A188377 -id:A188377 - cp's OEIS frontend

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

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

A198310 Moore lower bound on the order of a (10,g)-cage.

Original entry on oeis.org

11, 20, 101, 182, 911, 1640, 8201, 14762, 73811, 132860, 664301, 1195742, 5978711, 10761680, 53808401, 96855122, 484275611, 871696100, 4358480501, 7845264902, 39226324511, 70607384120, 353036920601, 635466457082, 3177332285411

Offset: 3

Jason Kimberley, Oct 30 2011

Paolo Xausa, Table of n, a(n) for n = 3..1000
Gordon Royle, Cages of higher valency
Index entries for linear recurrences with constant coefficients, signature (1,9,-9).

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), this sequence (k=10), A094626 (k=11); columns: A020725 (g=3), A005843 (g=4), A002522 (g=5), A051890 (g=6), A188377 (g=7).

A198310[n_] := (-3-(-3)^n+4*3^n)/12; Array[A198310, 30, 3] (* or *)
LinearRecurrence[{1, 9, -9}, {11, 20, 101}, 30] (* Paolo Xausa, Feb 21 2024 *)

a(n)=(-3-(-3)^n+4*3^n)/12 \\ Charles R Greathouse IV, Jul 06 2017

a(2i) = 2*Sum_{j=0..i-1} 9^j =  string "2"^i read in base 9.
a(2i+1) = 9^i +  2*Sum_{j=0..i-1} 9^j = string "1"*"2"^i read in base 9.
From Colin Barker, Feb 01 2013: (Start)
a(n) = (-3-(-3)^n+4*3^n)/12.
a(n) = a(n-1)+9*a(n-2)-9*a(n-3).
G.f.: -x^3*(18*x^2-9*x-11) / ((x-1)*(3*x-1)*(3*x+1)). (End)
E.g.f.: (3*(cosh(3*x) - cosh(x) - sinh(x)) + 5*sinh(3*x))/12 - x - x^2. - Stefano Spezia, Apr 09 2022

A191595 Order of smallest n-regular graph of girth 5.

Original entry on oeis.org

5, 10, 19, 30, 40, 50

Offset: 2

N. J. A. Sloane, Jun 07 2011

Current upper bounds for a(8)..a(20) are 80, 96, 124, 154, 203, 230, 288, 312, 336, 448, 480, 512, 576. - Corrected from "Lower" to "Upper" and updated, from Table 4 of the Dynamic cage survey, by Jason Kimberley, Dec 29 2012

Current lower bounds for a(8)..a(20) are 67, 86, 103, 124, 147, 174, 199, 230, 259, 294, 327, 364, 403. - from Table 4 of the Dynamic cage survey via Jason Kimberley, Dec 31 2012

M. Abreu et al., A family of regular graphs of girth 5, Discrete Math., 308 (2008), 1810-1815.
Andries E. Brouwer, Cages
Geoff Exoo, Regular graphs of given degree and girth
G. Exoo and R. Jajcay, Dynamic cage survey, Electr. J. Combin. (2008, 2011).
G. Royle, Cages of higher valency

Orders of cages: A054760 (n,k), A000066 (3,n), A037233 (4,n), A218553 (5,n), A218554 (6,n), A218555 (7,n), this sequence (n,5).

Moore lower bound on the orders of (k,g) cages: 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), A188377 (g=7). - Jason Kimberley, Nov 02 2011

a(n) >= A002522(n) with equality if and only if n = 2, 3, 7 or possibly 57. - Jason Kimberley, Nov 02 2011

a(2) = 5 prepended by Jason Kimberley, Jan 02 2013

A188947 a(n) = n^3 - 2n^2 + 2n + 1.

Original entry on oeis.org

2, 5, 16, 41, 86, 157, 260, 401, 586, 821, 1112, 1465, 1886, 2381, 2956, 3617, 4370, 5221, 6176, 7241, 8422, 9725, 11156, 12721, 14426, 16277, 18280, 20441, 22766, 25261, 27932, 30785, 33826, 37061, 40496, 44137, 47990, 52061, 56356, 60881, 65642, 70645

Offset: 1

Adeniji, Adenike, Apr 14 2011

The original definition was "Identity difference partial one - one transformation semigroup is a semigroup having the property that the difference between max im(alpha) and min im(alpha) is not greater than 1. This is denoted by S = IDI_n for each n." [Needs editing.]

