A068911 -id:A068911 - cp's OEIS frontend

A068912 Number of n step walks (each step +/-1 starting from 0) which are never more than 3 or less than -3.

1, 2, 4, 8, 14, 28, 48, 96, 164, 328, 560, 1120, 1912, 3824, 6528, 13056, 22288, 44576, 76096, 152192, 259808, 519616, 887040, 1774080, 3028544, 6057088, 10340096, 20680192, 35303296, 70606592, 120532992, 241065984, 411525376, 823050752, 1405035520, 2810071040

Offset: 0

Henry Bottomley, Mar 06 2002

The number of n step walks (each step +/-1 starting from 0) which are never more than k or less than -k is given by a(n,k) = 2^n/(k+1)*Sum_{r=1..k+1} (-1)^r*cos((Pi*(2*r-1))/(2*(k+1)))^n*cot((Pi*(1-2*r))/(4*(k+1))). Here we have k=3. - Herbert Kociemba, Sep 19 2020

T. Mansour and A. O. Munagi, Alternating subsets modulo m, Rocky Mt. J. Math. 42, No. 4, 1313-1325 (2012), Table 1 q=(0,1,1,1).
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-2).

Cf. A000007, A016116 (without initial term), A068911, A068913 for similar.

# From Peter Luschny, Sep 20 2020: (Start)
r := 1 + 2^(1/2): s := 1 - 2^(1/2):
c := n -> (1+r)^(n/2)*(r+(2*(1+r))^(1/2)+(-1)^n*(r-(2*(1+r))^(1/2))):
b := n -> (1+s)^(n/2)*(s-(2*(1+s))^(1/2)+(-1)^n*(s+(2*(1+s))^(1/2))):
a := n -> (c(n) + b(n))/4:
# Alternatively:
a := proc(n) local h; h := n -> add((1+x)*(2+x)^(n/2), x=[sqrt(2),-sqrt(2)]);
if n::even then h(n)/2 else h(n-1) fi end:
seq(simplify(a(n)), n=0..30); # (End)

nn=33; CoefficientList[Series[s+a + b + c + d + e +f/.Solve[{s ==1 + x a + x b, a==x s + x c, b==x s +x d, c==x a +x e, d== x b + x f, e==x c, f==x d,z==x e + x f },{s,a,b,c,d,e,f,z}],{x,0,nn}],x] (* Geoffrey Critzer, Jan 13 2014 *)
a[n_,k_]:=2^n /(k+1) Sum[(-1)^r Cos[(Pi (2r-1))/(2 (k+1))]^n Cot[(Pi (1-2r))/(4 (k+1))] ,{r,1,k+1}]
Table[a[n,3],{n,0,40}]//Round (* Herbert Kociemba, Sep 19 2020 *)
a[n_]:=Module[{r=2+Sqrt[2]},Floor[(r^(n/2) (-2 (-1+(-1)^n) Sqrt[r]+(1+(-1)^n) r))/(4 Sqrt[2])]]
Table[a[n],{n,0,40}] (* Herbert Kociemba, Sep 21 2020 *)

G.f.: (1+2*x)/(1-4*x^2+2*x^4).
a(n) = A068913(3, n).
a(n) = 4*a(n-2) - 2*a(n-4).
a(2*n) = A007070(n) = 2*a(2*n-1)-A060995(n); a(2*n+1) = 2*a(2*n).
a(n) = (2^n/4)*Sum_{r=1..4} (-1)^r*cos((Pi*(2*r-1))/8)^n*cot((Pi*(1-2*r))/16). - Herbert Kociemba, Sep 19 2020
Conjecture: a(n) = floor((1+r)^(n/2)*(r+(2*(1+r))^(1/2)+(-1)^n*(r-(2*(1+r))^(1/2)))/4) where r = 1 + 2^(1/2). - Peter Luschny, Sep 20 2020
From Herbert Kociemba, Sep 20 2020: (Start)
With the standard procedure to obtain an explicit formula for a(n) for a linear recurrence and r1=2-sqrt(2) and r2=2+sqrt(2) we get
a(n) = a1(n) + a2(n)  with
a1(n) = -(r1^(n/2)*(-2*(-1+(-1)^n)*sqrt(r1)+(1+(-1)^n)*r1))/(4*sqrt(2)) and
a2(n) = +(r2^(n/2)*(-2*(-1+(-1)^n)*sqrt(r2)+(1+(-1)^n)*r2))/(4*sqrt(2)).
We have -1

A077846 Expansion of g.f. 1/(1 - 3x + 2x^3).

