A210977 -id:A210977 - cp's OEIS frontend

A006578 Triangular numbers plus quarter squares: n*(n+1)/2 + floor(n^2/4) (i.e., A000217(n) + A002620(n)).

0, 1, 4, 8, 14, 21, 30, 40, 52, 65, 80, 96, 114, 133, 154, 176, 200, 225, 252, 280, 310, 341, 374, 408, 444, 481, 520, 560, 602, 645, 690, 736, 784, 833, 884, 936, 990, 1045, 1102, 1160, 1220, 1281, 1344, 1408, 1474, 1541, 1610, 1680, 1752, 1825, 1900, 1976, 2054

Offset: 0

N. J. A. Sloane

Equals (1, 2, 3, 4, ...) convolved with (1, 2, 1, 2, ...). a(4) = 14 = (1, 2, 3, 4) dot (2, 1, 2, 1) = (2 + 2 + 6 + 4). - Gary W. Adamson, May 01 2009
We observe that is the transform of A032766 by the following transform T: T(u_0,u_1,u_2,u_3,...) = (u_0, u_0+u_1, u_0+u_1+u_2, u_0+u_1+u_2+u_3+u_4,...). In other words, v_p = Sum_{k=0..p} u_k and the g.f. phi_v of is given by phi_v = phi_u/(1-z). - Richard Choulet, Jan 28 2010
Equals row sums of a triangle with (1, 4, 7, 10, ...) in every column, shifted down twice for columns > 1. - Gary W. Adamson, Mar 03 2010
Number of pairs (x,y) with x in {0,...,n}, y odd in {0,...,2n}, and x < y. - Clark Kimberling, Jul 02 2012
Also A049451 and positives A000567 interleaved. - Omar E. Pol, Aug 03 2012
Similar to A001082. Members of this family are A093005, A210977, this sequence, A210978, A181995, A210981, A210982. - Omar E. Pol, Aug 09 2012

			G.f. = x + 4*x^2 + 8*x^3 + 14*x^4 + 21*x^5 + 30*x^6 + 40*x^7 + 52*x^8 + 65*x^9 + ...

Marc LeBrun, personal communication.
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 = 0..10000
Bruno Berselli, Hexagonal spiral containing the initial terms.
Marc Le Brun, Email to N. J. A. Sloane, Jul 1991
Emanuele Munarini, Topological indices for the antiregular graphs, Le Mathematiche (2021) Vol. 76, No. 1, see p. 283.
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.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A001859, A077043, A002620, A002378.
Row sums of A104567.
Cf. A000034, A032766, A002717, A070893. - Richard Choulet, Jan 28 2010
Cf. A051125.

[(6*n^2+4*n-1+(-1)^n)/8: n in [0..50] ]; // Vincenzo Librandi, Aug 20 2011

with (combinat): seq(count(Partition((3*n+1)), size=3), n=0..52); # Zerinvary Lajos, Mar 28 2008
# 2nd program
A006578 := proc(n)
    (6*n^2 + 4*n - 1 + (-1)^n)/8 ;
end proc: # R. J. Mathar, Apr 28 2017

Accumulate[LinearRecurrence[{1,1,-1}, {0,1,3}, 100]] (* Harvey P. Dale, Sep 29 2013 *)
a[ n_] := Quotient[n + 1, 2] (Quotient[n, 2] 3 + 1); (* Michael Somos, Jun 09 2014 *)
a[ n_] := Quotient[3 (n + 1)^2 + 1, 4] - (n + 1); (* Michael Somos, Jun 10 2015 *)
LinearRecurrence[{2, 0, -2, 1},{0, 1, 4, 8},53] (* Ray Chandler, Aug 03 2015 *)

