A051890 -id:A051890 - cp's OEIS frontend

A020725 Integers >= 2. a(n) = n+1.

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75

Offset: 1

N. J. A. Sloane

This sequence is closed under multiplication by any integer k > 0. The primitive elements of the sequence (those not divisible by any smaller element) are the primes, A000040. - Franklin T. Adams-Watters, May 22 2006
Possible sums of the final scores of completed Chicago Bears football games. 1 point only is an impossible score in American football. But with the safety 2 and the field goal 3, we can construct the set of integers greater than 1. We can prove this by noting that if a score is even, we can build it with a series of safeties. Of course the other allowed scorings of 3, 6, and 1 after a touchdown, could also be used. Now if a score is odd it is of the form 2k+3. So for any odd number 2m+1, we subtract 3 (or 1 field goal) from it to make it even and divide by 2 to get the number of safeties we need to add back to the field goal. Symbolically, let the odd number be 2m+1; then (2m+1 - 3)/2 = m-1 safeties are needed. Add this to 3 and you will have the number. For example, say we want a score of 99. 99 = 2m+1 and m = 49. So m-1 = 48 safeties + 1 field goal = 99 points. - Cino Hilliard, Feb 03 2006
Possible nonnegative values of (a*b-c*d) where a,b,c and d are distinct positive integers and a+b=c+d. All positive values >=2 are possible: for even values 2n take a=m+n, b=m-n+1, c=m+n+1, d=m-n, where m>n; for odd values 2n+1 take a=m+n, b=m-n, c=m+n+1, d=m-n-1, where m>n+1. Elementary algebra shows that the values 0 and 1 are not possible without violating the assumption that a,b,c and d are distinct. - John Grint, Sep 26 2011
Also numbers n such that a semiprime is equal to the sum of n primes. Bachraoui proved that there is a prime between 2n and 3n for every n > 1, so every n > 1 is in this sequence since any number in that range is the sum of n integers each of which is either 2 or 3. - Charles R Greathouse IV, Oct 27 2011
From Jason Kimberley, Oct 30 2011: (Start)
Moore lower bound on the order of a (k,g) cage: A198300 (square); rows: A000027 (k=2), A027383 (k=3), A062318 (k=4), A061547 (k=5), A198306 (k=6), A198307 (k=7), A198308 (k=8), A198309 (k=9), A198310 (k=10), A094626 (k=11); columns: this sequence (g=3), A005843 (g=4), A002522 (g=5), A051890 (g=6), A188377 (g=7).
Digit string 12 read in base n-1 (for n>3 or by extending notation). (End)
Positive integers whose number of divisors is not 1. - Omar E. Pol, Aug 11 2012
Positive integers where the number of parts function on the set of 2-ary partitions is equidistributed mod 2. - Tom Edgar, Apr 26 2016
This sequence is also the Pierce Expansion of 1/exp(1). - G. C. Greubel, Nov 15 2016
Natural numbers with at least one prime factor. - Michal Bozon, Apr 24 2017
a(n) is the reciprocal of the Integral_{x=0..1} x^n dx. - Felix Huber, Aug 19 2023

M. El Bachraoui, Primes in the interval (2n, 3n), International Journal of Contemporary Mathematical Sciences 1:13 (2006), pp. 617-621.
Tanya Khovanova, Recursive Sequences
Antonio Gracia Llorente, Arithmetic Progression-Representing Constants, OSF Preprint, 2023.
Eric Weisstein's World of Mathematics, Pierce Expansion
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Column 1 of A210976.

Range[2,100] (* Harvey P. Dale, Aug 31 2015 *)
PierceExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@ NestList[{Floor[1/Expand[1 - #[[1]] #[[2]]]], Expand[1 - #[[1]] #[[2]]]} &, {Floor[1/(A - Floor[A])], A - Floor[A]}, n - 1]]; PierceExp[N[1/E , 7!], 50] (* G. C. Greubel, Nov 14 2016 *)

a(n)=n+1 \\ Charles R Greathouse IV, Aug 23 2011

From Franklin T. Adams-Watters, May 22 2006: (Start)
O.g.f.: (2*x - x^2)/(1 - x)^2.
E.g.f.: (1 + x)*exp(x)-1.
Dirichlet g.f.: zeta(s) + zeta(s-1).
a(n) = n + 1 for n>0. (End)

Edited by Jon E. Schoenfield, Sep 20 2013

A188377 a(n) = n^3 - 4n^2 + 6n - 2.

