A000982 -id:A000982 - cp's OEIS frontend

A008815 Expansion of (1+x^8)/((1-x)^2*(1-x^8)).

1, 2, 3, 4, 5, 6, 7, 8, 11, 14, 17, 20, 23, 26, 29, 32, 37, 42, 47, 52, 57, 62, 67, 72, 79, 86, 93, 100, 107, 114, 121, 128, 137, 146, 155, 164, 173, 182, 191, 200, 211, 222, 233, 244, 255, 266, 277, 288, 301, 314, 327, 340, 353, 366, 379, 392, 407, 422

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,1,-2,1).

Cf. Expansions of the form (1+x^m)/((1-x)^2*(1-x^m)): A000290 (m=1), A000982 (m=2), A008810 (m=3), A008811 (m=4), A008812 (m=5), A008813 (m=6), A008814 (m=7), this sequence (m=8), A008816 (m=9), A008817 (m=10).

a:=[1,2,3,4,5,6,7,8,11,14];; for n in [11..50] do a[n]:=2*a[n-1] -a[n-2]+a[n-8]-2*a[n-9]+a[n-10]; od; a; # G. C. Greubel, Sep 12 2019

I:=[1,2,3,4,5,6,7,8,11,14]; [n le 10 select I[n] else 2*Self(n-1) -Self(n-2)+Self(n-8)-2*Self(n-9)+Self(n-10): n in [1..50]]; // Vincenzo Librandi, May 14 2019

seq(coeff(series((1+x^8)/((1-x)^2*(1-x^8)), x, n+1), x, n), n = 0..50); # G. C. Greubel, Sep 12 2019

CoefficientList[Series[(1+x^8)/(1-x)^2/(1-x^8), {x, 0, 50}], x] (* or *) LinearRecurrence[{2,-1,0,0,0,0,0,1,-2,1}, {1,2,3,4,5,6,7,8,11,14}, 50] (* Harvey P. Dale, Dec 17 2016 *)

a(n)=(n*(n+2)+14+4*(n%4-1)*(-1)^(n\4))\8  \\ Tani Akinari, Jul 25 2013

def A008815_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+x^8)/((1-x)^2*(1-x^8))).list()
A008815_list(50) # G. C. Greubel, Sep 12 2019

G.f.: (1 + x^8)/((1 - x)^3*(1 + x)*(1 + x^2)*(1 + x^4)).
a(n) = floor( (n*(n+2) + 14 + 4*((n mod 4) - 1)*(-1)^floor(n/4))/8 ). - Tani Akinari, Jul 25 2013
a(n) = 2*a(n-1) - a(n-2) + a(n-8) - 2*a(n-9) + a(n-10). - Vincenzo Librandi, May 14 2019

A030978 Maximal number of non-attacking knights on an n X n board.

Original entry on oeis.org

0, 1, 4, 5, 8, 13, 18, 25, 32, 41, 50, 61, 72, 85, 98, 113, 128, 145, 162, 181, 200, 221, 242, 265, 288, 313, 338, 365, 392, 421, 450, 481, 512, 545, 578, 613, 648, 685, 722, 761, 800, 841, 882, 925, 968, 1013, 1058, 1105, 1152, 1201, 1250, 1301, 1352, 1405

Offset: 0

Eric W. Weisstein

In other words, independence number of the n X n knight graph. - Eric W. Weisstein, May 05 2017

H. E. Dudeney, The Knight-Guards, #319 in Amusements in Mathematics; New York: Dover, p. 95, 1970.
J. S. Madachy, Madachy's Mathematical Recreations, New York, Dover, pp. 38-39 1979.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 751.
Eric Weisstein's World of Mathematics, Independence Number
Eric Weisstein's World of Mathematics, Knight Graph
Eric Weisstein's World of Mathematics, Knights Problem
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Agrees with A000982 for n>1.

Cf. A244081.