{a(n) = (3*(n+1)^2 + 1)\4 - n - 1}; /* Michael Somos, Mar 10 2006 */

Expansion of x*(1+2*x) / ((1-x)^2*(1-x^2)). - Simon Plouffe in his 1992 dissertation
a(n) + A002620(n) = A002378(n) = n*(n+1).
Partial sums of A032766. - Paul Barry, May 30 2003
a(n) = a(n-1) + a(n-2) - a(n-3) + 3 = A002620(n) + A004526(n) = A001859(n) - A004526(n+1). - Henry Bottomley, Mar 08 2000
a(n) = (6*n^2 + 4*n - 1 + (-1)^n)/8. - Paul Barry, May 30 2003
a(n) = A001859(-1-n) for all n in Z. - Michael Somos, May 10 2006
a(n) = (A002378(n)/2 + A035608(n))/2. - Reinhard Zumkeller, Feb 07 2010
a(n) = (3*n^2 + 2*n - (n mod 2))/4. - Ctibor O. Zizka, Mar 11 2012
a(n) = Sum_{i=1..n} floor(3*i/2) = Sum_{i=0..n} (i + floor(i/2)). - Enrique Pérez Herrero, Apr 21 2012
a(n) = 3*n*(n+1)/2 - A001859(n). - Clark Kimberling, Jul 02 2012
a(n) = Sum_{i=1..n} (n - i + 1) * 2^( (i+1) mod 2 ). - Wesley Ivan Hurt, Mar 30 2014
a(n) = A002717(n) - A002717(n-1). - Michael Somos, Jun 09 2014
a(n) = Sum_{k=1..n} floor((n+k+1)/2). - Wesley Ivan Hurt, Mar 31 2017
a(n) = A002620(n+1)+2*A002620(n). - R. J. Mathar, Apr 28 2017
Sum_{n>=1} 1/a(n) = 3 - Pi/(4*sqrt(3)) - 3*log(3)/4. - Amiram Eldar, May 28 2022
E.g.f.: (x*(5 + 3*x)*cosh(x) - (1 - 5*x - 3*x^2)*sinh(x))/4. - Stefano Spezia, Aug 22 2023

Offset and description changed by N. J. A. Sloane, Nov 30 2006

A093005 a(n) = n * ceiling(n/2).

Original entry on oeis.org

1, 2, 6, 8, 15, 18, 28, 32, 45, 50, 66, 72, 91, 98, 120, 128, 153, 162, 190, 200, 231, 242, 276, 288, 325, 338, 378, 392, 435, 450, 496, 512, 561, 578, 630, 648, 703, 722, 780, 800, 861, 882, 946, 968, 1035, 1058, 1128, 1152, 1225, 1250, 1326, 1352, 1431, 1458

Offset: 1

Amarnath Murthy, Mar 29 2004

Old name was: The lone multiple of n among the next n numbers.
Another old name: a(n) = n*floor((n+1)/2).
Consider the triangle
   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
  ... Then sequence contains the multiple of n in the n-th row.
Interleaves A000384 and A001105. - Paul Barry, Jun 29 2006
Termwise products of the natural numbers and the natural numbers repeated.
Number of pairs (x,y) having the same parity, with x in {0,...,n} and y in {0,...,2n}. - Clark Kimberling, Jul 02 2012
Similar to generalized hexagonal numbers A000217. Other members of this family are A210977, A006578, A210978, A181995, A210981, A210982. - Omar E. Pol, Aug 09 2012
For even n, a(n) gives the sum of all the parts in the partitions of n into exactly two parts.  For odd n>1, a(n) gives n plus the sum of all the parts in the partitions of n into exactly two parts. - Wesley Ivan Hurt, Nov 14 2013
Number of regions of the plane that do not contain the origin, in the arrangement of lines with polar equations rho = 1/cos(theta-k*2*Pi/n), k=0..n-1; or, by inversion, number of bounded regions in the arrangement of circles with radius 1 and centers the n-th roots of unity. - Luc Rousseau, Feb 08 2019
Numbers k such that floor(sqrt(2k)+1/2) | k. - Wesley Ivan Hurt, Dec 01 2020

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Torleiv Kløve, Linear recurring sequences in boolean rings, Math. Scand., 33 (1973), 5-12.
Torleiv Kløve, Linear recurring sequences in boolean rings, Math. Scand., 33 (1973), 5-12. (Annotated scanned copy)
Luc Rousseau, Illustration, a(n) viewed as a number of regions in an arrangement of lines / of circles.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000217, A000384, A001105, A006578, A008619 (a(n)/n), A181995.

