A080856 -id:A080856 - cp's OEIS frontend

A386478 Array read by upward antidiagonals: T(k,n) = 1 (k = 0, n >= 0), T(k,n) = k^2n^2/2 - (3k-4)*n/2 + 1 (k >= 1, n >= 0).

1, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 3, 7, 7, 1, 1, 5, 14, 16, 11, 1, 1, 8, 25, 34, 29, 16, 1, 1, 12, 40, 61, 63, 46, 22, 1, 1, 17, 59, 97, 113, 101, 67, 29, 1, 1, 23, 82, 142, 179, 181, 148, 92, 37, 1, 1, 30, 109, 196, 261, 286, 265, 204, 121, 46, 1, 1, 38, 140, 259, 359, 416, 418, 365, 269, 154, 56, 1, 1, 47, 175, 331, 473, 571, 607, 575, 481, 343, 191, 67, 1

Offset: 0

N. J. A. Sloane, Jul 24 2025

A k-chain is a planar graph consisting of a continuous path made up of k straight segments. T(k,n) is the maximum number of regions the plane can be divided into by drawing n k-chains.

The array is almost symmetric: the difference between T(k,n) and T(n,k) is 2*|k-n| (which is exactly the difference between the numbers of infinite regions). All the rows and columns satisfy the recurrence u(n) = 3*u(n-1) - 3*u(n-2) + u(n-3).

			Array begins (the rows are T(0,n>=0),, T(1,n>=0), T(2,n>=0), ...):
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
  1, 2, 4, 7, 11, 16, 22, 29, 37, ...
  1, 2, 7, 16, 29, 46, 67, 92, 121, ...
  1, 3, 14, 34, 63, 101, 148, 204, 269, ...
  1, 5, 25, 61, 113, 181, 265, 365, 481, ...
  1, 8, 40, 97, 179, 286, 418, 575, 757, ...
  1, 12, 59, 142, 261, 416, 607, 834, 1097, ...
  1, 17, 82, 196, 359, 571, 832, 1142, 1501, ...
  1, 23, 109, 259, 473, 751, 1093, 1499, 1969, ...
  ...
The first few antidiagonals are:
  1,
  1, 1,
  1, 2, 1,
  1, 2, 4, 1,
  1, 3, 7, 7, 1,
  1, 5, 14, 16, 11, 1,
  1, 8, 25, 34, 29, 16, 1,
  1, 12, 40, 61, 63, 46, 22, 1,
  ...

David O. H. Cutler and Neil J. A. Sloane, Cutting a pancake with an exotic knife, Paper in preparation, Sep 05 2025

Paolo Xausa, Table of n, a(n) for n = 0..11324 (first 150 antidiagonals, flattened).

The first few rows are A000124, A130883, A140064, A080856, A383465.

The n=1 and 2 columns are A152948 and A386479.

A386478[k_, n_] := If[k == 0, 1, ((k*n - 3)*k + 4)*n/2 + 1];
Table[A386478[k - n, n], {k, 0, 12}, {n, 0, k}] (* Paolo Xausa, Jul 26 2025 *)

Row 0 added by N. J. A. Sloane, Jul 26 2025

A049061 Triangle a(n,k) (1<=k<=n) of "signed Eulerian numbers" read by rows.

Original entry on oeis.org

1, 1, -1, 1, 0, -1, 1, -1, -1, 1, 1, 2, -6, 2, 1, 1, 1, -8, 8, -1, -1, 1, 8, -19, 0, 19, -8, -1, 1, 7, -27, 19, 19, -27, 7, 1, 1, 22, -32, -86, 190, -86, -32, 22, 1, 1, 21, -54, -54, 276, -276, 54, 54, -21, -1, 1, 52, 27, -648, 1002, 0, -1002, 648, -27, -52, -1, 1, 51, -25

Offset: 1

N. J. A. Sloane

Identical to rows n=2..n of D(n,k) on page 2 of Tanimoto reference. - Jonathan Vos Post, Dec 11 2006

			Triangle begins:
  1;
  1, -1;
  1,  0, -1;
  1, -1, -1,  1;
  1,  2, -6,  2,  1;
  ...

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened
J. Desarmenien and D. Foata, The signed Eulerian Numbers
J. Desarmenien and D. Foata, The signed Eulerian numbers, Discrete Math. 99 (1992), no. 1-3, 49-58.
Shinji Tanimoto, Parity-Alternate Permutations and Signed Eulerian Numbers, arXiv:math/0612135 [math.CO], 2006.
S. Tanimoto, A new approach to signed Eulerian numbers, arXiv:math/0602263 [math.CO], 2006.

Cf. A008292.

Cf. A080856.

a049061 n k = a049061_tabl !! (n-1) !! (k-1)
a049061_row n = a049061_tabl !! (n-1)
a049061_tabl = map fst $ iterate t ([1], 1) where
   t (row, k) = (if odd k then us else vs, k + 1) where
     us = zipWith (-) (row ++ [0]) ([0] ++ row)
     vs = zipWith (+) ((zipWith (*) ks row) ++ [0])
                      ([0] ++ (zipWith (*) (reverse ks) row))
          where ks = [1..k]
-- Reinhard Zumkeller, Jan 30 2013

a[n_ /; EvenQ[n] && n > 0, k_] := a[n, k] = a[n - 1, k] - a[n - 1, k - 1]; a[n_ /; OddQ[n] && n > 0, k_] := a[n, k] = k*a[n - 1, k] + (n - k + 1)*a[n - 1, k - 1]; a[0,]=0; a[1,1]=1; Flatten[Table[a[n,k], {n,12}, {k, n}]] (* _Jean-François Alcover, May 02 2011 *)

a(2n, k) = a(2n-1, k) - a(2n-1, k-1), a(2n+1, k) = k*a(2n, k) + (2n-k+2)*a(2n, k-1).

