: x * (1 + x + x^2) / ((1 - x) * (1 - x^2)). - Michael Somos, Jun 08 2000
a(n) = (4-n)*a(n-1) + 2*a(n-2) + (n-3)*a(n-3) (from the Carlitz et al. article).
a(n) = floor((3*n-1)/2). [Corrected by Gary Detlefs]
a(1) = 1, a(n) = 2*a(n-1) - 3*floor(a(n-1)/3). - Benoit Cloitre, Aug 17 2002
a(n+1) = 1 + n - n mod 2 + (n + n mod 2)/2. - Reinhard Zumkeller, Dec 17 2002
a(1) = 1, a(n+1) = a(n) + (a(n) mod 3). - Reinhard Zumkeller, Mar 23 2003
a(1) = 1, a(n) = 3*(n-1) - a(n-1). - Benoit Cloitre, Apr 12 2003
a(n) = 3*(2*n-1)/4 - (-1)^n/4. - Benoit Cloitre, Jun 12 2003
Nearest integer to (Sum_{k>=n} 1/k^3)/(Sum_{k>=n} 1/k^4). - Benoit Cloitre, Jun 12 2003
Partial sums of A040001. a(n) = A032766(n-1)+1. - Paul Barry, Sep 02 2003
a(n) = T(n, 1) = T(n, n-1), where T is the array in A026386. - Emeric Deutsch, Feb 18 2004
a(n) = sqrt(3*A001082(n)+1). - Zak Seidov, Dec 12 2007
a(n) = A077043(n) - A077043(n-1). - Reinhard Zumkeller, Dec 28 2007
a(n) = A001477(n-1) + A008619(n-1). - Yosu Yurramendi, Aug 10 2008
Euler transform of length 3 sequence [2, 1, -1]. - Michael Somos, Sep 06 2008
A011655(a(n)) = 1. - Reinhard Zumkeller, Nov 30 2009
a(n) = n - 1 + ceiling(n/2). - Michael Somos, Jan 15 2011
a(n) = 3*A000217(n)+1 - 2*Sum_{i=1..n-1} a(i), for n>1. - Bruno Berselli, Nov 17 2010
a(n) = 3*floor(n/2) + (-1)^(n+1). - Gary Detlefs, Dec 29 2011
A215879(a(n)) > 0. - Reinhard Zumkeller, Dec 28 2012 [More precisely, A215879 is the characteristic function of A001651. - M. F. Hasler, Apr 07 2015]
a(n) = 2n - 1 - floor(n/2). - Wesley Ivan Hurt, Oct 25 2013
a(n) = (3n - 2 + (n mod 2)) / 2. - Wesley Ivan Hurt, Mar 31 2014
a(n) = A000217(n) - A000982(n-1). - Bui Quang Tuan, Mar 28 2015
1/1^3 - 1/2^3 + 1/4^3 - 1/5^3 + 1/7^3 - 1/8^3 + ... = 4 Pi^3/(3 sqrt(3)). - M. F. Hasler, Mar 29 2015
E.g.f.: (4 + sinh(x) - cosh(x) + 3*(2*x - 1)*exp(x))/4. - Ilya Gutkovskiy, May 24 2016
a(n) = a(n+k-1) + a(n-k) - a(n-1) for n > k >= 0. - Bob Selcoe, Feb 03 2017
a(n) = -a(1-n) for all n in Z. - Michael Somos, Jul 31 2018
a(n) = n + A004526(n-1). - David James Sycamore, Sep 06 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(3*sqrt(3)) (A073010). - Amiram Eldar, Dec 04 2021
From Amiram Eldar, Nov 22 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = 1.
Product_{n>=2} (1 + (-1)^n/a(n)) = 2*Pi/(3*sqrt(3)) (A248897). (End)

This is a list, so the offset should be 1. I corrected this and adjusted some of the comments and formulas. Other lines probably also need to be adjusted. - N. J. A. Sloane, Jan 01 2011

Offset of pre-2011 formulas verified or corrected by M. F. Hasler, Apr 07-18 2015 and by Danny Rorabaugh, Oct 23 2015

A268971 T(n,k)=Number of nXk 0..2 arrays with some element plus some horizontally or antidiagonally adjacent neighbor totalling two not more than once.

