A051890 -id:A051890 - cp's OEIS frontend

A217776 a(n) = n(n+1) + (n+2)(n+3) + (n+4)(n+5) + (n+6)(n+7).

68, 100, 140, 188, 244, 308, 380, 460, 548, 644, 748, 860, 980, 1108, 1244, 1388, 1540, 1700, 1868, 2044, 2228, 2420, 2620, 2828, 3044, 3268, 3500, 3740, 3988, 4244, 4508, 4780, 5060, 5348, 5644, 5948, 6260, 6580, 6908, 7244, 7588, 7940, 8300, 8668, 9044, 9428

Offset: 0

Jon Perry, Mar 24 2013

			a(1) = 1*2 + 3*4 + 5*6 + 7*8 = 2 + 12 + 30 + 56 = 100.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A020742, A027690, A051890 (two pairs), A217775 (3 pairs).

List([0..50], n-> (2*n+7)^2+19); # G. C. Greubel, Aug 27 2019

for (j=0;j<50;j++) {
a=j*(j+1)+(j+2)*(j+3)+(j+4)*(j+5)+(j+6)*(j+7);
document.write(a+", ");
}

[(2*n+7)^2+19: n in [0..50]]; // G. C. Greubel, Aug 27 2019

seq((2*n+7)^2+19, n=0..50); # G. C. Greubel, Aug 27 2019

(2*Range[50] +5)^2 +19 (* G. C. Greubel, Aug 27 2019 *)
Table[4n^2+28n+68,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{68,100,140},50] (* Harvey P. Dale, Jan 15 2020 *)

a(n)=4*n^2+28*n+68 \\ Charles R Greathouse IV, Jun 17 2017

[(2*n+7)^2+19 for n in (0..50)] # G. C. Greubel, Aug 27 2019

From Bruno Berselli, Mar 29 2013: (Start)
G.f.: 4*(17-26*x+11*x^2)/(1-x)^3.
a(n) = 4*n^2 + 28*n + 68.
a(n) = 4*A027690(n+3) = A020742(n)^2 + 19. (End)
E.g.f.: 4*(17 +8*x +x^2)*exp(x). - G. C. Greubel, Aug 27 2019

A254527 Total number of points on a sphere when both poles are on an x by x grid where x=8*n+1.

Original entry on oeis.org

6, 26, 62, 114, 182, 266, 366, 482, 614, 762, 926, 1106, 1302, 1514, 1742, 1986, 2246, 2522, 2814, 3122, 3446, 3786, 4142, 4514, 4902, 5306, 5726, 6162, 6614, 7082, 7566, 8066, 8582, 9114, 9662, 10226, 10806, 11402, 12014, 12642, 13286, 13946, 14622, 15314

Offset: 1

Thomas Olson, Jan 31 2015

Maximum number of regions formed by n circles and n ellipses in the plane. - Ivan N. Ianakiev, Sep 21 2019

Number of points on a sphere whose longitude and latitude are both multiples of (90 degrees)/n, including the poles. - Jianing Song, Aug 28 2022

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

Cf. A054554, A051890.

Table[8*n^2  - 4*n + 2,{n,1,44}] (* Ivan N. Ianakiev, Sep 21 2019 *)

vector(50, n, 8*n^2 - 4*n + 2) \\ Michel Marcus, Feb 08 2015

Vec(-2*x*(x+1)*(x+3)/(x-1)^3 + O(x^100)) \\ Colin Barker, Aug 09 2015

a(n) = 8*n^2 - 4*n + 2.
From Colin Barker, Aug 09 2015: (Start)
a(n) = 2*A054554(n+1).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>3.
G.f.: -2*x*(x+1)*(x+3) / (x-1)^3.
(End)
E.g.f.: -2 + exp(x)*(2 + 4*x + 8*x^2). - Stefano Spezia, Sep 21 2019
a(n) = A051890(2*n). - Jianing Song, Aug 28 2022

A294774 a(n) = 2n^2 + 2n + 5.