Original entry on oeis.org

7, 22, 53, 106, 187, 302, 457, 658, 911, 1222, 1597, 2042, 2563, 3166, 3857, 4642, 5527, 6518, 7621, 8842, 10187, 11662, 13273, 15026, 16927, 18982, 21197, 23578, 26131, 28862, 31777, 34882, 38183, 41686, 45397, 49322, 53467, 57838, 62441, 67282, 72367

Offset: 3

Adeniji, Adenike & Makanjuola, Samuel (somakanjuola(AT)unilorin.edu.ng) Apr 14 2011

Number of nilpotent elements in the identity difference partial one - one transformation semigroup, denoted by N(IDI_n). For n=3, #N(IDI_n) = 7.

a(n+1) is also the Moore lower bound on the order of an (n,7)-cage. - Jason Kimberley, Oct 20 2011

Vincenzo Librandi, Table of n, a(n) for n = 3..1000
A. Laradji and A. Umar, On the number of nilpotents in the partial symmetric semigroup, Comm. Algebra 32 (2004), 3017-3023.
Gordon Royle, Cages of higher valency
R. P. Sullivan, Semigroups generated by nilpotent transformations, Journal of Algebra 110 (1987), 324-345.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A188716, A188947.

Moore lower bound on the order of a (k,g) cage: A198300 (square); rows: A000027 (k=2), A027383 (k=3), A062318 (k=4), A061547 (k=5), A198306 (k=6), A198307 (k=7), A198308 (k=8), A198309 (k=9), A198310 (k=10), A094626 (k=11); columns: A020725 (g=3), A005843 (g=4), A002522 (g=5), A051890 (g=6), this sequence (g=7). - Jason Kimberley, Oct 30 2011

[n^3 - 4*n^2 + 6*n - 2: n in [3..50]]; // Vincenzo Librandi, May 01 2011

[SequenceToInteger([2^^3,1],n-2):n in [5..50]]; // Jason Kimberley, Oct 20 2011

Table[n^3 - 4*n^2 + 6*n - 2, {n, 3, 80}] (* Vladimir Joseph Stephan Orlovsky, Jul 07 2011 *)
LinearRecurrence[{4,-6,4,-1},{7,22,53,106},50] (* Harvey P. Dale, May 29 2019 *)

a(n)=n^3-4*n^2+6*n-2 \\ Charles R Greathouse IV, Apr 06 2012

a(n+1) = (n+1)^3 - 4*(n+1)^2 + 6*(n+1) - 2
       = (n-1)^3 + 2*(n-1)^2 + 2*(n-1) + 2
       = 1222 read in base n-1.
- Jason Kimberley, Oct 20 2011
From Colin Barker, Apr 06 2012: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
G.f.: x^3*(7 - 6*x + 7*x^2 - 2*x^3)/(1-x)^4. (End)
E.g.f.: 2 - x - x^2 + exp(x)*(x^3 - x^2 + 3*x - 2). - Stefano Spezia, Apr 09 2022

Edited by N. J. A. Sloane, Apr 23 2011

A094626 Expansion of x(1+x)/((1-x)(1-10*x^2)).

Original entry on oeis.org

0, 1, 2, 12, 22, 122, 222, 1222, 2222, 12222, 22222, 122222, 222222, 1222222, 2222222, 12222222, 22222222, 122222222, 222222222, 1222222222, 2222222222, 12222222222, 22222222222, 122222222222, 222222222222, 1222222222222, 2222222222222, 12222222222222

Offset: 0

Paul Barry, May 15 2004

Previous name: Sequence whose n-th term digits sum to n.
a(n) is the smallest integer with digits from {0,1,2} having digit sum n. Namely the base-10 reading of the ternary string of A062318. - Jason Kimberley, Nov 01 2011
a(n) is the Moore lower bound on the order of an (11,n)-cage. - Jason Kimberley, Oct 18 2011

Colin Barker, Table of n, a(n) for n = 0..1000
G. Royle, Cages of higher valency
Index entries for linear recurrences with constant coefficients, signature (1,10,-10).