For all n >= 3, a(n) expressed in base n has the three digits n-2, 2, and 1; for example, a(16) in hexadecimal is "E21". For all n >= 3, a(n+1) expressed in base n is "1112". For all n >= 7, a(n+2) expressed in base n is "1465". - Mathew Englander, Jan 07 2021

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. A027444, A053698, A056106 (first differences), A060354, A162607, A188377, A188716.

[n^3 - 2*n^2 + 2*n + 1: n in [1..30]]; // Vincenzo Librandi, May 01 2011

A188947[n_] := n^3 - 2*n^2 + 2*n + 1; Table[A188947[n], {n, 1, 42}] (* Robert P. P. McKone, Jan 31 2021 *)

a(n)=n^3-2*n^2+2*n+1 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = (n+1) + n*(n-1)^2 = n^3 - 2*n^2 + 2*n + 1 = 1 + A053698(n-1).
G.f.: ( -x*(-2 + 3*x - 8*x^2 + x^3) ) / ( (x-1)^4 ). - R. J. Mathar, Apr 14 2011
a(n) = A060354(n) + A162607(n+1). - Lechoslaw Ratajczak, Sep 24 2020
E.g.f.: exp(x)*(1 + x)*(1 + x^2) - 1. - Stefano Spezia, Apr 10 2022

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

A300401 Array T(n,k) = n(binomial(k, 2) + 1) + k(binomial(n, 2) + 1) read by antidiagonals.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 3, 4, 4, 3, 4, 7, 8, 7, 4, 5, 11, 14, 14, 11, 5, 6, 16, 22, 24, 22, 16, 6, 7, 22, 32, 37, 37, 32, 22, 7, 8, 29, 44, 53, 56, 53, 44, 29, 8, 9, 37, 58, 72, 79, 79, 72, 58, 37, 9, 10, 46, 74, 94, 106, 110, 106, 94, 74, 46, 10, 11, 56, 92, 119

Offset: 0

Franck Maminirina Ramaharo, Mar 05 2018

Antidiagonal sums are given by 2*A055795.
Rows/columns n are binomial transform of {n, A152947(n+1), n, 0, 0, 0, ...}.
Some primes in the array are
n = 1: {2, 7, 11, 29, 37, 67, 79, 137, 191, 211, 277, 379, ...} = A055469, primes of the form k*(k + 1)/2 + 1;
n = 3: {3, 7, 37, 53, 479, 653, 1249, 1619, 2503, 3727, 4349, 5737, 7109, 8179, 9803, 11839, 12107, ...};
n = 4: {11, 37, 79, 137, 211, 821, 991, 1597, 1831, 2081, 2347, ...} = A188382, primes of the form 8*(2*k - 1)^2 + 2*(2*k - 1) + 1.

			The array T(n,k) begins
0     1    2    3    4     5     6     7     8     9    10    11  ...
1     2    4    7   11    16    22    29    37    46    56    67  ...
2     4    8   14   22    32    44    58    74    92   112   134  ...
3     7   14   24   37    53    72    94   119   147   178   212  ...
4    11   22   37   56    79   106   137   172   211   254   301  ...
5    16   32   53   79   110   146   187   233   284   340   401  ...
6    22   44   72  106   146   192   244   302   366   436   512  ...
7    29   58   94  137   187   244   308   379   457   542   634  ...
8    37   74  119  172   233   302   379   464   557   658   767  ...
9    46   92  147  211   284   366   457   557   666   784   911  ...
10   56  112  178  254   340   436   542   658   784   920  1066  ...
11   67  134  212  301   401   512   634   767   911  1066  1232  ...
12   79  158  249  352   467   594   733   884  1047  1222  1409  ...
13   92  184  289  407   538   682   839  1009  1192  1388  1597  ...
14  106  212  332  466   614   776   952  1142  1346  1564  1796  ...
15  121  242  378  529   695   876  1072  1283  1509  1750  2006  ...
16  137  274  427  596   781   982  1199  1432  1681  1946  2227  ...
17  154  308  479  667   872  1094  1333  1589  1862  2152  2459  ...
18  172  344  534  742   968  1212  1474  1754  2052  2368  2702  ...
19  191  382  592  821  1069  1336  1622  1927  2251  2594  2956  ...
20  211  422  653  904  1175  1466  1777  2108  2459  2830  3221  ...
...
The inverse binomial transforms of the columns are
0     1    2    3    4     5     6     7     8     9    10    11  ...  A001477
1     1    2    4    7    11    22    29    37    45    56    67  ...  A152947
0     1    2    3    4     5     6     7     8     9    10    11  ...  A001477
0     0    0    0    0     0     0     0     0     0     0     0  ...
0     0    0    0    0     0     0     0     0     0     0     0  ...
0     0    0    0    0     0     0     0     0     0     0     0  ...
...