Original entry on oeis.org

5, 9, 17, 29, 45, 65, 89, 117, 149, 185, 225, 269, 317, 369, 425, 485, 549, 617, 689, 765, 845, 929, 1017, 1109, 1205, 1305, 1409, 1517, 1629, 1745, 1865, 1989, 2117, 2249, 2385, 2525, 2669, 2817, 2969, 3125, 3285, 3449, 3617, 3789, 3965, 4145, 4329, 4517, 4709, 4905

Offset: 0

Bruno Berselli, Nov 08 2017

This is the case k = 9 of 2*n^2 + (1-(-1)^k)*n + (2*k-(-1)^k+1)/4 (similar sequences are listed in Crossrefs section). Note that:

2*( 2*n^2 + (1-(-1)^k)*n + (2*k-(-1)^k+1)/4 ) - k = ( 2*n + (1-(-1)^k)/2 )^2. From this follows an alternative definition for the sequence: Numbers h such that 2*h - 9 is a square. Therefore, if a(n) is a square then its base is a term of A075841.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

1st diagonal of A154631, 3rd diagonal of A055096, 4th diagonal of A070216.
Second column of Mathar's array in A016813 (Comments section).
Subsequence of A001481, A001983, A004766, A020668, A046711 and A057653 (because a(n) = (n+2)^2 + (n-1)^2); A097268 (because it is also a(n) = (n^2+n+3)^2 - (n^2+n+2)^2); A047270; A243182 (for y=1).
Similar sequences (see the first comment): A161532 (k=-14), A181510 (k=-13), A152811 (k=-12), A222182 (k=-11), A271625 (k=-10), A139570 (k=-9), (-1)*A147973 (k=-8), A059993 (k=-7), A268581 (k=-6), A090288 (k=-5), A054000 (k=-4), A142463 or A132209 (k=-3), A056220 (k=-2), A046092 (k=-1), A001105 (k=0), A001844 (k=1), A058331 (k=2), A051890 (k=3), A271624 (k=4), A097080 (k=5), A093328 (k=6), A271649 (k=7), A255843 (k=8), this sequence (k=9).

seq(2*n^2 + 2*n + 5, n=0..100); # Robert Israel, Nov 10 2017

Table[2n^2+2n+5,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{5,9,17},50] (* Harvey P. Dale, Sep 18 2023 *)

Vec((5 - 6*x + 5*x^2) / (1 - x)^3 + O(x^50)) \\ Colin Barker, Nov 13 2017

O.g.f.: (5 - 6*x + 5*x^2)/(1 - x)^3.
E.g.f.: (5 + 4*x + 2*x^2)*exp(x).
a(n) = a(-1-n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = 5*A000217(n+1) - 6*A000217(n) + 5*A000217(n-1).
n*a(n) - Sum_{j=0..n-1} a(j) = A002492(n) for n>0.
a(n) = Integral_{x=0..2n+4} |3-x| dx. - Pedro Caceres, Dec 29 2020

A199855 Inverse permutation to A210521.

Original entry on oeis.org

1, 4, 2, 5, 3, 6, 11, 7, 12, 8, 13, 9, 14, 10, 15, 22, 16, 23, 17, 24, 18, 25, 19, 26, 20, 27, 21, 28, 37, 29, 38, 30, 39, 31, 40, 32, 41, 33, 42, 34, 43, 35, 44, 36, 45, 56, 46, 57, 47, 58, 48, 59, 49, 60, 50, 61, 51, 62, 52, 63, 53, 64, 54, 65, 55, 66, 79