Original entry on oeis.org

1, 3, 9, 25, 69, 189, 517, 1413, 3861, 10549, 28821, 78741, 215125, 587733, 1605717, 4386901, 11985237, 32744277, 89459029, 244406613, 667731285, 1824275797, 4984014165, 13616579925, 37201188181, 101635536213, 277673448789, 758617970005, 2072582837589, 5662401615189

Offset: 0

N. J. A. Sloane, Nov 17 2002

Number of (s(0), s(1), ..., s(n+2)) such that 0 < s(i) < 6 and |s(i) - s(i-1)| <= 1 for i = 1..n+2, s(0) = 1, s(n+2) = 3. - Herbert Kociemba, Jun 17 2004

A Whitney transform of 2^n (see Benoit Cloitre formula and A004070). The Whitney transform maps the sequence with g.f. g(x) to that with g.f. (1/(1-x))g(x(1+x)). - Paul Barry, Feb 16 2005

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Fan Chung and R. L. Graham, Primitive juggling sequences, Am. Math. Monthly 115 (3) (2008) 185-194.
William J. Keith, Partitions into parts simultaneously regular, distinct, and/or flat, Proceedings of CANT 2016; arXiv:1911.04755 [math.CO], 2019. Mentions this sequence.
Yassine Otmani, The 2-Pascal Triangle and a Related Riordan Array, J. Int. Seq. (2025) Vol. 28, Issue 3, Art. No. 25.3.5. See p. 12.
Index entries for linear recurrences with constant coefficients, signature (3,0,-2).

First differences are in A002605.

Cf. A028859, A052945, A057960, A068911,

CoefficientList[Series[1 / (1 - 3 x + 2 x^3), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 19 2013 *)
LinearRecurrence[{3,0,-2},{1,3,9},40] (* Harvey P. Dale, Apr 27 2014 *)

a(n)=sum(i=0,n,sum(j=0,n,2^j*binomial(j,i-j)))

Vec(1/(1-3*x+2*x^3) + O(x^100)) \\ Altug Alkan, Mar 24 2016

a(n) = 3*a(n-1) - 2*a(n-3) = 2*A057960(n) - 1 = round(2*A028859(n)/sqrt(3) - 1/3) = Sum_{i} b(n, i), where b(n, 0) = b(n, 6) = 0, b(0, 3) = 1, b(0, i) = 0 if i <> 3 and b(n+1, i) = b(n, i-1) + b(n, i) + b(n, i+1) if 0 < i < 6 (i.e., the number of three-choice paths along a corridor width 5, starting from the middle). - Henry Bottomley, Mar 06 2003
Binomial transform of A068911. a(n) = (1+sqrt(3))^n*(2+sqrt(3))/3 + (1-sqrt(3))^n*(2-sqrt(3))/3 - 1/3. - Paul Barry, Feb 17 2004
a(0)=1; for n >= 1, a(n) = ceiling((1+sqrt(3))*a(n-1)). - Benoit Cloitre, Jun 19 2004
a(n) = Sum_{i=0..n} Sum_{j=0..n} 2^j*binomial(j, i-j). - Benoit Cloitre, Oct 23 2004
a(n) = 2*(a(n-1) + a(n-2)) + 1, n > 1. - Gary Detlefs, Jun 20 2010
a(n) = (2*A052945(n+1) - 1)/3. - R. J. Mathar, Mar 31 2011
a(n) = floor((1+sqrt(3))^(n+2)/6). - Bruno Berselli, Feb 05 2013
a(n) = (-2 + (1-sqrt(3))^(n+2) + (1+sqrt(3))^(n+2))/6. - Alexander R. Povolotsky, Feb 13 2016
E.g.f.: exp(x)*(4*cosh(sqrt(3)*x) + 2*sqrt(3)*sinh(sqrt(3)*x) - 1)/3. - Stefano Spezia, Mar 02 2024

Name changed by Arkadiusz Wesolowski, Dec 06 2011

A216212 Number of n step walks (each step +-1 starting from 0) which are never more than 4 or less than -4.