Cf. A183207, A210977, A210978, A210981, A210982, A370980.

[n*(n+(n mod 2))/2: n in [1..70]]; // G. C. Greubel, Mar 14 2024

A093005:=n->n*ceil(n/2); seq(A093005(n), n=1..100); # Wesley Ivan Hurt, Nov 14 2013

a[n_Integer] := n*Floor[(n + 1)/2] (* Olivier Gérard, Jun 21 2007 *)
Table[n*Ceiling[n/2],{n,60}] (* or *) LinearRecurrence[{1,2,-2,-1,1},{1,2,6,8,15},60] (* Harvey P. Dale, May 08 2014 *)

a(n)=(n+1)\2*n \\ Charles R Greathouse IV, Jun 11 2015

for n in range(1,55):
    print(n*((n+1)//2), end=",")   # Alex Ratushnyak, Apr 26 2012

[n*(n +(n%2))/2 for n in range(1,71)] # G. C. Greubel, Mar 14 2024

a(n) = n*floor((n+1)/2).
a(n) = n*A008619(n).
a(2*n-1) = n*(2*n-1), a(2*n) = 2*n^2.
From Paul Barry, Jun 29 2006: (Start)
G.f.: x*(1+x+2*x^2)/((1+x)^2*(1-x)^3).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
a(n) = n*((2*n+1) - (-1)^n)/4. (End)
a(n) = n * ceiling(n/2). - Wesley Ivan Hurt, Nov 14 2013
E.g.f.: (1/2)*x*( (x+2)*cosh(x) + (x+1)*sinh(x) ). - G. C. Greubel, Mar 14 2024
Sum_{n>=1} 1/a(n) = Pi^2/12 + 2*log(2). - Amiram Eldar, Mar 15 2024
a(n) = A183207(n) - A370980(n) + 1, by Euler's formula. - Scott R. Shannon, Jul 07 2024

Corrected and extended by Joshua Zucker, May 08 2006
New name from Alex Ratushnyak, Apr 26 2012
New name from Wesley Ivan Hurt, Nov 14 2013

A210982 Zero together with A126264 and positive terms of A051624 interleaved.

Original entry on oeis.org

0, 1, 8, 12, 26, 33, 54, 64, 92, 105, 140, 156, 198, 217, 266, 288, 344, 369, 432, 460, 530, 561, 638, 672, 756, 793, 884, 924, 1022, 1065, 1170, 1216, 1328, 1377, 1496, 1548, 1674, 1729, 1862, 1920, 2060, 2121, 2268, 2332, 2486, 2553, 2714, 2784, 2952, 3025, 3200, 3276, 3458, 3537, 3726

Offset: 0

Omar E. Pol, Aug 03 2012

Vertex number of a square spiral similar to A195162.

This is the case k=5 of the formula b(n,k) = ( 2*(k+5)*n^2+2*(k+3)*n-(k+1)+(2*(k-1)*n+k+1)*(-1)^n )/16. Sequences of the same family: A093025 (k=-1, with an initial 0), A210977 (k=0), A006578 (k=1), A210978 (k=2), A181995 (k=3, with one 0 only), A210981 (k=4). - Luce ETIENNE, Oct 30 2014

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Members of this family are A093005, A210977, A006578, A210978, A181995, A210981, this sequence.

Cf. A001622, A051624, A126264, A195162.