Offset: 1

Boris Putievskiy, Feb 04 2013

Permutation of the natural numbers.
a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.
Enumeration table T(n,k). The order of the list:
  T(1,1)=1;
  T(2,1), T(2,2), T(1,2), T(1,3), T(3,1),
  ...
  T(2,n-1), T(4,n-3), T(6,n-5), ..., T(n,1),
  T(2,n),   T(4,n-2), T(6,n-4), ..., T(n,2),
  T(1,n),   T(3,n-2), T(5,n-4), ..., T(n-1,2),
  T(1,n+1), T(3,n-1), T(5,n-3), ..., T(n+1,1),
  ...
The order of the list elements of adjacent antidiagonals. Let m be a positive integer.
Movement by antidiagonal {T(1,2*m), T(2*m,1)}     from T(2,2*m-1) to T(2*m,1)   length of step is 2,
movement by antidiagonal {T(1,2*m+1), T(2*m+1,1)} from T(2,2*m)   to T(2*m,2)   length of step is 2,
movement by antidiagonal {T(1,2*m), T(2*m,1)}     from T(1,2*m)   to T(2*m-1,2) length of step is 2,
movement by antidiagonal {T(1,2*m+1), T(2*m+1,1)} from T(1,2*m+1) to T(2*m+1,1) length of step is 2.
Table contains:
  row  1 is alternation of elements A001844 and A084849,
  row  2 is alternation of elements A130883 and A058331,
  row  3 is alternation of elements A051890 and A096376,
  row  4 is alternation of elements A033816 and A005893,
  row  6 is alternation of elements A100037 and A093328;
  row  5  accommodates elements A097080 in odd places,
  row  7  accommodates elements A137882 in odd places,
  row 10  accommodates elements A100038 in odd places,
  row 14  accommodates elements A100039 in odd places;
  column 1 is A093005 and alternation of elements A000384 and A001105,
  column 2 is alternation of elements A046092 and A014105,
  column 3 is A105638 and alternation of elements A014106 and A056220,
  column 4 is alternation of elements A142463 and A014107,
  column 5 is alternation of elements A091823 and A054000,
  column 6 is alternation of elements A090288 and |A168244|,
  column 8 is alternation of elements A059993 and A033537;
  column  7 accommodates elements  A071355  in odd  places,
  column  9 accommodates elements |A147973| in even places,
  column 10 accommodates elements  A139570  in odd  places,
  column 13 accommodates elements  A130861  in odd  places.

			The start of the sequence as table:
   1,  4,  5,  11,  13,  22,  25,  37,  41,  56,  61, ...
   2,  3,  7,   9,  16,  19,  29,  33,  46,  51,  67, ...
   6, 12, 14,  23,  26,  38,  42,  57,  62,  80,  86, ...
   8, 10, 17,  20,  30,  34,  47,  52,  68,  74,  93, ...
  15, 24, 27,  39,  43,  58,  63,  81,  87, 108, 115, ...
  18, 21, 31,  35,  48,  53,  69,  75,  94, 101. 123, ...
  28, 40, 44,  59,  64,  82,  88, 109, 116, 140, 148, ...
  32, 36, 49,  54,  70,  76,  95, 102, 124, 132, 157, ...
  45, 60, 65,  83,  89, 110, 117, 141, 149, 176, 185, ...
  50, 55, 71,  77,  96, 103, 125, 133, 158, 167, 195, ...
  66, 84, 90, 111, 118, 142, 150, 177, 186, 216, 226, ...
  ...
The start of the sequence as triangle array read by rows:
   1;
   4,  2;
   5,  3,  6;
  11,  7, 12,  8;
  13,  9, 14, 10, 15;
  22, 16, 23, 17, 24, 18;
  25, 19, 26, 20, 27, 21, 28;
  37, 29, 38, 30, 39, 31, 40, 32;
  41, 33, 42, 34, 43, 35, 44, 36, 45;
  56, 46, 57, 47, 58, 48, 59, 49, 60, 50;
  61, 51, 62, 52, 63, 53, 64, 54, 65, 55, 66;
  ...
The start of the sequence as array read by rows, the length of row r is 4*r-3.
First 2*r-2 numbers are from the row number 2*r-2 of  triangle array, located above.
Last  2*r-1 numbers are from the row number 2*r-1 of  triangle array, located above.
   1;
   4, 2, 5, 3, 6;
  11, 7,12, 8,13, 9,14,10,15;
  22,16,23,17,24,18,25,19,26,20,27,21,28;
  37,29,38,30,39,31,40,32,41,33,42,34,43,35,44,36,45;
  56,46,57,47,58,48,59,49,60,50,61,51,62,52,63,53,64,54,65,55,66;
  ...
Row number r contains permutation numbers 4*r-3 from 2*r*r-5*r+4 to 2*r*r-r:
2*r*r-3*r+2,2*r*r-5*r+4, 2*r*r-3*r+3, 2*r*r-5*r+5, 2*r*r-3*r+4, 2*r*r-5*r+6, ..., 2*r*r-3*r+1, 2*r*r-r.
...