Original entry on oeis.org

3, 9, 9, 24, 60, 27, 60, 240, 336, 81, 144, 912, 2016, 1728, 243, 336, 3312, 11664, 15552, 8448, 729, 768, 11664, 63792, 136080, 114048, 39936, 2187, 1728, 40176, 339480, 1125360, 1504656, 808704, 184320, 6561, 3840, 136080, 1770048, 9093528, 18852912

Offset: 1

R. H. Hardin, Feb 16 2016

Table starts
.....3........9.........24...........60............144..............336
.....9.......60........240..........912...........3312............11664
....27......336.......2016........11664..........63792...........339480
....81.....1728......15552.......136080........1125360..........9093528
...243.....8448.....114048......1504656.......18852912........231730344
...729....39936.....808704.....16061328......305242992.......5712070032
..2187...184320....5598720....167226768.....4823705520.....137497776840
..6561...835584...38071296...1709114256....74858700528....3251386055664
.19683..3735552..255301632..17218688400..1145496747312...75828095546544
.59049.16515072.1693052928.171498136464.17332683832944.1748970953035272

			Some solutions for n=4 k=4
..1..0..0..1. .2..1..0..1. .2..1..2..1. .2..1..2..1. .1..2..1..2
..1..2..2..2. .0..0..2..2. .0..1..2..1. .1..2..2..1. .1..0..0..0
..2..2..2..2. .1..2..2..2. .2..1..0..0. .2..2..2..2. .1..0..1..2
..2..2..1..2. .2..1..2..2. .1..0..0..1. .2..2..1..0. .1..2..2..1

R. H. Hardin, Table of n, a(n) for n = 1..287

Column 1 is A000244.

Row 1 is A084858.

Empirical for column k:
k=1: a(n) = 3*a(n-1)
k=2: a(n) = 8*a(n-1) -16*a(n-2)
k=3: a(n) = 12*a(n-1) -36*a(n-2)
k=4: a(n) = 18*a(n-1) -81*a(n-2) for n>3
k=5: a(n) = 30*a(n-1) -261*a(n-2) +540*a(n-3) -324*a(n-4)
k=6: a(n) = 50*a(n-1) -805*a(n-2) +4662*a(n-3) -12150*a(n-4) +14580*a(n-5) -6561*a(n-6)
k=7: [order 8]
Empirical for row n:
n=1: a(n) = 4*a(n-1) -4*a(n-2)
n=2: a(n) = 6*a(n-1) -9*a(n-2) for n>4
n=3: a(n) = 10*a(n-1) -29*a(n-2) +20*a(n-3) -4*a(n-4) for n>6
n=4: [order 6] for n>12
n=5: [order 14] for n>18
n=6: [order 18] for n>26
n=7: [order 54] for n>60

A269052 T(n,k)=Number of nXk 0..2 arrays with some element plus some horizontally, diagonally or antidiagonally adjacent neighbor totalling two not more than once.

Original entry on oeis.org

3, 9, 9, 24, 42, 27, 60, 102, 174, 81, 144, 360, 594, 666, 243, 336, 1068, 3078, 3258, 2430, 729, 768, 3288, 13140, 24192, 17346, 8586, 2187, 1728, 9864, 58752, 149358, 183072, 90450, 29646, 6561, 3840, 29472, 253416, 971844, 1643376, 1350672, 464250

Offset: 1

R. H. Hardin, Feb 18 2016

Table starts
.....3.......9.......24.........60..........144............336.............768
.....9......42......102........360.........1068...........3288............9864
....27.....174......594.......3078........13140..........58752..........253416
....81.....666.....3258......24192.......149358.........971844.........6053094
...243....2430....17346.....183072......1643376.......15547380.......140497512
...729....8586....90450....1350672.....17696520......242861616......3193266318
..2187...29646...464250....9779808....187575858.....3726221592.....71430596250
..6561..100602..2353338...69793968...1964080920....56376679620...1577976495486
.19683..336798.11809746..492374976..20365312416...843461153880..34509932303172
.59049.1115370.58773858.3441051984.209472681102.12504078167988.748499855355192

			Some solutions for n=4 k=4
..1..2..2..1. .1..2..1..0. .0..1..2..2. .0..1..0..1. .2..1..0..1
..2..2..2..1. .1..2..1..0. .2..1..2..1. .0..1..2..1. .2..1..2..1
..1..2..2..2. .1..0..1..2. .1..2..2..1. .2..1..0..1. .2..1..2..1
..1..2..1..2. .0..2..1..2. .1..2..2..2. .0..1..2..0. .2..1..2..1

R. H. Hardin, Table of n, a(n) for n = 1..337

Column 1 is A000244.
Column 2 is A268622.
Row 1 is A084858.