Cf. A094623, A094624.

Moore lower bound on the order of a (k,g) cage: A198300 (square); rows: A000027 (k=2), A027383 (k=3), A062318 (k=4), A061547 (k=5), A198306 (k=6), A198307 (k=7), A198308 (k=8), A198309 (k=9), A198310 (k=10), this sequence (k=11); columns: A020725 (g=3), A005843 (g=4), A002522 (g=5), A051890 (g=6), A188377 (g=7). - Jason Kimberley, Nov 01 2011

LinearRecurrence[{1, 10, -10}, {0, 1, 2}, 30] (* Paolo Xausa, Feb 21 2024 *)

concat(0, Vec(x*(1+x)/((1-x)*(1-10*x^2)) + O(x^30))) \\ Colin Barker, Mar 17 2017

G.f.: x*(1+x)/((1-x)*(1-10*x^2)).
a(n) = 10^(n/2)*(11*sqrt(10)/180 + 1/9 - (11*sqrt(10)/180 - 1/9)*(-1)^n) - 2/9.
From Colin Barker, Mar 17 2017: (Start)
a(n) = 2*(10^(n/2) - 1)/9 for n even.
a(n) = (11*10^((n-1)/2) - 2)/9 for n odd. (End)
E.g.f.: (20*(cosh(sqrt(10)*x) - cosh(x) - sinh(x)) + 11*sqrt(10)*sinh(sqrt(10)*x))/90. - Stefano Spezia, Apr 09 2022

A185787 Sum of first k numbers in column k of the natural number array A000027; by antidiagonals.

Original entry on oeis.org

1, 7, 25, 62, 125, 221, 357, 540, 777, 1075, 1441, 1882, 2405, 3017, 3725, 4536, 5457, 6495, 7657, 8950, 10381, 11957, 13685, 15572, 17625, 19851, 22257, 24850, 27637, 30625, 33821, 37232, 40865, 44727, 48825, 53166, 57757, 62605, 67717, 73100, 78761, 84707, 90945, 97482, 104325, 111481, 118957, 126760, 134897, 143375

Offset: 1

Clark Kimberling, Feb 03 2011

This is one of many interesting sequences and arrays that stem from the natural number array A000027, of which a northwest corner is as follows:
  1....2.....4.....7...11...16...22...29...
  3....5.....8....12...17...23...30...38...
  6....9....13....18...24...31...39...48...
  10...14...19....25...32...40...49...59...
  15...20...26....33...41...50...60...71...
  21...27...34....42...51...61...72...84...
  28...35...43....52...62...73...85...98...
Blocking out all terms below the main diagonal leaves columns whose sums comprise A185787.  Deleting the main diagonal and then summing give A185787.  Analogous treatments to the left of the main diagonal give A100182 and A101165.  Further sequences obtained directly from this array are easily obtained using the following formula for the array: T(n,k)=n+(n+k-2)(n+k-1)/2.
Examples:
row 1:  A000124
row 2:  A022856
row 3:  A016028
row 4:  A145018
row 5:  A077169
col 1:  A000217
col 2:  A000096
col 3:  A034856
col 4:  A055998
col 5:  A046691
col 6:  A052905
col 7:  A055999
diag. (1,5,...) ...... A001844
diag. (2,8,...) ...... A001105
diag. (4,12,...)...... A046092
diag. (7,17,...)...... A056220
diag. (11,23,...) .... A132209
diag. (16,30,...) .... A054000
diag. (22,38,...) .... A090288
diag. (3,9,...) ...... A058331
diag. (6,14,...) ..... A051890
diag. (10,20,...) .... A005893
diag. (15,27,...) .... A097080
diag. (21,35,...) .... A093328
antidiagonal sums:  (1,5,15,34,...)=A006003=partial sums of A002817.
Let S(n,k) denote the n-th partial sum of column k.  Then
S(n,k)=n*(n^2+3k*n+3*k^2-6*k+5)/6.
S(n,1)=n(n+1)(n+2)/6
S(n,2)=n(n+1)(n+5)/6
S(n,3)=n(n+2)(n+7)/6
S(n,4)=n(n^2+12n+29)/6
S(n,5)=n(n+5)(n+10)/6
S(n,6)=n(n+7)(n+11)/6
S(n,7)=n(n+10)(n+11)/6
Weight array of T: A144112
Accumulation array of T: A185506
Second rectangular sum array of T: A185507
Third rectangular sum array of T: A185508
Fourth rectangular sum array of T: A185509