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Eric Weisstein's World of Mathematics, Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A210521, A001844, A084849, A130883, A058331, A051890, A096376, A033816, A005893, A100037, A093328, A097080, A137882, A100038, A100039, A093005, A000384, A001105, A046092, A014105, A105638, A014106, A056220, A142463, A014107, A091823, A054000, A090288, A168244, A059993, A033537, A071355, A147973, A139570, A130861.

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
result=(2*j**2+(4*i-5)*j+2*i**2-3*i+2+(2+(-1)**j)*((1-(t+1)*(-1)**i)))/4

T(n,k) = (2*k^2+(4*n-5)*k+2*n^2-3*n+2+(2+(-1)^k)*((1-(k+n-1)*(-1)^i)))/4.

a(n) = (2*j^2+(4*i-5)*j+2*i^2-3*i+2+(2+(-1)^j)*((1-(t+1)*(-1)^i)))/4, where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((sqrt(8*n-7) - 1)/2).

A221216 T(n,k) = ((n+k)^2-2(n+k)+4-(3n+k-2)*(-1)^(n+k))/2; n , k > 0, read by antidiagonals.

Original entry on oeis.org

1, 5, 6, 4, 3, 2, 12, 13, 14, 15, 11, 10, 9, 8, 7, 23, 24, 25, 26, 27, 28, 22, 21, 20, 19, 18, 17, 16, 38, 39, 40, 41, 42, 43, 44, 45, 37, 36, 35, 34, 33, 32, 31, 30, 29, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 56, 55, 54, 53, 52, 51, 50, 49, 48, 47, 46, 80

Offset: 1

Boris Putievskiy, Feb 22 2013

Permutation of the natural numbers.
a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.
Enumeration table T(n,k). Let m be natural number. The order of the list:
  T(1,1)=1;
  T(3,1), T(2,2), T(1,3);
  T(1,2), T(2,1);
  . . .
  T(2*m+1,1), T(2*m,2), T(2*m-1,3),...T(2,2*m), T(1,2*m+1);
  T(1,2*m), T(2,2*m-1), T(3,2*m-2),...T(2*m-1,2),T(2*m,1);
  . . .
First row  contains antidiagonal {T(1,2*m+1), ... T(2*m+1,1)}, read upwards.
Second row contains antidiagonal {T(1,2*m), ... T(2*m,1)},  read downwards.

			The start of the sequence as table:
  1....5...4..12..11..23..22...
  6....3..13..10..24..21..39...
  2...14...9..25..20..40..35...
  15...8..26..19..41..34..60...
  7...27..18..42..33..61..52...
  28..17..43..32..62..51..85...
  16..44..31..63..50..86..73...
  . . .
The start of the sequence as triangle array read by rows:
  1;
  5,6;
  4,3,2;
  12,13,14,15;
  11,10,9,8,7;
  23,24,25,26,27,28;
  22,21,20,19,18,17,16;
  . . .
Row number r consecutive contains r numbers.
If r is odd,  row is decreasing.
If r is even, row is increasing.