Miklós Bóna, Introduction to Enumerative Combinatorics, McGraw-Hill, 2007.
L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Reidel Publishing Company, 1974.
R. P. Stanley, Enumerative Combinatorics, second edition, Cambridge University Press, 2011.

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened).
Cheyne Homberger, Patterns in Permutations and Involutions: A Structural and Enumerative Approach, arXiv preprint 1410.2657 [math.CO], 2014.
Franck Ramaharo, A generating polynomial for the two-bridge knot with Conway's notation C(n,r), arXiv:1902.08989 [math.CO], 2019.

Cf. A000124, A001477, A006000, A008815, A014206, A051601, A055469, A077028, A081436, A084849, A131074, A134394, A139600, A141387, A179000, A188377, A188382, A273465.

T := (n, k) -> n*(binomial(k, 2) + 1) + k*(binomial(n, 2) + 1);
for n from 0 to 20 do seq(T(n, k), k = 0 .. 20) od;

T[n_, k_] := n (Binomial[k, 2] + 1) + k (Binomial[n, 2] + 1);
Table[T[n - k, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 07 2018 *)

T(n, k) := n*(binomial(k, 2) + 1) + k*(binomial(n, 2) + 1)$
for n:0 thru 20 do
  print(makelist(T(n, k), k, 0, 20));

T(n, k) = n*(binomial(k,2) + 1) + k*(binomial(n,2) + 1);
tabl(nn) = for (n=0, nn, for (k=0, nn, print1(T(n, k), ", ")); print); \\ Michel Marcus, Mar 12 2018

T(n,k) = T(k,n) = n*A152947(k+1) + k*A152947(n+1).
T(n,0) = A001477(n).
T(n,1) = A000124(n).
T(n,2) = A014206(n).
T(n,3) = A273465(3*n+2).
T(n,4) = A084849(n+1).
T(n,n) = A179000(n-1,n), n >= 1.
T(2*n,2*n) = 8*A081436(n-1), n >= 1.
T(2*n+1,2*n+1) = 2*A006000(2*n+1).
T(n,n+1) = A188377(n+3).
T(n,n+2) = A188377(n+2), n >= 1.
Sum_{k=0..n} T(k,n-k) = 2*(binomial(n, 4) + binomial(n, 2)).
G.f.: -((2*x*y - y - x)*(2*x*y - y - x + 1))/(((x - 1)*(y - 1))^3).
E.g.f.: (1/2)*(x + y)*(x*y + 2)*exp(x + y).

A189890 a(n) = (n^3 - 2n^2 + 3n + 2)/2.

Original entry on oeis.org

2, 4, 10, 23, 46, 82, 134, 205, 298, 416, 562, 739, 950, 1198, 1486, 1817, 2194, 2620, 3098, 3631, 4222, 4874, 5590, 6373, 7226, 8152, 9154, 10235, 11398, 12646, 13982, 15409, 16930, 18548, 20266, 22087, 24014, 26050, 28198, 30461, 32842, 35344, 37970, 40723, 43606, 46622

Offset: 1

Adeniji, Adenike and Samuel Makanjuola(somakanjuola(AT)unilorin.edu.ng), Apr 30 2011

Order preserving identity difference partial one - one transformation semigroup, OIDI_n is defined if for each transformation, alpha, x<= y implies xalpha <= yalpha, for all x,y in X_n (set of natural numbers) and also the absolute value of the difference between max(Im(alpha)) and min(Im(alpha)) is less than or equal to one with non-isolation property.

			For n = 4, a(4) = (4^3-2*4^2+3*4+2)/2 = 46/2 = 23.

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. A188947, A188377.

[(n^3-2*n^2+3*n+2)/2: n in [1..50]]; // Vincenzo Librandi, May 07 2011

Table[(n^3-2*n^2+3*n+2)/2, {n,1,50}] (* or *) LinearRecurrence[{4,-6,4, -1}, {2,4,10,23}, 50] (* G. C. Greubel, Jan 13 2018 *)

a(n)=(n^3-2*n^2+3*n+2)/2 \\ Charles R Greathouse IV, Oct 16 2015

G.f.: -x*(-2+4*x-6*x^2+x^3) / (x-1)^4. - R. J. Mathar, Jun 20 2011
E.g.f.: 4*(-2 + (2 + 2*x + x^2 + x^3)*exp(x)). - G. C. Greubel, Jan 13 2018
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Wesley Ivan Hurt, Apr 23 2021

A188716 a(n) = n + (n-1)*(2^n-2).