CoefficientList[Series[x (2 x^5 - 4 x^4 + 3 x^2 - 2 x - 1)/((x - 1)^3 (x + 1)), {x, 0, 60}], x] (* Vincenzo Librandi, Oct 19 2013 *)
Join[{0, 1, 4}, Table[If[EvenQ[n], n^2/2, (n^2 + 1)/2], {n, 3, 60}]] (* Harvey P. Dale, Nov 28 2014 *)
Join[{0, 1, 4}, LinearRecurrence[{2, 0, -2, 1}, {5, 8, 13, 18}, 60]] (* Harvey P. Dale, Nov 28 2014 *)
Table[If[n == 2, 4, (1 - (-1)^n + 2 n^2)/4], {n, 20}] (* Eric W. Weisstein, May 05 2017 *)
Table[Length[FindIndependentVertexSet[KnightTourGraph[n, n]][[1]]], {n, 20}] (* Eric W. Weisstein, Jun 27 2017 *)

a(n) = 4 if n = 2, n^2/2 if n even > 2, (n^2+1)/2 if n odd > 1.
a(n) = 4 if n = 2, (1 + (-1)^(1 + n) + 2 n^2)/4 otherwise.
G.f.: x*(2*x^5-4*x^4+3*x^2-2*x-1) / ((x-1)^3*(x+1)). [Colin Barker, Jan 09 2013]

More terms from Erich Friedman

Definition clarified by Vaclav Kotesovec, Sep 16 2014

A188211 T(n,k)=Number of nondecreasing arrangements of n numbers in -(n+k-2)..(n+k-2) with sum zero.

Original entry on oeis.org

1, 1, 2, 1, 3, 5, 1, 4, 8, 18, 1, 5, 13, 33, 73, 1, 6, 18, 55, 141, 338, 1, 7, 25, 86, 252, 676, 1656, 1, 8, 32, 126, 414, 1242, 3370, 8512, 1, 9, 41, 177, 649, 2137, 6375, 17575, 45207, 1, 10, 50, 241, 967, 3486, 11322, 33885, 94257, 246448, 1, 11, 61, 318, 1394, 5444

Offset: 1

R. H. Hardin Mar 24 2011

Table starts
......1......1.......1.......1.......1.......1........1........1........1
......2......3.......4.......5.......6.......7........8........9.......10
......5......8......13......18......25......32.......41.......50.......61
.....18.....33......55......86.....126.....177......241......318......410
.....73....141.....252.....414.....649.....967.....1394.....1944.....2649
....338....676....1242....2137....3486....5444.....8196....11963....17002
...1656...3370....6375...11322...19138...30982....48417....73316...108108
...8512..17575...33885...61731..107233..178870...288100...450096...684572
..45207..94257..184717..343363..610358.1043534..1724882..2767118..4323349
.246448.517971.1028172.1943488.3521260.6147894.10388788.17052653.27273240

			Some solutions for n=5 k=3
.-5...-5...-4...-4...-6...-3...-6...-4...-2...-6...-2...-6...-4...-5...-5...-4
.-2...-1...-4...-2...-5...-3...-1...-3...-1...-6...-1...-1...-3...-4...-3...-4
..0....0...-1...-2...-1...-1....0...-1...-1....1...-1....0....1...-1....0....1
..1....0....3....3....6....2....2....3....1....5....0....1....1....5....4....3
..6....6....6....5....6....5....5....5....3....6....4....6....5....5....4....4

R. H. Hardin, Table of n, a(n) for n = 1..430

Column 1 is A039744
Column 2 is A109655
Row 3 is A000982(n+2)
Row 5 is A188183(n+2)
Row 7 is A188185(n+3)

A199847 T(n,k)=Number of -k..k arrays x(0..n-1) of n elements with zero sum and no element more than one greater than the previous.

Original entry on oeis.org

1, 1, 2, 1, 3, 5, 1, 4, 8, 11, 1, 5, 13, 23, 26, 1, 6, 18, 42, 66, 63, 1, 7, 25, 68, 132, 192, 153, 1, 8, 32, 103, 239, 428, 575, 376, 1, 9, 41, 149, 399, 845, 1393, 1739, 931, 1, 10, 50, 206, 630, 1532, 2958, 4561, 5307, 2317, 1, 11, 61, 276, 944, 2600, 5754, 10349, 15032

Offset: 1

R. H. Hardin Nov 11 2011

Table starts
....1.....1.....1......1......1......1.......1.......1.......1.......1........1
....2.....3.....4......5......6......7.......8.......9......10......11.......12
....5.....8....13.....18.....25.....32......41......50......61......72.......85
...11....23....42.....68....103....149.....206.....276.....361.....461......578
...26....66...132....239....399....630.....944....1367....1913....2612.....3482
...63...192...428....845...1532...2600....4188....6465....9634...13932....19636
..153...575..1393...2958...5754..10426...17879...29268...46126...70322...104253
..376..1739..4561..10349..21383..41079...74424..128462..212862..340592...528705
..931..5307.15032..36111..78735.159330..303400..549404..953172.1594101..2581937
.2317.16304.49889.126121.288580.611698.1218440.2304210.4169150.7260993.12230924