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Eric Weisstein's World of Mathematics, Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A211394, A221215, A002260, A004736, A003057;
table T(n,k) contains: in rows A084849, A096376, A014105, A014107, A168244, A033537, A100040, A100041;
in columns A130883, A000384, A014106, A033816, A100037, A091823, A071355, A100038, A100039,A130861;
main diagonal and parallel diagonals are  A058331, A001844, A005893,A046092, A093328, A142463, A090288, A059993, A051890, A001105, A097080, A056220, A137882, A054000.

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
result=((t+2)**2-2*(t+2)+4-(3*i+j-2)*(-1)**t)/2

As table
T(n,k) = ((n+k)^2-2*(n+k)+4-(3*n+k-2)*(-1)^(n+k))/2.
As linear sequence
a(n) = (A003057(n)^2-2*A003057(n)+4-(3*A002260(n)+A004736(n)-2)*(-1)^A003056(n))/2;  a(n) = ((t+2)^2-2*(t+2)+4-(i+3*j-2)*(-1)^t)/2,
  where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2).

A221217 T(n,k) = ((n+k)^2-2n+3-(n+k-1)(1+2*(-1)^(n+k)))/2; n , k > 0, read by antidiagonals.

Original entry on oeis.org

1, 6, 5, 4, 3, 2, 15, 14, 13, 12, 11, 10, 9, 8, 7, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 66, 65, 64, 63, 62, 61, 60, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50, 49, 48, 47, 46, 91

Offset: 1

Boris Putievskiy, Feb 22 2013

Permutation of the natural numbers.
a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.
Enumeration table T(n,k). Let m be natural number. The order of the list:
  T(1,1)=1;
  T(3,1), T(2,2), T(1,3);
  T(2,1), T(1,2);
  . . .
  T(2*m+1,1), T(2*m,2), T(2*m-1,3),...T(1,2*m+1);
  T(2*m,1), T(2*m-1,2), T(2*m-2,3),...T(1,2*m);
  . . .
First row  contains antidiagonal {T(1,2*m+1), ... T(2*m+1,1)}, read upwards.
Second row contains antidiagonal {T(1,2*m), ... T(2*m,1)},  read upwards.

			The start of the sequence as table:
  1....6...4..15..11..28..22...
  5....3..14..10..27..21..44...
  2...13...9..26..20..43..35...
  12...8..25..19..42..34..63...
  7...24..18..41..33..62..52...
  23..17..40..32..61..51..86...
  16..39..31..60..50..85..73...
  . . .
The start of the sequence as triangle array read by rows:
  1;
  6,5;
  4,3,2;
  15,14,13,12;
  11,10,9,8,7;
  28,27,26,25,24,23;
  22,21,20,19,18,17,16;
  . . .
Row number r consecutive contains r numbers in decreasing order.

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Eric Weisstein's World of Mathematics, Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A211394, A221215, A002260, A004736, A003057, A002024;
table T(n,k) contains: in rows A084849, A000384, A014106, A014105, A014107, A091823, A071355, A091823, A071355,  A100040, A130861, A100041;
in columns A130883, A096376, A033816, A100037, A100038, A100039;
main diagonal and parallel diagonals are A058331, A051890, A005893, A097080, A093328, A137882, A001844, A001105, A056220, A142463, A054000, A090288, A059993.

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
result=((t+2)**2-2*i+3-(t+1)*(1+2*(-1)**t))/2

As table
T(n,k) = ((n+k)^2-2*n+3-(n+k-1)*(1+2*(-1)^(n+k)))/2.
As linear sequence
a(n) = (A003057(n)^2-2*A002260(n)+3-A002024(n)*(1+2*(-1)^A003056(n)))/2;
a(n) = ((t+2)^2-2*i+3-(t+1)*(1+2*(-1)**t))/2, where i=n-t*(t+1)/2,
  j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2).