Empirical for column k:
k=1: a(n) = 3*a(n-1)
k=2: a(n) = 6*a(n-1) -9*a(n-2) for n>3
k=3: a(n) = 10*a(n-1) -29*a(n-2) +20*a(n-3) -4*a(n-4) for n>5
k=4: a(n) = 14*a(n-1) -57*a(n-2) +56*a(n-3) -16*a(n-4) for n>5
k=5: [order 12] for n>13
k=6: [order 18] for n>19
k=7: [order 38] for n>39
Empirical for row n:
n=1: a(n) = 4*a(n-1) -4*a(n-2)
n=2: a(n) = 4*a(n-1) -8*a(n-3) -4*a(n-4)
n=3: a(n) = 6*a(n-1) -a(n-2) -28*a(n-3) -4*a(n-4) +16*a(n-5) -4*a(n-6) for n>8
n=4: [order 12] for n>14
n=5: [order 20] for n>22
n=6: [order 46] for n>48
n=7: [order 92] for n>94

A268628 T(n,k)=Number of nXk 0..2 arrays with some element plus some horizontally or vertically adjacent neighbor totalling two no more than once.

Original entry on oeis.org

3, 9, 9, 24, 42, 24, 60, 174, 174, 60, 144, 666, 1086, 666, 144, 336, 2430, 6300, 6300, 2430, 336, 768, 8586, 34890, 55452, 34890, 8586, 768, 1728, 29646, 187224, 467190, 467190, 187224, 29646, 1728, 3840, 100602, 982086, 3819654, 6000978, 3819654

Offset: 1

R. H. Hardin, Feb 09 2016

Table starts
....3.......9........24..........60...........144.............336
....9......42.......174.........666..........2430............8586
...24.....174......1086........6300.........34890..........187224
...60.....666......6300.......55452........467190.........3819654
..144....2430.....34890......467190.......6000978........74914554
..336....8586....187224.....3819654......74914554......1430057208
..768...29646....982086....30553014.....915847266.....26758514760
.1728..100602...5063964...240364746...11018667294....493042858032
.3840..336798..25764066..1866503592..130903914954...8974328440044
.8448.1115370.129678528.14342680944.1539375100362.161737670836314

			Some solutions for n=4 k=4
..2..1..0..0. .1..2..1..0. .0..0..0..0. .1..2..2..2. .0..0..0..1
..1..0..0..0. .0..1..0..0. .0..0..0..0. .2..1..2..2. .0..0..1..0
..2..0..1..0. .1..0..0..0. .1..0..0..0. .1..2..2..2. .0..0..0..1
..2..1..0..1. .0..0..1..1. .2..1..1..0. .0..2..2..2. .0..1..1..2

R. H. Hardin, Table of n, a(n) for n = 1..420

Column 1 is A084858.

Empirical for column k:
k=1: a(n) = 4*a(n-1) -4*a(n-2)
k=2: a(n) = 6*a(n-1) -9*a(n-2) for n>3
k=3: a(n) = 10*a(n-1) -29*a(n-2) +20*a(n-3) -4*a(n-4)
k=4: [order 6] for n>7
k=5: [order 10]
k=6: [order 14] for n>15
k=7: [order 26]

A268809 T(n,k)=Number of nXk 0..2 arrays with some element plus some horizontally, vertically or antidiagonally adjacent neighbor totalling two not more than once.

Original entry on oeis.org

3, 9, 9, 24, 34, 24, 60, 104, 104, 60, 144, 290, 332, 290, 144, 336, 772, 1202, 1202, 772, 336, 768, 1972, 4158, 5848, 4158, 1972, 768, 1728, 4914, 14308, 28452, 28452, 14308, 4914, 1728, 3840, 12010, 48460, 135912, 195384, 135912, 48460, 12010, 3840, 8448

Offset: 1

R. H. Hardin, Feb 13 2016

Table starts
....3.....9......24.......60........144.........336...........768
....9....34.....104......290........772........1972..........4914
...24...104.....332.....1202.......4158.......14308.........48460
...60...290....1202.....5848......28452......135912........640926
..144...772....4158....28452.....195384.....1316226.......8734264
..336..1972...14308...135912....1316226....12432856.....115671422
..768..4914...48460...640926....8734264...115671422....1508087180
.1728.12010..162722..2990786...57302798..1062318610...19390335102
.3840.28922..541744.13835892..372342650..9657289546..246666802206
.8448.68836.1791504.63544542.2400532536.87052567448.3110082281974

			Some solutions for n=4 k=4
..1..2..1..2. .1..0..0..0. .2..2..1..2. .2..1..0..1. .2..2..1..2
..1..2..2..2. .0..0..1..0. .1..2..2..2. .0..0..0..0. .1..2..1..2
..2..2..2..1. .0..0..0..1. .2..1..2..1. .0..1..0..0. .2..2..2..2
..2..2..2..2. .0..0..0..0. .2..2..1..2. .0..0..0..0. .2..2..1..2

R. H. Hardin, Table of n, a(n) for n = 1..637

Column 1 is A084858.