			Some solutions for n=7 k=6
..5....4....2....2....6....5....2...-1....3....3....6....4....6....6....5....3
..4....2....1....3....6....2....1....0....3....1....3....1....6....1....5...-1
..1....2....1....3....0....3....1....1...-1....1....4....1...-2....1...-1...-1
..0...-1...-1....0...-4...-2....1....0....0...-1....1....2...-2....1...-3....0
.-2....0...-2...-1...-3...-3....1....1....1...-1...-2....2...-2....0...-3...-1
.-2...-1...-1...-1...-2...-3....0....0....0....0...-6...-5...-3...-3...-2....0
.-6...-6....0...-6...-3...-2...-6...-1...-6...-3...-6...-5...-3...-6...-1....0

R. H. Hardin, Table of n, a(n) for n = 1..958

Column 1 is A051286

Row 3 is A000982(n+2)

A277239 Number A(n,k) of factorizations of m^n into exactly k factors, where m is a product of two distinct primes; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 2, 5, 1, 0, 1, 2, 8, 8, 1, 0, 1, 2, 9, 19, 13, 1, 0, 1, 2, 9, 27, 42, 18, 1, 0, 1, 2, 9, 30, 74, 78, 25, 1, 0, 1, 2, 9, 31, 95, 168, 139, 32, 1, 0, 1, 2, 9, 31, 105, 248, 363, 224, 41, 1, 0, 1, 2, 9, 31, 108, 300, 614, 703, 350, 50, 1, 0

Offset: 0

Alois P. Heinz, Oct 06 2016

			Square array A(n,k) begins:
  1, 1,  1,   1,    1,    1,    1,    1,    1, ...
  0, 1,  2,   2,    2,    2,    2,    2,    2, ...
  0, 1,  5,   8,    9,    9,    9,    9,    9, ...
  0, 1,  8,  19,   27,   30,   31,   31,   31, ...
  0, 1, 13,  42,   74,   95,  105,  108,  109, ...
  0, 1, 18,  78,  168,  248,  300,  325,  335, ...
  0, 1, 25, 139,  363,  614,  814,  938, 1002, ...
  0, 1, 32, 224,  703, 1367, 1996, 2457, 2741, ...
  0, 1, 41, 350, 1297, 2879, 4642, 6128, 7168, ...

Alois P. Heinz, Antidiagonals n = 0..45, flattened

Columns k=0-10 give: A000007, A000012, A000982(n+1), A101427, A277240, A277241, A277242, A277243, A277244, A277245, A277246.
Main diagonal gives A254686.
A(n,2n) gives A002774.

A008816 Expansion of (1+x^9)/((1-x)^2*(1-x^9)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 41, 46, 51, 56, 61, 66, 71, 76, 81, 88, 95, 102, 109, 116, 123, 130, 137, 144, 153, 162, 171, 180, 189, 198, 207, 216, 225, 236, 247, 258, 269, 280, 291, 302, 313, 324, 337, 350, 363, 376, 389, 402

Offset: 0

N. J. A. Sloane

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,0,1,-2,1).

Cf. Expansions of the form (1+x^m)/((1-x)^2*(1-x^m)): A000290 (m=1), A000982 (m=2), A008810 (m=3), A008811 (m=4), A008812 (m=5), A008813 (m=6), A008814 (m=7), A008815 (m=8), this sequence (m=9), A008817 (m=10).

a:=[1,2,3,4,5,6,7,8,9,12,15];; for n in [12..70] do a[n]:=2*a[n-1] -a[n-2]+a[n-9]-2*a[n-10]+a[n-11]; od; a; # G. C. Greubel, Sep 12 2019

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( (1+x^9)/((1-x)^2*(1-x^9)) )); // G. C. Greubel, Sep 12 2019

seq(coeff(series((1+x^9)/((1-x)^2*(1-x^9)), x, n+1), x, n), n = 0..50); # G. C. Greubel, Sep 12 2019