A356799 Table read by antidiagonals: T(n,k) (n >= 2, k >= 1) is the number of regions formed in a regular 2n-gon by straight line segments when connecting the k+1 points that divide each side into k equal parts to the equivalent point on the side diagonally opposite.

Original entry on oeis.org

1, 4, 13, 9, 24, 25, 16, 55, 48, 41, 25, 66, 105, 70, 61, 36, 121, 144, 171, 108, 85, 49, 126, 233, 220, 253, 140, 113, 64, 211, 288, 381, 312, 351, 192, 145, 81, 204, 409, 450, 565, 448, 465, 234, 181, 100, 325, 480, 671, 636, 785, 608, 595, 300, 221, 121, 300, 633, 760, 997, 924, 1041, 738, 741, 352, 265

Offset: 2

Scott R. Shannon, Aug 28 2022

Many rows and columns in the table appear to be given by a quadratic in even and odd values of k and n; see the Formula section. The exceptions are for rows with n mod 6 = 0 for even k, and for columns with even k, formulas for which are unknown.

			The table begins:
    1,   4,    9,   16,   25,   36,   49,    64,    81,   100,   121,   144, ...
   13,  24,   55,   66,  121,  126,  211,   204,   325,   300,   463,   414, ...
   25,  48,  105,  144,  233,  288,  409,   480,   633,   720,   905,  1008, ...
   41,  70,  171,  220,  381,  450,  671,   760,  1041,  1150,  1491,  1620, ...
   61, 108,  253,  312,  565,  636,  997,  1056,  1549,  1596,  2221,  2232, ...
   85, 140,  351,  448,  785,  924, 1387,  1568,  2157,  2380,  3095,  3360, ...
  113, 192,  465,  608, 1041, 1248, 1841,  2112,  2865,  3200,  4113,  4512, ...
  145, 234,  595,  738, 1333, 1512, 2359,  2556,  3673,  3870,  5275,  5454, ...
  181, 300,  741,  960, 1661, 1980, 2941,  3360,  4581,  5100,  6581,  7200, ...
  221, 352,  903, 1144, 2025, 2376, 3587,  4048,  5589,  6160,  8031,  8712, ...
  265, 432, 1081, 1344, 2425, 2784, 4297,  4704,  6697,  7152,  9625, 10080, ...
  313, 494, 1275, 1612, 2861, 3354, 5071,  5720,  7905,  8710, 11363, 12324, ...
  365, 588, 1485, 1904, 3333, 3948, 5909,  6720,  9213, 10220, 13245, 14448, ...
  421, 660, 1711, 2130, 3841, 4410, 6811,  7500, 10621, 11400, 15271, 16110, ...
  481, 768, 1953, 2496, 4385, 5184, 7777,  8832, 12129, 13440, 17441, 19008, ...
  545, 850, 2211, 2788, 4965, 5814, 8807,  9928, 13737, 15130, 19755, 21420, ...
  613, 972, 2485, 3096, 5581, 6444, 9901, 10944, 15445, 16668, 22213, 23544, ...
  .
  .

Scott R. Shannon, Table for n=2..35, k=1..50.
Scott R. Shannon, Image for T(2,8) = 64.
Scott R. Shannon, Image for T(3,6) = 126.
Scott R. Shannon, Image for T(3,7) = 211.
Scott R. Shannon, Image for T(4,8) = 480.
Scott R. Shannon, Image for T(4,9) = 633.
Scott R. Shannon, Image for T(6,12) = 2232.
Scott R. Shannon, Image for T(6,13) = 3013.
Scott R. Shannon, Image for T(10,10) = 5100.
Scott R. Shannon, Image for T(10,11) = 6581.
Scott R. Shannon, Image for T(18,1) = 613.
Scott R. Shannon, Image for T(18,2) = 972.
Scott R. Shannon, Image for T(18,3) = 2485.
Scott R. Shannon, Image for T(18,4) = 3096.
Scott R. Shannon, Image for T(18,5) = 5581.
Scott R. Shannon, Image for T(18,6) = 6444.