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000027, A185788, A100182, A101165, A079824.

[n*(7*n^2-6*n+5)/6: n in [1..50]]; // Vincenzo Librandi, Jul 04 2012

f[n_,k_]:=n+(n+k-2)(n+k-1)/2;
s[k_]:=Sum[f[n,k],{n,1,k}];
Factor[s[k]]
Table[s[k],{k,1,70}]  (* A185787 *)
CoefficientList[Series[(3*x^2+3*x+1)/(1-x)^4,{x,0,50}],x] (* Vincenzo Librandi, Jul 04 2012 *)

a(n)=n*(7*n^2-6*n+5)/6.

G.f.: x*(3*x^2+3*x+1)/(1-x)^4. - Vincenzo Librandi, Jul 04 2012

Edited by Clark Kimberling, Feb 25 2023

A061925 a(n) = ceiling(n^2/2) + 1.

Original entry on oeis.org

1, 2, 3, 6, 9, 14, 19, 26, 33, 42, 51, 62, 73, 86, 99, 114, 129, 146, 163, 182, 201, 222, 243, 266, 289, 314, 339, 366, 393, 422, 451, 482, 513, 546, 579, 614, 649, 686, 723, 762, 801, 842, 883, 926, 969, 1014, 1059, 1106, 1153, 1202, 1251, 1302, 1353, 1406

Offset: 0

Henry Bottomley, May 17 2001

a(n+1) gives index of the first occurrence of n in A100795. - Amarnath Murthy, Dec 05 2004
First term in each group in A074148. - Amarnath Murthy, Aug 28 2002
From Christian Barrientos, Jan 01 2021: (Start)
For n >= 3, a(n) is the number of square polyominoes with at least 2n - 2 cells whose bounding box has size 2 X n.
For n = 3, there are 6 square polyominoes with a bounding box of size 2 X 3:
      _       _       _       _       _      
   |||_|   |||_|   |||_|   |||_|   |||_|   |||_
   |||_|   |||     || ||       ||     ||       |||
(End)
Except for a(2), a(n) agrees with the lower matching number of the (n+1) X (n+1) bishop graph up to at least n = 13. - Eric W. Weisstein, Dec 23 2024

Harry J. Smith, Table of n, a(n) for n = 0..1000
Kassie Archer, Ethan Borsh, Jensen Bridges, Christina Graves, and Millie Jeske, Cyclic permutations avoiding patterns in both one-line and cycle forms, arXiv:2312.05145 [math.CO], 2023. See p. 2.
Eric Weisstein's World of Mathematics, Bishop Graph.
Eric Weisstein's World of Mathematics, Lower Matching Number.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A100795, A074147-A074149.