LinearRecurrence[{2,-1,0,0,0,0,0,0,1,-2,1}, {1,2,3,4,5,6,7,8,9,12,15}, 70] (* or *) CoefficientList[Series[(1+x^9)/((1-x)^2*(1-x^9)), {x,0, 70}], x] (* G. C. Greubel, Sep 12 2019 *)

my(x='x+O('x^70)); Vec((1+x^9)/((1-x)^2*(1-x^9))) \\ G. C. Greubel, Sep 12 2019

def A008815_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+x^8)/((1-x)^2*(1-x^8))).list()
A008815_list(70) # G. C. Greubel, Sep 12 2019

G.f.: (1+x^9)/((1-x)^2*(1-x^9)). - G. C. Greubel, Sep 12 2019

More terms added by G. C. Greubel, Sep 12 2019

A008817 Expansion of (1+x^10)/((1-x)^2*(1-x^10)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 97, 104, 111, 118, 125, 132, 139, 146, 153, 160, 169, 178, 187, 196, 205, 214, 223, 232, 241, 250, 261, 272, 283, 294, 305, 316, 327, 338, 349, 360

Offset: 0

N. J. A. Sloane

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,0,0,1,-2,1).

Cf. Expansions of the form (1+x^m)/((1-x)^2*(1-x^m)): A000290 (m=1), A000982 (m=2), A008810 (m=3), A008811 (m=4), A008812 (m=5), A008813 (m=6), A008814 (m=7), A008815 (m=8), A008816 (m=9), this sequence (m=10).

a:=[1,2,3,4,5,6,7,8,9,10, 13,16];; for n in [13..80] do a[n]:=2*a[n-1]-a[n-2]+a[n-10]-2*a[n-11]+a[n-12]; od; a; # G. C. Greubel, Sep 12 2019

R:=PowerSeriesRing(Integers(), 80); Coefficients(R!( (1+x^10)/((1-x)^2*(1-x^10)) )); // G. C. Greubel, Sep 12 2019

seq(coeff(series((1+x^10)/((1-x)^2*(1-x^10)), x, n+1), x, n), n = 0..80); # G. C. Greubel, Sep 12 2019

CoefficientList[Series[(1+x^10)/(1-x)^2/(1-x^10), {x,0,80}], x] (* or *) LinearRecurrence[{2,-1,0,0,0,0,0,0,0,1,-2,1}, {1,2,3,4,5,6,7,8,9,10, 13,16}, 80] (* Harvey P. Dale, Jul 31 2014 *)

my(x='x+O('x^80)); Vec((1+x^10)/((1-x)^2*(1-x^10))) \\ G. C. Greubel, Sep 12 2019

def A008817_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+x^10)/((1-x)^2*(1-x^10))).list()
A008817_list(80) # G. C. Greubel, Sep 12 2019

G.f.: (1+x^10)/((1-x)^2*(1-x^10)).

a(0)=1, a(1)=2, a(2)=3, a(3)=4, a(4)=5, a(5)=6, a(6)=7, a(7)=8, a(8)=9, a(9)=10, a(10)=13, a(11)=16, a(n) = 2*a(n-1) - a(n-2) + a(n-10) - 2*a(n-11) + a(n-12). - Harvey P. Dale, Jul 31 2014

A081352 Main diagonal of square maze arrangement of natural numbers A081349.

Original entry on oeis.org

1, 7, 11, 21, 29, 43, 55, 73, 89, 111, 131, 157, 181, 211, 239, 273, 305, 343, 379, 421, 461, 507, 551, 601, 649, 703, 755, 813, 869, 931, 991, 1057, 1121, 1191, 1259, 1333, 1405, 1483, 1559, 1641, 1721, 1807, 1891, 1981, 2069, 2163, 2255, 2353, 2449, 2551