Original entry on oeis.org

1, 2, 4, 8, 16, 30, 60, 110, 220, 400, 800, 1450, 2900, 5250, 10500, 19000, 38000, 68750, 137500, 248750, 497500, 900000, 1800000, 3256250, 6512500, 11781250, 23562500, 42625000, 85250000, 154218750, 308437500, 557968750, 1115937500, 2018750000, 4037500000

Offset: 0

Philippe Deléham, Mar 13 2013

The number of n step walks (each step +-1 starting from 0) which are never more than k or less than -k is given by a(n,k) = 2^n/(k+1)*Sum_{r=1..k+1} (-1)^r*cos((Pi*(2*r-1))/(2*(k+1)))^n*cot((Pi*(1-2*r))/(4*(k+1))), n<>0 if k even. Here we have k=4. - Herbert Kociemba, Sep 22 2020

Index entries for linear recurrences with constant coefficients, signature (0,5,0,-5).

Cf. A068911, A068912, A068913, A178381 (starting from 4).

nn=30;CoefficientList[Series[(1+x-x^2)^2/(1-5x^2+5x^4),{x,0,nn}],x] (* Geoffrey Critzer, Jan 14 2014 *)
a[0,4]=1; a[n_,k_]:=2^n/(k+1) Sum[(-1)^r Cos[(Pi (2r-1))/(2 (k+1))]^n Cot[(Pi (1-2r))/(4 (k+1))],{r,1,k+1}]
Table[a[n,4],{n,0,40}]//Round (* Herbert Kociemba, Sep 22 2020 *)

a(n) = A068913(4,n).
G.f.: (1+2*x-x^2-2*x^3+x^4)/(1-5*x^2+5*x^4).
a(n) = 5*a(n-2) - 5*a(n-4), a(0) = 1, a(1) = 2, a(2) = 4, a(3) = 8, a(4) = 16.
a(2*n+1) = 2*A039717(n+1), a(2*n+2) = 4*A039717(n+1).
a(n) = (2^n/5)*Sum_{r=1..5} (-1)^r*cos(Pi*(2*r-1)/10)^n*cot(Pi*(1-2*r)/20), n>0. - Herbert Kociemba, Sep 22 2020

A365825 Number of integer partitions of n that are not of length 2 and do not contain n/2.

Original entry on oeis.org

1, 1, 1, 2, 2, 5, 6, 12, 14, 26, 31, 51, 61, 95, 114, 169, 202, 289, 347, 481, 576, 782, 936, 1244, 1487, 1946, 2323, 2997, 3570, 4551, 5414, 6827, 8103, 10127, 11997, 14866, 17575, 21619, 25507, 31166, 36692, 44563, 52362, 63240, 74152, 89112, 104281, 124731

Offset: 0

Gus Wiseman, Sep 19 2023

nonn

Also the number of integer partitions of n with no two possibly equal parts summing to n.

			The a(1) = 1 through a(8) = 14 partitions:
  (1)  (2)  (3)    (4)     (5)      (6)       (7)        (8)
            (111)  (1111)  (221)    (222)     (322)      (332)
                           (311)    (411)     (331)      (521)
                           (2111)   (2211)    (421)      (611)
                           (11111)  (21111)   (511)      (2222)
                                    (111111)  (2221)     (3221)
                                              (3211)     (3311)
                                              (4111)     (5111)
                                              (22111)    (22211)
                                              (31111)    (32111)
                                              (211111)   (221111)
                                              (1111111)  (311111)
                                                         (2111111)
                                                         (11111111)

First condition alone is A058984, complement A004526, ranks A100959.
Second condition alone is A086543, complement A035363, ranks !A344415.
The complement is counted by A238628.
The strict case is A365826, complement A365659.
A000041 counts integer partitions, strict A000009.
A046663 counts partitions with no submultiset summing to k, strict A365663.
A140106 counts strict partitions of length 2, complement A365827.
A182616 counts partitions of 2n that do not contain n, strict A365828.
Cf. A005408, A008967, A068911, A365377, A365544, A365543.