Empirical for column k:
k=1: a(n) = 4*a(n-1) -4*a(n-2)
k=2: a(n) = 2*a(n-1) +3*a(n-2) -2*a(n-3) -6*a(n-4) -4*a(n-5) -a(n-6) for n>7
k=3: [order 10] for n>12
k=4: [order 16] for n>19
k=5: [order 26] for n>29
k=6: [order 42] for n>45
k=7: [order 68] for n>71

A084860 Expansion of (1 - 2x + 2x^2 - x^3)/(1 - 2x)^2.

Original entry on oeis.org

1, 2, 6, 15, 36, 84, 192, 432, 960, 2112, 4608, 9984, 21504, 46080, 98304, 208896, 442368, 933888, 1966080, 4128768, 8650752, 18087936, 37748736, 78643200, 163577856, 339738624, 704643072, 1459617792, 3019898880, 6241124352

Offset: 0

Paul Barry, Jun 12 2003

Partial sums are A084858. Partial sums of A084860.

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

Cf. A026741.

Cf. A139633.

CoefficientList[Series[(1-2x+2x^2-x^3)/(1-2x)^2,{x,0,50}],x]  (* Harvey P. Dale, Mar 30 2011 *)

a(0)=1, a(n+1) = 3*2^(n-2)*(n+3) - 0^n/4.
Equals binomial transform of nonzero terms of A026741: (1, 1, 3, 2, 5, 3, 7, 4, ...). - Gary W. Adamson, Apr 25 2008
Equals row sums of triangle A139633. - Gary W. Adamson, Apr 27 2008

A165241 Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,1,0,0,0,0,0,0,0,...] DELTA [1,0,1,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 4, 9, 6, 1, 8, 24, 25, 10, 1, 16, 60, 85, 55, 15, 1, 32, 144, 258, 231, 105, 21, 1, 64, 336, 728, 833, 532, 182, 28, 1, 128, 768, 1952, 2720, 2241, 1092, 294, 36, 1, 256, 1728, 5040, 8280, 8361, 5301, 2058, 450, 45, 1

Offset: 0

Philippe Deléham, Sep 09 2009

Rows sums: A006012; Diagonal sums: A052960.

The sums of each column of A117317 with its subsequent column, treated as a lower triangular matrix with an initial null column attached, or, equivalently, the products of the row polynomials p(n,y) of A117317 with (1+y) with the initial first row below added to the final result. The reversal of A117317 is A056242 with several combinatorial interpretations. - Tom Copeland, Jan 08 2017

			Triangle begins:
  1;
  1,  1;
  2,  3,  1;
  4,  9,  6,  1;
  8, 24, 25, 10,  1; ...

Cf. A000012, A000217, A005582, A011782, A084858.

Cf. A056242, A117317.

Sum_{k=0..n} T(n,k)*x^k = A009116(n), A000007(n), A011782(n), A006012(n), A083881(n), A081335(n), A090139(n), A145301(n), A145302(n), A145303(n), A143079(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, respectively. Sum_{k=0..n} T(n,k)*x^(n-k) = A123335(n), A000007(n), A000012(n), A006012(n), A084120(n), A090965(n), A165225(n), A165229(n), A165230(n), A165231(n), A165232(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, respectively.
G.f.: (1-(1+y)*x)/(1-2(1+y)*x+(y+y^2)*x^2). - Philippe Deléham, Dec 19 2011
T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-1) - T(n-2,k-2) with T(0,0) = T(1,0) = T(1,1) = 1 and T(n,k) = 0 if k<0 or if nPhilippe Deléham, Dec 19 2011

O.g.f. corrected by Tom Copeland, Jan 15 2017

A202481 Column k = 3 of triangular array in A165241.

Original entry on oeis.org

1, 10, 55, 231, 833, 2720, 8280, 23920, 66352, 178176, 465920, 1191680, 2991360, 7389184, 17999872, 43315200, 103116800, 243138560, 568393728, 1318518784, 3037265920, 6952058880, 15820390400, 35809918976, 80659611648

Offset: 0

Philippe Deléham, Dec 20 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-24,32,-16).