Offset: 0

Paul Barry, Mar 19 2003

Conjecture: let a and b be integers such that 0 < a < b so that 0 < a/b is a proper fraction. Define the map f(a,b,D) = a/b + gcd(a,b)/D. Of course, all such a/b can be partially ordered by value, i.e., 1/2 = 0.5 < 2/3 = 4/6 = 6/9 = 0.6666... < 3/4 = 6/8 = 0.75 < 4/5 = 0.8 etc. The map f appears to specify a total strict order on the co-domain for all a/b that is consistent with the given partial order of the domain, i.e., the partial order remains intact, while equivalent fractions are given a total strict order themselves. Moreover, equivalent fractions are strictly ordered by numerator (or denominator), e.g., 1/2 < 2/4 < 3/6 etc. The conditions are that for n >= 4 all of the fractions with denominator b <= n are listed and the minimum integer value of D to achieve the total strict order of the co-domain is 2*C(n-1,2) - (-1)^(n-1). So, a(n-3) = D for n >= 4. Example: given n = 4, we have D = 2*(4-1,2) - (-1)^(4-1) = 2*3 + 1 = 7 = a(4-3) = a(1). Partial order of domain. 1/4 < 1/3 < 1/2 = 2/4 < 2/3 < 3/4. Total order of co-domain. f(1,4,7) = 1/4 + 1/7 = 33/84 < f(1,3,7) = 1/3 + 1/7 = 40/84 < f(1,2,7) = 1/2 + 1/7 = 54/84 < f(2,4,7) = 2/4 + 2/7 = 66/84 < f(2,3,7) = 2/3 + 2/7 = 68/84 < f(3,4,7) = 3/4 + 1/7 = 75/84. Observe that if D = 6, then f(2,4,6) = 2/4 + 2/6 = 10/12 = f(2,3,6) = 2/3 + 1/6. Computation shows the same failure to achieve total strict order of the co-domain for D = 2..5. (As a >= 1, then b >=2, from the above). Computation also shows that the conjecture holds for n = 4..17. - Ross La Haye, Oct 02 2016

B. Berselli, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A081350, A081351, A054569.

I:=[1,7,11,21]; [n le 4 select I[n] else 2*Self(n-1)-2*Self(n-3)+Self(n-4): n in [1..50]]; // Vincenzo Librandi, Aug 08 2013

A081352:=n->(n + 1)*(n + 2) - (-1)^n; seq(A081352(n), n=0..50); # Wesley Ivan Hurt, Feb 26 2014

CoefficientList[Series[(1 + 5 x - 3 x^2 + x^3) / ((1 + x) (1 - x)^3), {x, 0, 60}], x] (* Vincenzo Librandi, Aug 08 2013 *)

x='x+O('x^99); Vec((1+5*x-3*x^2+x^3)/((1+x)*(1-x)^3)) \\ Altug Alkan, Mar 26 2016

a(n) = (n + 1)*(n + 2) - (-1)^n = 2*C(n+2, 2) - (-1)^n.
G.f.: (1 +5*x -3*x^2 +x^3) / ((1+x)*(1-x)^3). [Bruno Berselli, Aug 01 2010]
a(n) -2*a(n-1) +2*a(n-3) -a(n-4) = 0 with n>3. [Bruno Berselli, Aug 01 2010]
a(n) = 3*A000982(n + 2) - A000982(n + 3). - Miko Labalan, Mar 26 2016
a(n) = A116940(n) + A236283(n + 1). - Miko Labalan, Dec 04 2016
a(n) = (2*n^2 + 6*n - 2*(-1)^n + (-1)^(2*n) + 3)/2. - Kritsada Moomuang, Oct 24 2019

A116940 Greatest m such that A116939(m) = n.

Original entry on oeis.org

0, 3, 6, 11, 16, 23, 30, 39, 48, 59, 70, 83, 96, 111, 126, 143, 160, 179, 198, 219, 240, 263, 286, 311, 336, 363, 390, 419, 448, 479, 510, 543, 576, 611, 646, 683, 720, 759, 798, 839, 880, 923, 966, 1011, 1056, 1103, 1150, 1199, 1248, 1299, 1350, 1403, 1456

Offset: 0

Reinhard Zumkeller, Feb 27 2006

From Andrew Rupinski, Nov 30 2009: (Start)
For n > 0, a(n) appears to be the set such that binomial(2*a(n),r) - binomial(2*a(n),r-2) = binomial(2*a(n),s) - binomial(2*a(n),s-2) for some r != s.
As a consequence of the Weyl Dimension Formula and the above comment, a(n) also appears to be the indices k such that the Symplectic Group Sp(k) has two fundamental irreducible representations of the same dimension. (End)

			a(n) = A000982(n) + A005843(n).
From _Andrew Rupinski_, Nov 30 2009: (Start)
a(1) = 3 and binomial(6,3)-binomial(6,1) = binomial(6,2)-binomial(6,0).
a(1) = 3 and the fundamental representations of Sp(3) are of dimensions 6, 14 and 14. a(2) = 6 and the fundamental representations of Sp(6) are of dimensions 12, 65, 208, 429, 572, and 429. (End)