Table[Length[Select[IntegerPartitions[n],Length[#]!=2&&FreeQ[#,n/2]&]],{n,0,15}]

from sympy import npartitions
def A365825(n): return npartitions(n)-(m:=n>>1)-(0 if n&1 else npartitions(m)-1) # Chai Wah Wu, Sep 23 2023

Heinz numbers are A100959 /\ !A344415.

a(n) = A000041(n)-(n-1)/2 if n is odd. a(n) = A000041(n)-n/2-A000041(n/2)+1 if n is even. - Chai Wah Wu, Sep 23 2023

a(31)-a(47) from Chai Wah Wu, Sep 23 2023

A366131 Number of subsets of {1..n} with two elements (possibly the same) summing to n.

Original entry on oeis.org

0, 0, 2, 2, 10, 14, 46, 74, 202, 350, 862, 1562, 3610, 6734, 14926, 28394, 61162, 117950, 249022, 484922, 1009210, 1979054, 4076206, 8034314, 16422922, 32491550, 66045982, 131029082, 265246810, 527304974, 1064175886, 2118785834, 4266269482, 8503841150, 17093775742, 34101458042, 68461196410, 136664112494

Offset: 0

Gus Wiseman, Oct 07 2023

			The a(0) = 0 through a(5) = 14 subsets:
  .  .  {1}    {1,2}    {2}        {1,4}
        {1,2}  {1,2,3}  {1,2}      {2,3}
                        {1,3}      {1,2,3}
                        {2,3}      {1,2,4}
                        {2,4}      {1,3,4}
                        {1,2,3}    {1,4,5}
                        {1,2,4}    {2,3,4}
                        {1,3,4}    {2,3,5}
                        {2,3,4}    {1,2,3,4}
                        {1,2,3,4}  {1,2,3,5}
                                   {1,2,4,5}
                                   {1,3,4,5}
                                   {2,3,4,5}
                                   {1,2,3,4,5}

Index entries for linear recurrences with constant coefficients, signature (2,3,-6).

The complement is counted by A117855.
For pairs summing to n + 1 we have A167936.
A068911 counts subsets of {1..n} w/o two distinct elements summing to n.
A093971/A088809/A364534 count certain types of sum-full subsets.
Cf. A008967, A167762, A238628, A365376, A365377, A365381, A365541, A365544.

Table[Length[Select[Subsets[Range[n]],MemberQ[Total/@Tuples[#,2],n]&]],{n,0,10}]

def A366131(n): return (1<>1)<<1) if n else 0 # Chai Wah Wu, Nov 14 2023

From Chai Wah Wu, Nov 14 2023: (Start)
a(n) = 2*a(n-1) + 3*a(n-2) - 6*a(n-3) for n > 3.
G.f.: 2*x^2*(1 - x)/((2*x - 1)*(3*x^2 - 1)). (End)

A060647 Number of alpha-beta evaluations in a tree of depth n and branching factor b=3.

Original entry on oeis.org

1, 3, 5, 11, 17, 35, 53, 107, 161, 323, 485, 971, 1457, 2915, 4373, 8747, 13121, 26243, 39365, 78731, 118097, 236195, 354293, 708587, 1062881, 2125763, 3188645, 6377291, 9565937, 19131875, 28697813, 57395627, 86093441, 172186883, 258280325, 516560651, 774840977

Offset: 0

Frank Ellermann, Apr 17 2001

			a(2n+1) = 2*a(2n) + 1, a(15) = a(2*7+1) = 2*a(14) + 1 = 2*4373 + 1 = 8747.