Original entry on oeis.org

1, 1, 4, 15, 46, 125, 316, 763, 1786, 4089, 9208, 20471, 45046, 98293, 212980, 458739, 983026, 2097137, 4456432, 9437167, 19922926, 41943021, 88080364, 184549355, 385875946, 805306345, 1677721576, 3489660903, 7247757286, 15032385509, 31138512868, 64424509411, 133143986146, 274877906913, 566935683040, 1168231104479

Offset: 0

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

Number of elements in the semigroup IDT_n.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (6,-13,12,-4).

Cf. A000337, A188377, A188947.

[n + (n-1)*(2^n-2): n in [0..50]]; // Vincenzo Librandi, May 01 2011

Table[n+(n-1)(2^n-2),{n,0,40}] (* or *) LinearRecurrence[{6,-13,12,-4},{1,1,4,15},40] (* Harvey P. Dale, Aug 03 2024 *)

a(n)=(n-1)<Charles R Greathouse IV, Apr 06 2012

From Colin Barker, Apr 06 2012: (Start)
a(n) = 6*a(n-1)-13*a(n-2)+12*a(n-3)-4*a(n-4).
G.f.: (1-5*x+11*x^2-8*x^3)/((1-x)^2*(1-2*x)^2). (End)
a(n) = A000337(n) - (n-1). - Andrew Penland , Mar 24 2016
E.g.f.: exp(x)*(2 - x + exp(x)*(2*x - 1)). - Stefano Spezia, Apr 10 2022

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

Offset changed from 1 to 0 by Vincenzo Librandi, May 01 2011

A191821 a(n) = n(2^n - n + 1) + 2^(n-1)(n^2 - 3*n + 2).

Original entry on oeis.org

2, 6, 26, 100, 332, 994, 2774, 7368, 18872, 47014, 114578, 274300, 647012, 1507146, 3473198, 7929616, 17956592, 40369870, 90177194, 200277636, 442498652, 973078066, 2130705926, 4647288280, 10099883432, 21877489014, 47244639554, 101737037068

Offset: 1

Adeniji, Adenike, Jun 17 2011

Conjecture: generating function = -((2 (-1+6 x-19 x^2+31 x^3-22 x^4+4 x^5))/(1-3 x+2 x^2)^3) - Harvey P. Dale, May 10 2021

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (9,-33,63,-66,36,-8).

Cf. A188377, A188947, A189890.

[n*(2^n-n+1)+2^(n-1)*(n^2-3*n+2): n in [1..40]]; // Vincenzo Librandi, Nov 25 2011

LinearRecurrence[{9,-33,63,-66,36,-8},{2,6,26,100,332,994},50] (* Vincenzo Librandi, Nov 25 2011 *)
Table[n(2^n-n+1)+2^(n-1) (n^2-3n+2),{n,30}] (* Harvey P. Dale, May 10 2021 *)

a(n)=(n^2-n+2)<<(n-1)-n*(n-1) \\ Charles R Greathouse IV, Jul 13 2011

G.f.: -2*x*(-1 + 6*x - 19*x^2 + 31*x^3 - 22*x^4 + 4*x^5) / ( (2*x-1)^3*(x-1)^3 ). - R. J. Mathar, Aug 26 2011

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

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

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

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A198310 Moore lower bound on the order of a (10,g)-cage.

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A191595 Order of smallest n-regular graph of girth 5.

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A188947 a(n) = n^3 - 2*n^2 + 2*n + 1.

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A300401 Array T(n,k) = n*(binomial(k, 2) + 1) + k*(binomial(n, 2) + 1) read by antidiagonals.

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A189890 a(n) = (n^3 - 2*n^2 + 3*n + 2)/2.

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A188947 a(n) = n^3 - 2n^2 + 2n + 1.

A300401 Array T(n,k) = n(binomial(k, 2) + 1) + k(binomial(n, 2) + 1) read by antidiagonals.

A189890 a(n) = (n^3 - 2n^2 + 3n + 2)/2.

A191821 a(n) = n(2^n - n + 1) + 2^(n-1)(n^2 - 3*n + 2).