A093005 -id:A093005 - cp's OEIS frontend

A370980 If n is even, (n^2-2*n+2)/2, otherwise (n^2-n+2)/2.

1, 1, 1, 4, 5, 11, 13, 22, 25, 37, 41, 56, 61, 79, 85, 106, 113, 137, 145, 172, 181, 211, 221, 254, 265, 301, 313, 352, 365, 407, 421, 466, 481, 529, 545, 596, 613, 667, 685, 742, 761, 821, 841, 904, 925, 991, 1013, 1082, 1105, 1177, 1201, 1276, 1301, 1379, 1405, 1486, 1513, 1597, 1625, 1712, 1741, 1831, 1861, 1954

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Mar 23 2024

nonn

Total number of circles in A371373 and A371253, if in the later all the circular arcs are completed to form full circles.

The sequence also gives the number of vertices created from circle intersections when a circle of radius r is drawn around each of n equally spaced points on the circumference of a circle of radius r. The number of regions in these constructions is A093005(n) and the number of edges is A183207(n). See the attached images. - Scott R. Shannon, Jul 06 2024.

			a(n) = 1+n*floor((n-1)/2) = 1+n*A004526(n-1). - _Chai Wah Wu_, Mar 23 2024

Scott R. Shannon, Image for n = 3. In this and other images the center of each circle of shown as a white dot.
Scott R. Shannon, Image for n = 4.
Scott R. Shannon, Image for n = 5
Scott R. Shannon, Image for n = 10.
Scott R. Shannon, Image for n = 20.

Bisections are A001844 and A084849.
Cf. A371373, A371374, A371375.
Cf. A004526, A371253, A371254, A371253, A183297, A093005, A183207.

def A370980(n): return n*(n-1>>1)+1 # Chai Wah Wu, Mar 23 2024

a(n) = A183207(n) - A093005(n) + 1, by Euler's formula. - Scott R. Shannon, Jul 07 2024

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

A374826 Place n equally spaced points on the circumference of a circle of radius r and then connect each pair of points with straight lines whose intersections create A007569(n) - n additional points. Draw a circle of radius r around each of the A007569(n) points. The sequence gives the total number of regions formed from all circle intersections.

Original entry on oeis.org

1, 2, 6, 16, 80, 324, 1666, 3120, 17703, 28780, 115401, 96624, 528073, 589708

Offset: 1

Scott R. Shannon, Jul 21 2024

Scott R. Shannon, Image for n = 2. In this and other images the starting circles' centers are shown as white dots.
Scott R. Shannon, Image for n = 3.
Scott R. Shannon, Image for n = 4.
Scott R. Shannon, Image for n = 5.
Scott R. Shannon, Image for n = 6.
Scott R. Shannon, Image for n = 8.
Scott R. Shannon, Image for n = 10.

Cf. A374825 (vertices), A374827 (edges), A374828 (k-gons), A007569 (total circles), A093005, A374337.

a(n) = A374827(n) - A374825(n) + 1, by Euler's formula.

A128621 A127648 * A128174 as an infinite lower triangular matrix.

Original entry on oeis.org

1, 0, 2, 3, 0, 3, 0, 4, 0, 4, 5, 0, 5, 0, 5, 0, 6, 0, 6, 0, 6, 7, 0, 7, 0, 7, 0, 7, 0, 8, 0, 8, 0, 8, 0, 8, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 10, 0, 10, 0, 10, 0, 10, 0, 10, 11, 0, 11, 0, 11, 0, 11, 0, 11, 0, 11, 0, 12, 0, 12, 0, 12, 0, 12, 0, 12, 0, 12, 13, 0, 13, 0, 13, 0, 13, 0, 13, 0, 13, 0, 13

Offset: 1

Gary W. Adamson, Mar 14 2007

			First few rows of the triangle:
  1;
  0, 2;
  3, 0, 3;
  0, 4, 0, 4;
  5, 0, 5, 0, 5;
  ...

G. C. Greubel, Rows n = 1..100 of the triangle, flattened

Cf. A000326, A123684, A127648, A128174.

Cf. A093005 (row sums).

[n*(1+(-1)^(n+k))/2: k in [1..n], n in [1..15]]; // G. C. Greubel, Mar 13 2024

Table[n*(1+(-1)^(n+k))/2, {n,15}, {k,n}]//Flatten (* G. C. Greubel, Mar 13 2024 *)

flatten([[n*(1+(-1)^(n+k))//2 for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Mar 13 2024

Odd rows: n terms of n, 0, n, ...; even rows, n terms of 0, n, 0, ...
T(n,k) = n if n+k even, T(n,k) = 0 if n+k odd.
Sum_{k=1..n} T(n, k) = A093005(n) (row sums).
From G. C. Greubel, Mar 13 2024: (Start)
T(n, k) = n*(1 + (-1)^(n+k))/2.
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (-1)^(n+1)*A093005(n).
Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = (1/2)*(1-(-1)^n) * A000326(floor((n+1)/2)).
Sum_{k=1..floor((n+1)/2)} (-1)^(k-1)*T(n-k+1, k) = (1/2)*(1 - (-1)^n)*A123684(floor((n+1)/2)). (End)

More terms added by G. C. Greubel, Mar 13 2024

A128623 Triangle read by rows: A128621 * A000012 as infinite lower triangular matrices.