[(10*n^2+8*n-3+(4*n+3)*(-1)^n )/8: n in [0..60]]; // Vincenzo Librandi, Oct 31 2014

Table[(10*n^2 + 8*n - 3 + (4*n + 3)*(-1)^n)/8, {n, 0, 50}] (* G. C. Greubel, Aug 23 2017 *)

my(x='x+O('x^50)); Vec(x*(1+7*x+2*x^2)/((1+x)^2*(1-x)^3)) \\ G. C. Greubel, Aug 23 2017

G.f.: x*(1+7*x+2*x^2) / ( (1+x)^2*(1-x)^3 ). - R. J. Mathar, Aug 07 2012
a(n) = (10*n^2 +8*n -3 +(4*n+3)*(-1)^n)/8. - Luce ETIENNE, Oct 14 2014
E.g.f.: (1/8)*((10*x^3 + 18*x -3)*exp(x) - (4*x - 3)*exp(-x)). - G. C. Greubel, Aug 23 2017
Sum_{n>=1} 1/a(n) = 5/9 + (sqrt(1-2/sqrt(5))/6 + sqrt(1+2/sqrt(5))/8)*Pi + 7*log(phi)*sqrt(5)/24 - 5*log(5)/48, where phi is the golden ratio (A001622). - Amiram Eldar, Aug 21 2022

A181995 a(n) = if n mod 2 = 1 then n*(n - 1) else (n - 1)^2 + (n - 2)/2.

Original entry on oeis.org

0, 0, 1, 6, 10, 20, 27, 42, 52, 72, 85, 110, 126, 156, 175, 210, 232, 272, 297, 342, 370, 420, 451, 506, 540, 600, 637, 702, 742, 812, 855, 930, 976, 1056, 1105, 1190, 1242, 1332, 1387, 1482, 1540, 1640, 1701, 1806, 1870, 1980, 2047, 2162, 2232, 2352, 2425, 2550, 2626, 2756, 2835, 2970, 3052, 3192, 3277, 3422, 3510, 3660, 3751, 3906, 4000, 4160, 4257, 4422

Offset: 0

N. J. A. Sloane, Apr 05 2012

Decagonal numbers (A001107) and twice second hexagonal numbers (A002943) interleaved. - Omar E. Pol, Aug 03 2012
Similar to A074377. Members of this family are A093005, A210977, A006578, A210978, this sequence, A210981, A210982. - Omar E. Pol, Aug 09 2012
Number of kites whose vertices are the vertices a regular 2n-gon. - Halil Ibrahim Kanpak, Nov 08 2018

H. L. Abbott, D. Hanson, N. Sauer, Intersection theorems for systems of sets, J. Combinatorial Theory Ser. A 12 (1972), 381--389.MR0297579 (45 #6633).
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A001107, A002943, A006578, A074377, A093005.

Cf. A210977, A210978, A210981, A210982.

[n*(4*n - 5 - (-1)^n)/4 : n in [0..80]]; // Wesley Ivan Hurt, Apr 11 2016

f:=n->if n mod 2 = 1 then n*(n-1) else (n-1)^2+(n-2)/2; fi;
[seq(f(n),n=0..130)];

Table[n*(4*n - 5 - (-1)^n)/4, {n, 0, 80}] (* Wesley Ivan Hurt, Apr 11 2016 *)

a(n)=n*(4*n-5-(-1)^n)/4 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: -x^2*(1 + 5*x + 2*x^2)/((1 + x)^2*(x - 1)^3). - R. J. Mathar, Apr 06 2012
a(n) = n*(4*n - 5 - (-1)^n)/4. - Luce ETIENNE, Oct 04 2014
From Wesley Ivan Hurt, Apr 11 2016: (Start)
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>4.
a(n) = Sum_{i=floor((n-1)/2)..floor(3*(n-1)/2)} i. (End)
E.g.f.: x^2*cosh(x) - x*(1 - 2*x)*sinh(x)/2. - Franck Maminirina Ramaharo, Nov 08 2018

A210978 A186029 and positive terms of A001106 interleaved.

Original entry on oeis.org

0, 1, 5, 9, 17, 24, 36, 46, 62, 75, 95, 111, 135, 154, 182, 204, 236, 261, 297, 325, 365, 396, 440, 474, 522, 559, 611, 651, 707, 750, 810, 856, 920, 969, 1037, 1089, 1161, 1216, 1292, 1350, 1430, 1491, 1575, 1639, 1727, 1794, 1886, 1956, 2052, 2125, 2225, 2301

Offset: 0

Omar E. Pol, Aug 03 2012

Vertex number of a square spiral similar to A118277.