P. H. Winston, Artificial Intelligence, Addison-Wesley, 1977, pp. 115-122, (alpha-beta technique).

Harry J. Smith, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (1, 3, -3).

For b=2 see A052955.

Cf. A068911.

A060647 := proc(n,b) option remember: if n mod 2 = 0 then RETURN(2*b^(n/2)-1) else RETURN(b^((n-1)/2) +b^((n+1)/2)-1) fi: end: for n from 0 to 60 do printf(`%d,`, A060647(n,3)) od:
a[0]:=1:a[1]:=3:for n from 2 to 100 do a[n]:=3*a[n-2]+2 od: seq(a[n], n=0..33); # Zerinvary Lajos, Mar 17 2008

f[n_] := Simplify[Sqrt[3]^n(1 + 2/Sqrt[3]) + (1 - 2/Sqrt[3])(-Sqrt[3])^n - 1]; Table[ f[n], {n, 0, 34}] (* or *)
f[n_] := If[ EvenQ[n], 2(3^(n/2)) - 1, 3^((n - 1)/2) + 3^((n + 1)/2) - 1]; Table[ f[n], {n, 0, 34}] (* or *)
CoefficientList[ Series[(1 + 2x - x^2)/((1 - x)(1 - 3x^2)), {x, 0, 35}], x] (* Robert G. Wilson v, Nov 17 2005 *)

a(n) = { if (n%2==0, 2*(3^(n/2)) - 1, my(m=(n - 1)/2); 3^m + 3^(m + 1) - 1) } \\ Harry J. Smith, Jul 09 2009

a(2n) = 2*(3^n) - 1, a(2n+1) = 3^n + 3^(n+1) - 1.
Formula for b branches: a(2n) = 2*(b^n)-1, a(2n+1) = b^n+b^(n+1)-1.
a(n) = A068911(n+1) - 1.
G.f.: (1+2*z-z^2)/((1-z)*(1-3*z^2)). - Emeric Deutsch, Nov 18 2002
a(n) = (sqrt(3))^n(1+2/sqrt(3))+(1-2/sqrt(3))(-sqrt(3))^n-1. - Paul Barry, Apr 17 2004
a(2n+1) = 3*a(2n-1) + 2;  a(2n) = (a(2n-1) + a(2n+1))/2, with a(1)= 1. See A062318 for case where a(1)= 0.
a(n) = (2^((1+(-1)^n)/2))*(b^((2*n-1+(-1)^n)/4))+((1-(-1)^n)/2)*(b^((2*n+1-(-1)^n)/4))-1, with b=3. - Luce ETIENNE, Aug 30 2014

More terms from James Sellers, Apr 19 2001

A306293 Number of binary words of length n such that in every prefix and in every suffix the number of 0's and the number of 1's differ by at most two.

Original entry on oeis.org

1, 2, 4, 6, 10, 16, 26, 42, 70, 110, 194, 288, 550, 754, 1586, 1974, 4630, 5168, 13634, 13530, 40390, 35422, 120146, 92736, 358390, 242786, 1071074, 635622, 3205030, 1664080, 9598706, 4356618, 28763350, 11405774, 86224514, 29860704, 258542470, 78176338

Offset: 0

Alois P. Heinz, Feb 04 2019

All terms with index n > 0 are even.

			a(3) = 6: 001, 010, 011, 100, 101, 110.
a(4) = 10: 0010, 0011, 0100, 0101, 0110, 1001, 1010, 1011, 1100, 1101.
a(5) = 16: 00101, 00110, 01001, 01010, 01011, 01100, 01101, 01110, 10001, 10010, 10011, 10100, 10101, 10110, 11001, 11010.
a(6) = 26: 001010, 001011, 001100, 001101, 001110, 010010, 010011, 010100, 010101, 010110, 011001, 011010, 011100, 100011, 100101, 100110, 101001, 101010, 101011, 101100, 101101, 110001, 110010, 110011, 110100, 110101.
a(7) = 42: 0010101, 0010110, 0011001, ..., 1100110, 1101001, 1101010.
a(8) = 70: 00101010, ..., 00111100, ..., 11000011, ..., 11010101.