Original entry on oeis.org

1, 2, 2, 6, 3, 3, 8, 8, 4, 4, 15, 10, 10, 5, 5, 18, 18, 12, 12, 6, 6, 28, 21, 21, 14, 14, 7, 7, 32, 32, 24, 24, 16, 16, 8, 8, 45, 36, 36, 27, 27, 18, 18, 9, 9, 50, 50, 40, 40, 30, 30, 20, 20, 10, 10, 66, 55, 55, 44, 44, 33, 33, 22, 22, 11, 11, 72, 72, 60, 60, 48, 48, 36, 36, 24, 24, 12, 12, 91, 78, 78, 65, 65, 52, 52, 39, 39, 26, 26, 13, 13

Offset: 1

Gary W. Adamson, Mar 14 2007

			First few rows of the triangle are:
   1;
   2,  2;
   6,  3,  3;
   8,  8,  4,  4;
  15, 10, 10,  5,  5;
  18, 18, 12, 12,  6, 6;
  28, 21, 21, 14, 14, 7, 7;
  ...

G. C. Greubel, Rows n = 1..100 of the triangle, flattened

Cf. A000027, A093005, A093353, A115514, A128621, A245524.

Cf. A128624 (row sums).

[n*Floor((n-k+2)/2): k in [1..n], n in [1..15]]; // G. C. Greubel, Mar 13 2024

Table[n*Floor[(n-k+2)/2], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Mar 13 2024 *)

flatten([[n*((n-k+2)//2) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Mar 13 2024

Sum_{k=1..n} T(n, k) = A128624(n) (row sums).
T(n,k) = n*(1+floor((n-k)/2)), 1 <= k <= n. - R. J. Mathar, Jun 27 2012
From G. C. Greubel, Mar 13 2024: (Start)
T(n, k) = n*A115514(n, k).
T(n, k) = Sum_{j=k..n} A128621(n, j).
T(n, 1) = A093005(n).
T(n, 2) = A093353(n-1), n >= 2.
T(n, n) = A000027(n).
T(2*n-1, n) = A245524(n).
Sum_{k=1..n} (-1)^k*T(n, k) = (1/2)*(1-(-1)^n)*A000384(floor((n+1)/2)). (End)

a(41) = 27 inserted and more terms from Georg Fischer, Jun 05 2023

A005984 Related to recurrences over rings.

Original entry on oeis.org

1, 2, 5, 6, 10, 14, 21, 22, 27, 32, 42, 48, 59, 70, 85

Offset: 1

N. J. A. Sloane

In his paper, Kløve wants to find, in a Boolean ring, the least integer P(r) such that, for any linear recurring sequence {x(n)} of order r, we have x(n+P(r)) = x(n), for all n >= 0. First, he proves that P(r) = 2^v(r)* lcm_{j=1..r} (2^j - 1), where v(r) = floor(log_2(r)) when 1 <= r < 6 and r <= 2^v(r) < 2*r*floor((r+1)/2) for r >= 1. Then, a(n) is defined to be sigma(1-2^r,1,1), being the exact power of X+1 dividing a recursively defined polynomial g(m,X), that is shown to be an upper bound to v(r). He proves also that a(n) <= A093005(n). - Michel Marcus, Mar 02 2013

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. Kløve, Linear recurring sequences in Boolean rings, Math. Scand., 33 (1973), 5-12.
T. Kløve, Linear recurring sequences in Boolean rings, Math. Scand., 33 (1973), 5-12. (Annotated scanned copy)

Cf. A093005.

lambda(m) = {return (floor(log(m)/log(2)));}
D(m) = {local(vb, vbl, j); vb = binary(m); vbl = length(vb); vj = []; for (j=1, lambda(m)+1, if (vb[j] == 1, vj = concat(vj, vbl - j + 1););); return (vj);}
Q(m) = {local(i, xp, vb); xp = lambda(m)+1; q = 1; vb = binary(m); for (i=1, length(vb), q += (vb[i]*Mod(1,2))*x^xp; xp--;); return (q);}
G(n, vG) = {local(vn, vs, vp, vec, i, vi); if (vG[n] != 0, return (vG)); vn = binary(n); vs = sum(i=1, length(vn), vn[i]); if (vs == 1, vG[n] = Q(n); return (vG); ); vp = 1; vec = D(n); for (i=1, length(vec), vi = n-2^(vec[i]-1); vG = G(vi, vG); vp = lcm(vp, vG[vi]);); vG[n] = vp*Q(n); return (vG);}
a(r) = {n = 2^r-1; vG = vector(n); vG = G(n, vG); g = vG[n]; phi = Mod(1,2)*(x + 1); dphi = phi; np = 1; while (1, if (type(g/dphi) != "t_POL", break;); dphi *= phi; np++;); return (np-1);}
\\ Michel Marcus, Mar 03 2013

a(15) from Sean A. Irvine, Nov 05 2016

A130656 Interlacing n^3/2 and n^2(n + 1)/2.