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Members of this family are A093005, A210977, A006578, this sequence, A181995, A210981, A210982.

Cf. A001106, A118277, A186029, A195020, A195024.

Vec(-x*(2*x^2+4*x+1)/((x-1)^3*(x+1)^2) + O(x^100)) \\ Colin Barker, Sep 15 2013

a(n) = (3*(-1+(-1)^n)+2*(5+(-1)^n)*n+14*n^2)/16. a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5). G.f.: -x*(2*x^2+4*x+1) / ((x-1)^3*(x+1)^2). - Colin Barker, Sep 15 2013

More terms from Colin Barker, Sep 15 2013

A210981 A062725 and positive terms of A051682 interleaved.

Original entry on oeis.org

0, 1, 7, 11, 23, 30, 48, 58, 82, 95, 125, 141, 177, 196, 238, 260, 308, 333, 387, 415, 475, 506, 572, 606, 678, 715, 793, 833, 917, 960, 1050, 1096, 1192, 1241, 1343, 1395, 1503, 1558, 1672, 1730, 1850, 1911, 2037, 2101, 2233, 2300, 2438, 2508, 2652, 2725, 2875, 2951, 3107, 3186, 3348

Offset: 0

Omar E. Pol, Aug 03 2012

Vertex number of a square spiral similar to A195160.

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Members of this family are A093005, A210977, A006578, A210978, A181995, this sequence, A210982.

Cf. A051682, A062725, A195160.

LinearRecurrence[{1,2,-2,-1,1},{0,1,7,11,23},70] (* Harvey P. Dale, Jun 29 2023 *)

G.f.: -x*(1+6*x+2*x^2) / ( (1+x)^2*(x-1)^3 ). - R. J. Mathar, Aug 07 2012

a(n) = ( 18*n^2+14*n-5+(6*n+5)*(-1)^n )/16. - Luce ETIENNE, Oct 14 2014

A212970 Number of (w,x,y) with all terms in {0,...,n} and w != x and x < range(w,x,y).

Original entry on oeis.org

0, 2, 8, 22, 44, 80, 128, 196, 280, 390, 520, 682, 868, 1092, 1344, 1640, 1968, 2346, 2760, 3230, 3740, 4312, 4928, 5612, 6344, 7150, 8008, 8946, 9940, 11020, 12160, 13392, 14688, 16082, 17544, 19110, 20748, 22496, 24320, 26260, 28280

Offset: 0

Clark Kimberling, Jun 02 2012

For a guide to related sequences, see A212959.

Twice the partial sums of A210977. - J. M. Bergot, Aug 10 2013

Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A088003, A210977, A212959.

Cf. A045991, A212969.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w != x < (Max[w, x, y] - Min[w, x, y]),
   s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 60]]   (* A212970 *)
m/2 (* essentially A088003 *)

a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6).
G.f.: f(x)/g(x), where f(x) = 2*x*(1 + 2*x + 2*x^2) and g(x) = ((1-x)^4)(1+x)^2.
a(n) = 2 * A088003(n) for n>0.
From Ayoub Saber Rguez, Mar 31 2023: (Start)
a(n) + A212969(n+1) = A045991(n+1).
a(n) = (10*n^3 + 24*n^2 + 8*n + (6*n)*(n mod 2))/24. (End)

Typo in name corrected by Ayoub Saber Rguez, Mar 31 2023

cp's OEIS Frontend

A006578 Triangular numbers plus quarter squares: n*(n+1)/2 + floor(n^2/4) (i.e., A000217(n) + A002620(n)).

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A093005 a(n) = n * ceiling(n/2).

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A210982 Zero together with A126264 and positive terms of A051624 interleaved.

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A181995 a(n) = if n mod 2 = 1 then n*(n - 1) else (n - 1)^2 + (n - 2)/2.

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A210978 A186029 and positive terms of A001106 interleaved.

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A210981 A062725 and positive terms of A051682 interleaved.

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A212970 Number of (w,x,y) with all terms in {0,...,n} and w != x and x < range(w,x,y).