Cf. A000217, A265225, A000096, A000290, A005563, A001844, A001105, A051890, A249127, A356044.

T(2,k) = k^2.
Conjectured formula for the rows for odd values of k for n>=3:
T(n,k) = A000217(n-1)*k^2 + n^2*k + A000217(n-2) = (n^2 - n)*k^2/2 + n^2*k + (n^2 - 3n + 2)/2.
E.g., T(7,k) = A000217(6)*k^2 + 7^2*k + A000217(5) = 21k^2 + 49k + 15.
Conjectured formula for the rows for even values of k for n>=3:
For n mod 3 = 1 or n mod 3 = 2, T(n,k) = A000217(n-1)*k^2 + A265225(n-1)*k = (n^2 - n)*k^2/2 + (floor(n/2) + 1)*n*k.
E.g., T(10,k) = A000217(9)*k^2 + A265225(9)*k = 45k^2 + 60k.
For n mod 6 = 0, no formula is currently known.
For (n - 3) mod 6 = 0, T(n,k) = A000096(2n-3)*k^2/4 + A005563(n)*k/2 = (2n^2 - 3n)*k^2/4 + (n^2 + 2n)*k/2.
E.g., T(15,k) = 405k^2/4 + 255k/2.
Conjectured formula for the columns for odd values of k for n>=3:
T(n,k) = A001105((k+1)/2)*n^2 - A051890((k+1)/2)*n + 1 = (k^2 + 2k + 1)*n^2/2 - (k^2 + 3)*n/2 + 1.
E.g., T(n,9) =  50n^2 - 42n + 1.
Conjectured formula for T(n,2):
T(n,2) = 2*A249127(n) = 2*floor(3n/2)*n, for n>=3.
No formula is current known for the columns for even values of k for k>=4.

A361070 a(n) is the number of occurrences of n in A360923.

Original entry on oeis.org

1, 1, 1, 2, 4, 5, 7, 10, 12, 15, 19, 21, 27, 30, 35, 40, 44, 52, 56, 63, 70, 75, 85, 90, 100, 107, 115, 126, 132, 145, 153, 163, 175, 182, 199, 206, 220, 232, 242, 259, 268, 285, 297, 310, 328, 337, 359, 370, 387, 404, 416, 440, 451, 472, 489, 504, 528, 540

Offset: 0

Rémy Sigrist, Mar 01 2023

nonn

For n > 0, the number of starting positions from Z^2 at distance n from (0, 0) appears to be A051890(n):
  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
  . . . . . . . . . . . . . . . . . . . . . . . . . 5 6 6 6 6 6
  . . . . . . . . . . . . . . . . . . . . 6 4 5 5 5 5 6 6 6 6 6
  . . . . . . . . . . . . . . . . 6 5 3 4 4 4 5 5 5 5 6 6 6 6 .
  . . . . . . . . . . . . 6 6 5 4 2 3 3 4 4 4 5 5 5 6 6 6 6 . .
  . . . . . . . . . 6 6 5 5 4 3 1 2 3 3 4 4 5 5 5 6 6 6 . . . .
  . . . . . . 6 6 6 5 5 4 4 3 2 0 2 3 4 4 5 5 6 6 6 . . . . . .
  . . . . 6 6 6 5 5 5 4 4 3 3 2 1 3 4 5 5 6 6 . . . . . . . . .
  . . 6 6 6 6 5 5 5 4 4 4 3 3 2 4 5 6 6 . . . . . . . . . . . .
  . 6 6 6 6 5 5 5 5 4 4 4 3 5 6 . . . . . . . . . . . . . . . .
  6 6 6 6 6 5 5 5 5 4 6 . . . . . . . . . . . . . . . . . . . .
  6 6 6 6 6 5 . . . . . . . . . . . . . . . . . . . . . . . . .
  6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The largest column of A360923 containing n appears to have index A002620(n).