Original entry on oeis.org

1, 4, 18, 32, 75, 108, 196, 256, 405, 500, 726, 864, 1183, 1372, 1800, 2048, 2601, 2916, 3610, 4000, 4851, 5324, 6348, 6912, 8125, 8788, 10206, 10976, 12615, 13500, 15376, 16384, 18513, 19652, 22050, 23328, 26011, 27436, 30420, 32000, 35301, 37044

Offset: 1

Olivier Gérard, Jun 21 2007

Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Cf. A093005 (quadratic equivalent), A065423 (linear equivalent).

A130656:=n->n^2 * floor((n + 1)/2): seq(A130656(n), n=1..100); # Wesley Ivan Hurt, Jan 21 2017

a[n_Integer] := n^2 * Floor[(n + 1)/2]
LinearRecurrence[{1,3,-3,-3,3,1,-1},{1,4,18,32,75,108,196},50] (* Harvey P. Dale, Feb 18 2015 *)

a(n) = n^2 * floor((n + 1)/2).
G.f.: x*(1+3*x+11*x^2+5*x^3+4*x^4)/((1-x)^4*(1+x)^3). - R. J. Mathar, Sep 09 2008
a(n) = a(n-1)+ 3*a(n-2)-3*a(n-3)-3*a(n-4)+3*a(n-5)+a(n-6)-a(n-7), a(1)=1, a(2)=4, a(3)=18, a(4)=32, a(5)=75, a(6)=108, a(7)=196. - Harvey P. Dale, Feb 18 2015
Sum_{n>=1} 1/a(n) = zeta(3)/4 + Pi^2/4 - 2*log(2). - Amiram Eldar, Mar 15 2024

A159469 Maximum remainder when (k + 1)^n + (k - 1)^n is divided by k^2 for variable n and k > 2.

Original entry on oeis.org

6, 8, 20, 24, 42, 48, 72, 80, 110, 120, 156, 168, 210, 224, 272, 288, 342, 360, 420, 440, 506, 528, 600, 624, 702, 728, 812, 840, 930, 960, 1056, 1088, 1190, 1224, 1332, 1368, 1482, 1520, 1640, 1680, 1806, 1848, 1980, 2024, 2162, 2208, 2352, 2400, 2550, 2600

Offset: 3

Gaurav Kumar, Apr 13 2009

			For n = 3, maxr => 3*3 - 3 = 6 since 3 is odd.
For n = 4, maxr => 4*4 - 2*4 = 8 since 4 is even.

Iain Fox, Table of n, a(n) for n = 3..10000
Project Euler, Problem 120: Square remainders
Iain Fox, Proof for recursive definition and generating function
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A050187.

LinearRecurrence[{1,2,-2,-1,1},{6,8,20,24,42},50] (* Harvey P. Dale, Apr 18 2018 *)

a(n) = if (n % 2, n^2 - n, n^2 - 2*n); \\ Michel Marcus, Aug 26 2013

first(n) = Vec(x^3*(-6-2*x)/((x+1)^2*(x-1)^3) + O(x^(n+3))) \\ Iain Fox, Nov 26 2017

maxr(n) = n*n - 2*n if n is even, and n*n - n if n is odd.
G.f.: x^3*(-6-2*x)/((x+1)^2*(x-1)^3). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 26 2009 (proved by Iain Fox, Nov 26 2017)
a(n) = 2*A050187(n). - R. J. Mathar, Aug 08 2009 (proved by Iain Fox, Nov 27 2017)
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 7. - Colin Barker, Oct 29 2017 (proved by Iain Fox, Nov 26 2017)
a(n) = n^2 - n*(3 + (-1)^n)/2. - Iain Fox, Nov 26 2017
From Iain Fox, Nov 27 2017: (Start)
a(n) = A000290(n) - A022998(n).
a(n) = 2*A093005(n-2) + A168273(n-1).
a(n) = (4*(A152749(n-2)) + A091574(n-1) - A010719(n-1))/3.
E.g.f.: x*(exp(x)*x - sinh(x)).
(End)

cp's OEIS Frontend

A370980 If n is even, (n^2-2*n+2)/2, otherwise (n^2-n+2)/2.

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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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A128621 A127648 * A128174 as an infinite lower triangular matrix.

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A128623 Triangle read by rows: A128621 * A000012 as infinite lower triangular matrices.

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A005984 Related to recurrences over rings.

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