Alois P. Heinz, Table of n, a(n) for n = 0..4193
Index entries for linear recurrences with constant coefficients, signature (0,8,0,-22,0,23,0,-6).

Bisections of a(n+2)/2 give: A007689 (even part), A001906(n+2) (odd part).

Cf. A068911, A128588, A303696, A306306, A306315.

a:= n-> `if`(n<2, 1+n, 2*(<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>,
        <-6|23|-22|8>>^iquo(n-2, 2, 'r').[<<2, 5, 13, 35>>,
        <<3, 8, 21, 55>>][1+r])[1, 1]):
seq(a(n), n=0..50);

G.f.: -(x+1)*(4*x^7-4*x^6-7*x^5-5*x^4+5*x^3+5*x^2-x-1) / ((3*x^2-1) *(2*x^2-1) *(x^2+x-1) *(x^2-x-1)).

a(n) <= A306306(n).

A366130 Number of subsets of {1..n} with a subset summing to n + 1.

Original entry on oeis.org

0, 0, 1, 2, 7, 15, 38, 79, 184, 378, 823, 1682, 3552, 7208, 14948, 30154, 61698, 124302, 252125, 506521, 1022768, 2051555, 4127633, 8272147, 16607469, 33258510, 66680774, 133467385, 267349211, 535007304, 1071020315, 2142778192, 4288207796

Offset: 0

Gus Wiseman, Oct 07 2023

			The subset S = {1,2,4} has subset {1,4} with sum 4+1 and {2,4} with sum 5+1 and {1,2,4} with sum 6+1, so S is counted under a(4), a(5), and a(6).
The a(0) = 0 through a(5) = 15 subsets:
  .  .  {1,2}  {1,3}    {1,4}      {1,5}
               {1,2,3}  {2,3}      {2,4}
                        {1,2,3}    {1,2,3}
                        {1,2,4}    {1,2,4}
                        {1,3,4}    {1,2,5}
                        {2,3,4}    {1,3,5}
                        {1,2,3,4}  {1,4,5}
                                   {2,3,4}
                                   {2,4,5}
                                   {1,2,3,4}
                                   {1,2,3,5}
                                   {1,2,4,5}
                                   {1,3,4,5}
                                   {2,3,4,5}
                                   {1,2,3,4,5}

For pairs summing to n + 1 we have A167762, complement A038754.
For n instead of n + 1 we have A365376, for pairs summing to n A365544.
The complement is counted by A365377 shifted.
The complement for pairs summing to n is counted by A365377.
A068911 counts subsets of {1..n} w/o two distinct elements summing to n.
A093971/A088809/A364534 count certain types of sum-full subsets.
Cf. A004526, A004737, A008967, A046663, A238628, A365381, A365541.

Table[Length[Select[Subsets[Range[n]],MemberQ[Total/@Subsets[#],n+1]&]],{n,0,10}]

from itertools import combinations
from sympy.utilities.iterables import partitions
def A366130(n):
    a = tuple(set(p.keys()) for p in partitions(n+1,k=n) if max(p.values(),default=0)==1)
    return sum(1 for k in range(2,n+1) for w in (set(d) for d in combinations(range(1,n+1),k)) if any(s<=w for s in a)) # Chai Wah Wu, Nov 24 2023

Diagonal k = n + 1 of A365381.

a(20)-a(32) from Chai Wah Wu, Nov 24 2023

A216241 Number of n-step walks (each step +-1 starting from 0) which are never more than 5 or less than -5.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 62, 124, 236, 472, 890, 1780, 3340, 6680, 12502, 25004, 46732, 93464, 174554, 349108, 651740, 1303480, 2432918, 4865836, 9080956, 18161912, 33892954, 67785908, 126494956, 252989912, 472095062, 944190124, 1761901676, 3523803352, 6575544410, 13151088820

Offset: 0

Philippe Deléham, Mar 15 2013

Cf. Rows of A068913: A000007, A016116 (without initial term), A068911, A068912, A214846, A216212.