seq(floor((n^2+3)/2),n=0..25); # Gary Detlefs, Feb 13 2010

Table[Ceiling[n^2/2]+1,{n,0,60}] (* Vladimir Joseph Stephan Orlovsky, Apr 02 2011 *)
LinearRecurrence[{2,0,-2,1},{1,2,3,6},60] (* Harvey P. Dale, Jan 03 2024 *)

a(n) = { ceil(n^2/2) + 1 } \\ Harry J. Smith, Jul 29 2009

a(n) = a(n-1) + 2*floor((n-1)/2) + 1 = A061926(3, k) = 2*A002620(n+1) - (n-1) = A000982(n) + 1.
a(2*n) = a(2*n-1) + 2*n - 1 = 2*n^2 + 1 = A058331(n).
a(2*n+1) = a(2*n) + 2*n + 1 = 2*(n^2 + n + 1) = A051890(n+1).
a(n) = floor((n^2+3)/2). - Gary Detlefs, Feb 13 2010
From R. J. Mathar, Feb 19 2010: (Start)
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
G.f.: (1-x^2+2*x^3)/((1+x) * (1-x)^3). (End)
a(n) = (2*n^2 - (-1)^n + 5)/4. - Bruno Berselli, Sep 29 2011
a(n) = A007590(n+1) - n + 1. - Wesley Ivan Hurt, Jul 15 2013
a(n) + a(n+1) = A027688(n). a(n+1) - a(n) = A109613(n). - R. J. Mathar, Jul 20 2013
E.g.f.: ((2 + x + x^2)*cosh(x) + (3 + x + x^2)*sinh(x))/2. - Stefano Spezia, May 07 2021

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 09 2007

A198300 Square array M(k,g), read by antidiagonals, of the Moore lower bound on the order of a (k,g)-cage.

Original entry on oeis.org

3, 4, 4, 5, 6, 5, 6, 8, 10, 6, 7, 10, 17, 14, 7, 8, 12, 26, 26, 22, 8, 9, 14, 37, 42, 53, 30, 9, 10, 16, 50, 62, 106, 80, 46, 10, 11, 18, 65, 86, 187, 170, 161, 62, 11, 12, 20, 82, 114, 302, 312, 426, 242, 94, 12, 13, 22, 101, 146, 457, 518, 937, 682, 485, 126, 13

Offset: 1

Jason Kimberley, Oct 27 2011

k >= 2; g >= 3.
The base k-1 reading of the base 10 string of A094626(g).
Exoo and Jajcay Theorem 1: M(k,g) <= A054760(k,g) with equality if and only if: k = 2 and g >= 3; g = 3 and k >= 2; g = 4 and k >= 2; g = 5 and k = 2, 3, 7 or possibly 57; or g = 6, 8, or 12, and there exists a symmetric generalized n-gon of order k - 1.

			This is the table formed from the antidiagonals for k+g = 5..20:
3   4   5   6    7    8    9     10    11    12    13    14    15   16  17 18
4   6  10  14   22   30    46    62    94   126   190   254   382  510 766
5   8  17  26   53   80   161   242   485   728  1457  2186  4373 6560
6  10  26  42  106  170   426   682  1706  2730  6826 10922 27306
7  12  37  62  187  312   937  1562  4687  7812 23437 39062
8  14  50  86  302  518  1814  3110 10886 18662 65318
9  16  65 114  457  800  3201  5602 22409 39216
10 18  82 146  658 1170  5266  9362 42130
11 20 101 182  911 1640  8201 14762
12 22 122 222 1222 2222 12222
13 24 145 266 1597 2928
14 26 170 314 2042
15 28 197 366
16 30 226
17 32
18

E. Bannai and T. Ito, On finite Moore graphs, J. Fac. Sci. Tokyo, Sect. 1A, 20 (1973) 191-208.
R. M. Damerell, On Moore graphs, Proc. Cambridge Phil. Soc. 74 (1973) 227-236.