Cf. A011782, A084858, A165241, A202480

LinearRecurrence[{8,-24,32,-16},{1,10,55,231,833},30] (* Harvey P. Dale, Dec 04 2018 *)

G.f.: (1+2x-x^2-x^3+x^4)/(1-2x)^4.
a(n) = A165241(n+3,3).
a(n) = 2^n*(n+3)*(3*n+2)*(3*n+5)/32 for n>0. - Bruno Berselli, Dec 21 2011

A202493 Column k = 4 of triangular array in A165241.

Original entry on oeis.org

1, 15, 105, 532, 2241, 8361, 28610, 91740, 279624, 818272, 2315712, 6372480, 17123840, 45082368, 116596224, 296879104, 745543680, 1849344000, 4536958976, 11020075008, 26526547968, 63329075200, 150057123840

Offset: 0

Philippe Deléham, Dec 20 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-40,80,-80,32).

Cf. A011782, A084858, A165241, A202480, A202481.

a(n) = A165241(n+4,4).
G.f.: (1+5x-5x^2+2x^3+x^4-x^5)/(1-2x)^5.
a(n) = 2^n*(n+4)*(27*n^3+126*n^2+169*n+62)/256 for n>0. - Bruno Berselli, Dec 21 2011

a(12) added by Bruno Berselli, Dec 21 2011

A308663 Partial sums of A097805.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 4, 5, 7, 8, 8, 9, 12, 15, 16, 16, 17, 21, 27, 31, 32, 32, 33, 38, 48, 58, 63, 64, 64, 65, 71, 86, 106, 121, 127, 128, 128, 129, 136, 157, 192, 227, 248, 255, 256, 256, 257, 265, 293, 349, 419, 475, 503, 511, 512

Offset: 0

Paul Curtz, Jun 15 2019

Curtz (1965), page 15, from right to left, gives (F1):
  1/2;
  1/4,  3/4;
  1/8,  4/8,   7/8;
  1/16, 5/16, 11/16, 15/16;
  ... .
Numerators + Denominators = (C) =
   3;
   5,  7;
   9, 12, 15;
  17, 21, 27, 31;
  ... .
This is the current sequence without powers of 2.
The triangle (P) for a(n) is
  1;
  1, 2;
  2, 3,  4;
  4, 5,  7,  8;
  8, 9, 12, 15, 16;
  ... .
(C) is the core of (P).
Extension of (F1). (F2) =
  0/1;
  0/1, 1/1;
  0/2, 1/2, 2/2;
  0/4, 1/4, 3/4, 4/4;
  0/8, 1/8, 4/8, 7/8, 8/8;
  ... .
(Mentioned, without 0's, op. cit., page 16.)
a(n) = Numerators + Denominators.
Row sums of triangle (P): A084858(n).
From right to left, with alternating signs: 1, 1, 3, 2, 12, 8, 48, 32, ..., see A098646.
For triangle (C), row sums give A167667(n+1).
From right to left, with alternating signs: A098646(n).
Rank of A016116(n): 0 together with A117142.

Paul Curtz, Accélération de la convergence de certaines séries alternées à l'aide des fonctions de sommation, Thèse de 3ème Cycle d'Analyse Numérique, Faculté des Sciences de l'Université de Paris, 4 mai 1965.

Cf. A097805.

T(n,k) = ceiling(2^(n-1)) + Sum_{j=0..k-1} binomial(n-1,j). - Alois P. Heinz, Jun 15 2019

a(n+1) = a(n) + A097805(n+1) for n >= 0.

Edited by N. J. A. Sloane, Sep 15 2019

cp's OEIS Frontend

A001651 Numbers not divisible by 3.

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A268971 T(n,k)=Number of nXk 0..2 arrays with some element plus some horizontally or antidiagonally adjacent neighbor totalling two not more than once.

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A269052 T(n,k)=Number of nXk 0..2 arrays with some element plus some horizontally, diagonally or antidiagonally adjacent neighbor totalling two not more than once.

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A268628 T(n,k)=Number of nXk 0..2 arrays with some element plus some horizontally or vertically adjacent neighbor totalling two no more than once.

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A268809 T(n,k)=Number of nXk 0..2 arrays with some element plus some horizontally, vertically or antidiagonally adjacent neighbor totalling two not more than once.

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A084860 Expansion of (1 - 2x + 2x^2 - x^3)/(1 - 2x)^2.

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A165241 Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,1,0,0,0,0,0,0,0,...] DELTA [1,0,1,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

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A202481 Column k = 3 of triangular array in A165241.

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