			Square array A360923 begins as follows:
     0 2 3 4 4 5 5 6 6 6 7 7 7 8 8 8 8 .
     1 3 4 5 5 6 6 7 7 7 8 8 8 . . . . .
     4 5 6 6 7 7 8 8 8 . . . . .
     7 7 8 8 . . . . . .
     . . . . .
Hence a(0) = 1, a(1) = 1, a(2) = 1, a(3) = 2, a(4) = 4, a(5) = 5, a(6) = 7, a(7) = 10 and a(8) = 12.

Rémy Sigrist, C++ program

Cf. A002620, A051890, A360923.

A386485 a(0) = 1; thereafter a(n) = 5n^2 - 5n + 2.

Original entry on oeis.org

1, 2, 12, 32, 62, 102, 152, 212, 282, 362, 452, 552, 662, 782, 912, 1052, 1202, 1362, 1532, 1712, 1902, 2102, 2312, 2532, 2762, 3002, 3252, 3512, 3782, 4062, 4352, 4652, 4962, 5282, 5612, 5952, 6302, 6662, 7032, 7412, 7802, 8202, 8612, 9032, 9462, 9902, 10352, 10812, 11282, 11762, 12252, 12752, 13262, 13782, 14312

Offset: 0

N. J. A. Sloane, Aug 18 2025

Maximum number of regions that can be formed in the plane by drawing n regular pentagons (of any size). Differs from A062786 and A124080 by a small constant shift, but is included here because of its geometrical applications.

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

Cf. A051890, A069894, A077588, A386480.

A077074 Least k such that Z(k,3) <= Z(n,4) where Z(m,s) = Sum_{i>=m} 1/i^s.

Original entry on oeis.org

2, 2, 3, 6, 9, 13, 17, 21, 26, 31, 37, 43, 49, 55, 62, 69, 76, 83, 91, 99, 107, 115, 123, 132, 141, 150, 159, 168, 178, 187, 197, 207, 218, 228, 239, 249, 260, 271, 282, 294, 305, 317, 328, 340, 352, 365

Offset: 0

Benoit Cloitre, Nov 29 2002

nonn

Cf. A051890 for least k such that Z(k,2) <= Z(n,3).

u=3; v=4; a(n)=if(n<0,0,k=1; while((zeta(u)-sum(k=1,k-1,1/k^u))>(zeta(v)-sum(i=1,n-1,1/i^v)),k++); k)

cp's OEIS Frontend

A217776 a(n) = n*(n+1) + (n+2)*(n+3) + (n+4)*(n+5) + (n+6)*(n+7).

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A294774 a(n) = 2*n^2 + 2*n + 5.

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A199855 Inverse permutation to A210521.

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A221216 T(n,k) = ((n+k)^2-2*(n+k)+4-(3*n+k-2)*(-1)^(n+k))/2; n , k > 0, read by antidiagonals.

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A221217 T(n,k) = ((n+k)^2-2*n+3-(n+k-1)*(1+2*(-1)^(n+k)))/2; n , k > 0, read by antidiagonals.

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A356799 Table read by antidiagonals: T(n,k) (n >= 2, k >= 1) is the number of regions formed in a regular 2n-gon by straight line segments when connecting the k+1 points that divide each side into k equal parts to the equivalent point on the side diagonally opposite.

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A217776 a(n) = n(n+1) + (n+2)(n+3) + (n+4)(n+5) + (n+6)(n+7).

A294774 a(n) = 2n^2 + 2n + 5.

A221216 T(n,k) = ((n+k)^2-2(n+k)+4-(3n+k-2)*(-1)^(n+k))/2; n , k > 0, read by antidiagonals.

A221217 T(n,k) = ((n+k)^2-2n+3-(n+k-1)(1+2*(-1)^(n+k)))/2; n , k > 0, read by antidiagonals.

A386485 a(0) = 1; thereafter a(n) = 5n^2 - 5n + 2.