Jason Kimberley, Table of n, a(n) for n = 1..20100 (k+g = 5..204)
Jason Kimberley, Table of n, k+g, k, g, M(k,g)=a(n) for k+g = 5..204 (n = 1..20100)
G. Exoo and R. Jajcay, Dynamic cage survey, Electr. J. Combin. (2008, 2011).
Gordon Royle, Cages of higher valency

Moore lower bound on the order of a (k,g) cage: this sequence (square); rows: A000027 (k=2), A027383 (k=3), A062318 (k=4), A061547 (k=5), A198306 (k=6), A198307 (k=7), A198308 (k=8), A198309 (k=9), A198310 (k=10), A094626 (k=11); columns: A020725 (g=3), A005843 (g=4), A002522 (g=5), A051890 (g=6), A188377 (g=7), 2*A053698 (g=8), 2*A053699 (g=10), 2*A053700 (g=12), 2*A053716 (g=14), 2*A053716 (g=16), 2*A102909 (g=18), 2*A103623 (g=20), 2*A060885 (g=22), 2*A105067 (g=24), 2*A060887 (g=26), 2*A104376 (g=28), 2*A104682 (g=30), 2*A105312 (g=32).

Cf. A054760 (the actual order of a (k,g)-cage).

ExtendedStringToInt:=func;
M:=func;
k_:=2;g_:=3;
anti:=func;
[anti(kg):kg in[5..15]];

Table[Function[g, FromDigits[#, k - 1] &@ IntegerDigits@ SeriesCoefficient[x (1 + x)/((1 - x) (1 - 10 x^2)), {x, 0, g}]][n - k + 3], {n, 2, 12}, {k, n, 2, -1}] // Flatten (* Michael De Vlieger, May 15 2017 *)

M(k,2i) = 2 sum_{j=0}^{i-1}(k-1)^j =  string "2"^i read in base k-1.
M(k,2i+1) = (k-1)^i +  2 sum_{j=0}^{i-1}(k-1)^j = string "1"*"2"^i read in base k-1.
Recurrence:
M(k,3) = k + 1,
M(k,2i) = M(k,2i-1) + (k-1)^(i-1),
M(k,2i+1) = M(k,2i) + (k-1)^i.

A198306 Moore lower bound on the order of a (6,g)-cage.

Original entry on oeis.org

7, 12, 37, 62, 187, 312, 937, 1562, 4687, 7812, 23437, 39062, 117187, 195312, 585937, 976562, 2929687, 4882812, 14648437, 24414062, 73242187, 122070312, 366210937, 610351562, 1831054687, 3051757812, 9155273437, 15258789062

Offset: 3

Jason Kimberley, Oct 30 2011

Gordon Royle, Cages of higher valency
Index entries for linear recurrences with constant coefficients, signature (1,5,-5).

Moore lower bound on the order of a (k,g) cage: A198300 (square); rows: A000027 (k=2), A027383 (k=3), A062318 (k=4), A061547 (k=5), this sequence (k=6), A198307 (k=7), A198308 (k=8), A198309 (k=9), A198310 (k=10), A094626 (k=11); columns: A020725 (g=3), A005843 (g=4), A002522 (g=5), A051890 (g=6), A188377 (g=7).

LinearRecurrence[{1,5,-5},{7,12,37},30] (* Harvey P. Dale, Jun 28 2015 *)

a(2*i) = 2*Sum_{j=0..i-1} 5^j = string "2"^i read in base 5.
a(2*i+1) = 5^i + 2*Sum_{j=0..i-1} 5^j = string "1"*"2"^i read in base 5.
a(n) <= A218554(n). - Jason Kimberley, Dec 21 2012
a(n) = a(n-1)+5*a(n-2)-5*a(n-3). G.f.: -x^3*(10*x^2-5*x-7) / ((x-1)*(5*x^2-1)). - Colin Barker, Feb 01 2013
From Colin Barker, Nov 25 2016: (Start)
a(n) = (5^(n/2) - 1)/2 for n>2 and even.
a(n) = (3*5^((n-1)/2) - 1)/2 for n>2 and odd. (End)
E.g.f.: (5*cosh(sqrt(5)*x) - 5*cosh(x) - 5*sinh(x) + 3*sqrt(5)*sinh(sqrt(5)*x) - 10*x*(1 + x))/10. - Stefano Spezia, Apr 07 2022

A198307 Moore lower bound on the order of a (7,g)-cage.