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
John Tyler Rascoe, Illustration of the first 10 terms.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A116939.
Cf. A081352, A236283.
Cf. A028884, A099392.
Cf. A000217, A001859.

import Data.List (elemIndices)
a116940 n = last $ elemIndices n $ takeWhile (<= n + 1) a116939_list
-- Reinhard Zumkeller, Jun 28 2013

[(2*n*(n+4) -(-1)^n +1)/4: n in [0..55]]; // G. C. Greubel, Jan 26 2020

seq( (2*(n+2)^2 -(-1)^n -7)/4, n=0..55); # G. C. Greubel, Jan 26 2020

a = {0}; Do[AppendTo[a, If[Count[a, #-1] > #-1, #+1, #-1]] &@ a[[n]], {n, 1500}]; Most@ Values@ Map[Last, PositionIndex@ a] - 1 (* Michael De Vlieger, Dec 07 2016, Version 10 *)
Table[(2*(n+2)^2 -(-1)^n -7)/4, {n,0,55}] (* G. C. Greubel, Jan 26 2020 *)

vector(56, n, (2*(n+1)^2 +(-1)^n -7)/4) \\ G. C. Greubel, Jan 26 2020

[(2*n*(n+4) -(-1)^n +1)/4 for n in (0..55)] # G. C. Greubel, Jan 26 2020

a(0) = 0, a(n+1) = a(n) + 2*floor(n/2) + 3.
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4). - Joerg Arndt, Apr 02 2011
G.f.: x*(3 - x^2)/((1 + x)*(1 - x)^3). - Arkadiusz Wesolowski, Jan 01 2012
a(n) = 2n + ceiling(n^2/2). - Wesley Ivan Hurt, Jun 14 2013
a(n) = (2*n*(n + 4) - (-1)^n + 1)/4. - Bruno Berselli, Jun 14 2013
a(n) = A081352(n) - A236283(n + 1). - Miko Labalan, Dec 04 2016
From Klaus Purath, Jan 26 2020: (Start)
a(n) = binomial(n+2, 2) + floor((n-1)/2).
a(n) = floor(A028884(n)/2) - n.
a(n) = (n+1)^2 - A099392(n+1).
a(2*n)^2 - a(2*n-1)*a(2*n+1) = 3, n > 0.
a(2*n+1)^2 - a(2*n)*a(2*n+2) = (2*n+3)^2. (End)
E.g.f.: (1/2)*(x*(5 + x)*cosh(x) + (1 + 5*x + x^2)*sinh(x)). - Stefano Spezia, Jan 26 2020
a(n) = A000217(2*n) - 2*A001859(n-1) for n>0. - John Tyler Rascoe, Jul 31 2022
Sum_{n>=1} 1/a(n) = 11/8 + tan(sqrt(3)*Pi/2)*Pi/(2*sqrt(3)). - Amiram Eldar, Sep 16 2022

A131474 a(n) = ceiling(n/2)*ceiling(n^2/2).

Original entry on oeis.org

0, 1, 2, 10, 16, 39, 54, 100, 128, 205, 250, 366, 432, 595, 686, 904, 1024, 1305, 1458, 1810, 2000, 2431, 2662, 3180, 3456, 4069, 4394, 5110, 5488, 6315, 6750, 7696, 8192, 9265, 9826, 11034, 11664, 13015, 13718, 15220, 16000, 17661

Offset: 0

Mohammad K. Azarian, Jul 27 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Cf. A000982, A004526, A008619, A033431, A059722.

[Ceiling(n/2)*Ceiling(n^2/2) : n in [0..50]]; // Vincenzo Librandi, Oct 01 2011

Table[Ceiling[n/2]*Ceiling[n^2/2], {n, 0, 50}] (* Paolo Xausa, Aug 21 2024 *)

a(n)=ceil(n/2)*ceil(n^2/2) \\ Charles R Greathouse IV, Jun 11 2015

From R. J. Mathar, Dec 19 2008: (Start)
G.f.: x*(1 + x + 5*x^2 + 2*x^4 + 3*x^3)/((1+x)^3*(x-1)^4).
a(n) = A000982(n)*A004526(n+1). (End)
a(2*n) = A033431(n). - Bernard Schott, Mar 29 2020
a(2*n-1) = A059722(n). - Michel Marcus, Mar 29 2020

cp's OEIS Frontend

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A199847 T(n,k)=Number of -k..k arrays x(0..n-1) of n elements with zero sum and no element more than one greater than the previous.

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