nn=35;CoefficientList[Series[(1+2x)(1-x^2)^2/(1-6x^2+9x^4-2x^6),{x,0,nn}],x] (* Geoffrey Critzer, Jan 14 2014 *)

a(n) = A068913(5,n).
a(n) = 6*a(n-2) - 9*a(n-4) + 2*a(n-6).
a(n) = 2^n for n < 6.
G.f.: ((1-x)^2*(1+x)^2*(1+2*x)) / ((1-2*x^2)*(1-4*x^2+x^4)).
a(2*n+1) = 2*a(2*n).
a(n) = Sum_{k=0..n} A214846(n-k, k). - Philippe Deléham, Mar 25 2013

a(34) corrected by Sean A. Irvine, May 19 2019

A228879 a(n+2) = 3*a(n), starting 4,7.

Original entry on oeis.org

4, 7, 12, 21, 36, 63, 108, 189, 324, 567, 972, 1701, 2916, 5103, 8748, 15309, 26244, 45927, 78732, 137781, 236196, 413343, 708588, 1240029, 2125764, 3720087, 6377292, 11160261, 19131876, 33480783, 57395628, 100442349, 172186884, 301327047, 516560652

Offset: 0

Richard R. Forberg, Sep 06 2013

Successive terms are the square roots of expressions of prior terms. The same recursive formula (see below) can be applied using any term of A001353 as the initializing value to produce the family of sequences, as given in the array A227418. This sequence uses A001353(2) = 4, and is the third row of that array.
a(4n) are the squares of A008776(n).
Binomial transform of a(n) is A021006.
First differences of a(n) = A083658 (without initial two terms).
2nd differences of a(n) = A068911 (with initial term).
a(n-1) is the number of n-digit base 10 numbers where all the digits are even numbers, and each digit differs by 2 from the previous and the next digit. - Graeme McRae, Jun 09 2014

Paolo Xausa, Table of n, a(n) for n = 0..1000
Twitter / MathQuizzes, Puzzle relating to this sequence
Index entries for linear recurrences with constant coefficients, signature (0,3).

Cf. A001353, A008776, A021006, A068911, A083658, A227418.

Row n = 5 of A377000.

LinearRecurrence[{0, 3}, {4, 7}, 50] (* Paolo Xausa, Oct 14 2024 *)

Vec(-(7*x+4)/(3*x^2-1) + O(x^100)) \\ Colin Barker, Jun 09 2014

a(n) = sqrt(3*a(n-1)^2 + (-3)^(n-1)), a(0) = 4.
This divisibility relation also applies: a(n+3) = a(n+2)*a(n+1)/a(n).
G.f.: -(7*x+4) / (3*x^2-1). - Colin Barker, Jun 09 2014
From Stefano Spezia, Mar 20 2022: (Start)
a(n) = 3^((n-1)/2)*(4*sqrt(3) + 7 + (-1)^n*(4*sqrt(3) - 7))/2.
E.g.f.: 4*cosh(sqrt(3)*x) + 7*sinh(sqrt(3)*x)/sqrt(3). (End)

More terms from Colin Barker, Jun 09 2014

Previous Showing 21-30 of 34 results. Next

cp's OEIS Frontend

A068912 Number of n step walks (each step +/-1 starting from 0) which are never more than 3 or less than -3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A077846 Expansion of g.f. 1/(1 - 3*x + 2*x^3).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

A216212 Number of n step walks (each step +-1 starting from 0) which are never more than 4 or less than -4.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A365825 Number of integer partitions of n that are not of length 2 and do not contain n/2.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Python

Formula

Extensions

A366131 Number of subsets of {1..n} with two elements (possibly the same) summing to n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Python

Formula

A060647 Number of alpha-beta evaluations in a tree of depth n and branching factor b=3.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A306293 Number of binary words of length n such that in every prefix and in every suffix the number of 0's and the number of 1's differ by at most two.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A077846 Expansion of g.f. 1/(1 - 3x + 2x^3).