Original entry on oeis.org

8, 14, 50, 86, 302, 518, 1814, 3110, 10886, 18662, 65318, 111974, 391910, 671846, 2351462, 4031078, 14108774, 24186470, 84652646, 145118822, 507915878, 870712934, 3047495270, 5224277606, 18284971622, 31345665638, 109709829734, 188073993830, 658258978406

Offset: 3

Jason Kimberley, Oct 30 2011

Colin Barker, Table of n, a(n) for n = 3..1000
Gordon Royle, Cages of higher valency
Index entries for linear recurrences with constant coefficients, signature (1,6,-6).

Moore lower bound on the order of a (k,g) cage: A198300 (square); rows: A000027 (k=2), A027383 (k=3), A062318 (k=4), A061547 (k=5), A198306 (k=6), this sequence (k=7), A198308 (k=8), A198309 (k=9), A198310 (k=10), A094626 (k=11); columns: A020725 (g=3), A005843 (g=4), A002522 (g=5), A051890 (g=6), A188377 (g=7).

DeleteCases[CoefficientList[Series[2 x^3*(4 + 3 x - 6 x^2)/((1 - x) (1 - 6 x^2)), {x, 0, 31}], x], 0] (* Michael De Vlieger, Mar 17 2017 *)
LinearRecurrence[{1,6,-6},{8,14,50},30] (* or *) CoefficientList[ Series[ -((2 (-4-3 x+6 x^2))/(1-x-6 x^2+6 x^3)),{x,0,30}],x] (* Harvey P. Dale, Aug 03 2021 *)

Vec(2*x^3*(4 + 3*x - 6*x^2) / ((1 - x)*(1 - 6*x^2)) + O(x^40)) \\ Colin Barker, Mar 17 2017

a(2*i) = 2*Sum_{j=0..i-1}6^j =  string "2"^i read in base 6.
a(2*i+1) = 6^i +  2*Sum_{j=0..i-1}6^j = string "1"*"2"^i read in base 6.
a(n) <= A218555(n).
From Colin Barker, Feb 01 2013: (Start)
a(n) = a(n-1) + 6*a(n-2) - 6*a(n-3) for n>5.
G.f.: 2*x^3*(4 + 3*x - 6*x^2) / ((1 - x)*(1 - 6*x^2)). (End)
From Colin Barker, Mar 17 2017: (Start)
a(n) = 2*(6^(n/2) - 1)/5 for n>2 and even.
a(n) = (7*6^(n/2-1/2) - 2)/5 for n>2 and odd. (End)
E.g.f.: (12*(cosh(sqrt(6)*x) - cosh(x)) + 7*sqrt(6)*sinh(sqrt(6)*x) - 12*sinh(x) - 30*x*(1 + x))/30. - Stefano Spezia, Apr 07 2022

A198308 Moore lower bound on the order of an (8,g)-cage.

Original entry on oeis.org

9, 16, 65, 114, 457, 800, 3201, 5602, 22409, 39216, 156865, 274514, 1098057, 1921600, 7686401, 13451202, 53804809, 94158416, 376633665, 659108914, 2636435657, 4613762400, 18455049601, 32296336802, 129185347209, 226074357616, 904297430465, 1582520503314

Offset: 3

Jason Kimberley, Oct 30 2011

Colin Barker, Table of n, a(n) for n = 3..1000
Gordon Royle, Cages of higher valency
Index entries for linear recurrences with constant coefficients, signature (1,7,-7).

Moore lower bound on the order of a (k,g) cage: A198300 (square); rows: A000027 (k=2), A027383 (k=3), A062318 (k=4), A061547 (k=5), A198306 (k=6), A198307 (k=7), this sequence (k=8), A198309 (k=9), A198310 (k=10), A094626 (k=11); columns: A020725 (g=3), A005843 (g=4), A002522 (g=5), A051890 (g=6), A188377 (g=7).

LinearRecurrence[{1,7,-7},{9,16,65},40] (* Harvey P. Dale, Oct 14 2019 *)

Vec(x^3*(9 + 7*x - 14*x^2) / ((1 - x)*(1 - 7*x^2)) + O(x^40)) \\ Colin Barker, Mar 17 2017

a(2*i) = 2 Sum_{j=0..i-1} 7^j = string "2"^i read in base 7.
a(2*i+1) = 7^i + 2 Sum_{j=0..i-1} 7^j = string "1"*"2"^i read in base 7.
From Colin Barker, Feb 01 2013: (Start)
a(n) = a(n-1) + 7*a(n-2) - 7*a(n-3) for n>5.
G.f.: x^3*(9 + 7*x - 14*x^2) / ((1 - x)*(1 - 7*x^2)). (End)
From Colin Barker, Mar 17 2017: (Start)
a(n) = (7^(n/2) - 1)/3 for n even.
a(n) = (4*7^(n/2-1/2) - 1)/3 for n odd. (End)
E.g.f.: (7*(cosh(sqrt(7)*x) - cosh(x) - sinh(x)) + 4*sqrt(7)*sinh(sqrt(7)*x) - 21*x*(1 + x))/21. - Stefano Spezia, Apr 09 2022

A198309 Moore lower bound on the order of a (9,g)-cage.

Original entry on oeis.org

10, 18, 82, 146, 658, 1170, 5266, 9362, 42130, 74898, 337042, 599186, 2696338, 4793490, 21570706, 38347922, 172565650, 306783378, 1380525202, 2454267026, 11044201618, 19634136210, 88353612946, 157073089682, 706828903570, 1256584717458, 5654631228562

Offset: 3

Jason Kimberley, Oct 30 2011

Colin Barker, Table of n, a(n) for n = 3..1000
Gordon Royle, Cages of higher valency
Index entries for linear recurrences with constant coefficients, signature (1,8,-8).

Moore lower bound on the order of a (k,g) cage: A198300 (square); rows: A000027 (k=2), A027383 (k=3), A062318 (k=4), A061547 (k=5), A198306 (k=6), A198307 (k=7), A198308 (k=8), this sequence (k=9), A198310 (k=10), A094626 (k=11); columns: A020725 (g=3), A005843 (g=4), A002522 (g=5), A051890 (g=6), A188377 (g=7).

LinearRecurrence[{1,8,-8},{10,18,82},30] (* Harvey P. Dale, Apr 03 2015 *)

Vec(2*x^3*(5 + 4*x - 8*x^2) / ((1 - x)*(1 - 8*x^2)) + O(x^40)) \\ Colin Barker, Mar 17 2017

a(2*i) = 2 Sum_{j=0..i-1} 8^j =  string "2"^i read in base 8.
a(2*i+1) = 8^i + 2 Sum_{j=0..i-1} 8^j = string "1"*"2"^i read in base 8.
From Colin Barker, Feb 01 2013: (Start)
a(n) = a(n-1) + 8*a(n-2) - 8*a(n-3) for n>5.
G.f.: 2*x^3*(5 + 4*x - 8*x^2) / ((1 - x)*(1 - 8*x^2)). (End)
From Colin Barker, Mar 17 2017: (Start)
a(n) = 2*(2^(3*n/2) - 1)/7 for n even.
a(n) = (9*2^((3*(n-1))/2) - 2)/7 for n odd. (End)
E.g.f.: (8*(cosh(2*sqrt(2)*x) - cosh(x) - sinh(x)) + 9*sqrt(2)*sinh(2*sqrt(2)*x) - 28*x*(1 + x))/28. - Stefano Spezia, Apr 09 2022

cp's OEIS Frontend

A020725 Integers >= 2. a(n) = n+1.

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A188377 a(n) = n^3 - 4n^2 + 6n - 2.

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A094626 Expansion of x*(1+x)/((1-x)*(1-10*x^2)).

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A185787 Sum of first k numbers in column k of the natural number array A000027; by antidiagonals.

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A061925 a(n) = ceiling(n^2/2) + 1.

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A198300 Square array M(k,g), read by antidiagonals, of the Moore lower bound on the order of a (k,g)-cage.

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A198306 Moore lower bound on the order of a (6,g)-cage.

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A094626 Expansion of x(1+x)/((1-x)(1-10*x^2)).