More terms from Larry Reeves (larryr(AT)acm.org), Apr 12 2000

ArXiv URL replaced by its non-cached version - R. J. Mathar, Oct 23 2009

A190816 a(n) = 5n^2 - 4n + 1.

Original entry on oeis.org

1, 2, 13, 34, 65, 106, 157, 218, 289, 370, 461, 562, 673, 794, 925, 1066, 1217, 1378, 1549, 1730, 1921, 2122, 2333, 2554, 2785, 3026, 3277, 3538, 3809, 4090, 4381, 4682, 4993, 5314, 5645, 5986, 6337, 6698, 7069, 7450, 7841, 8242, 8653, 9074

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 20 2011

For n >= 2, hypotenuses of primitive Pythagorean triangles with m = 2*n-1, where the sides of the triangle are a = m^2 - n^2, b = 2*n*m, c = m^2 + n^2; this sequence is the c values, short sides (a) are A045944(n-1), and long sides (b) are A002939(n).

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Short sides (a) A045944(n-1), long sides (b) A002939(n).
Cf. A017281 (first differences), A051624 (a(n)-1), A202141.
Sequences of the form m*n^2 - 4*n + 1: -A131098 (m=0), A028872 (m=1), A056220 (m=2), A045944 (m=3), A016754 (m=4), this sequence (m=5), A126587 (m=6), A339623 (m=7), A080856 (m=8).

[5*n^2 - 4*n + 1: n in [0..50]]; // Vincenzo Librandi, Jun 19 2011

Table[5*n^2 - 4*n + 1, {n, 0, 100}]
LinearRecurrence[{3,-3,1},{1,2,13},100] (* or *) CoefficientList[ Series[ (-10 x^2+x-1)/(x-1)^3,{x,0,100}],x] (* Harvey P. Dale, May 24 2011 *)

a(n)=5*n^2-4*n+1 \\ Charles R Greathouse IV, Oct 16 2015

[5*n^2-4*n+1 for n in range(41)] # G. C. Greubel, Dec 03 2023

From Harvey P. Dale, May 24 2011: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=1, a(1)=2, a(2)=13.
G.f.: (1 - x + 10*x^2)/(1-x)^3. (End)
E.g.f.: (1 + x + 5*x^2)*exp(x). - G. C. Greubel, Dec 03 2023

Edited by Franklin T. Adams-Watters, May 20 2011

A360418 Numbers k such that, in a listing of all congruence classes of positive integers, the k-th congruence class contains k. Here the class r mod m (with r in {1,...,m}) precedes the class a' mod b' (with r' in {1,...,m'}) iff m < m' or r > r'.

Original entry on oeis.org

1, 2, 3, 5, 13, 17, 20, 25, 41, 48, 53, 61, 85, 95, 102, 113, 145, 158, 167, 181, 221, 237, 248, 265, 313, 332, 345, 365, 421, 443, 458, 481, 545, 570, 587, 613, 685, 713, 732, 761, 841, 872, 893, 925, 1013, 1047, 1070, 1105, 1201, 1238, 1263, 1301, 1405, 1445, 1472, 1513, 1625, 1668, 1697, 1741, 1861

Offset: 1

James Propp, Feb 06 2023

The sequence appears to be the interleaving of the four sequences A080856, A102083, A360416, A360417. This has been verified for values of k up to one million as of February 06 2023.

Above conjecture confirmed with more terms and linear recurrence. See supporting formula below. - Ray Chandler, Feb 10 2025

			The 1st congruence class in the list (with m=1 and r=1) is {1,2,3,...} which contains 1, so 1 is in the sequence. The 2nd congruence class (with m=2 and r=2) is {2,4,6,...} which contains 2, so 2 is in the sequence. The 3rd congruence class (with m=2 and r=1) is {1,3,5,...} which contains 3, so 3 is in the sequence. The 4th congruence class (with m=3 and r=3) is {3,6,9,...} which does not contain 4, so 4 is not in the sequence.

Ray Chandler, Table of n, a(n) for n = 1..14142 (terms up to 10^8)
Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 2, -2, 0, 0, -1, 1).

Cf. A002024, A004736, A007607, A080856, A102083, A360416, A360417.

mval[n_] := Floor[Sqrt[2 n] + 1/2]; (* A002024 *)
rval[n_] := (2 - 2 n + Round[Sqrt[2 n]] + Round[Sqrt[2 n]]^2)/2; (* A004736 *)
test[n_] := Mod[n - rval[n], mval[n]] == 0;
Select[Range[10000], test[#] &]

From Ray Chandler, Feb 10 2025: (Start)
a(n) = a(n-1) + 2*a(n-4) - 2*a(n-5) - a(n-8) + a(n-9) for n > 8.
A080856(n) = A360417(n-1) + 2*A080856(n-1) - 2*A360417(n-2) - A080856(n-2) + A360417(n-3).
A102083(n) = A080856(n) + 2*A102083(n-1) - 2*A080856(n-1) - A102083(n-2) + A080856(n-2).
A360416(n) = A102083(n) + 2*A360416(n-1) - 2*A102083(n-1) - A360416(n-2) + A102083(n-2).
A360417(n) = A360416(n) + 2*A360417(n-1) - 2*A360416(n-1) - A360417(n-2) + A360416(n-2). (End)

A185438 a(n) = 8n^2 - 2n + 1.

Original entry on oeis.org

1, 7, 29, 67, 121, 191, 277, 379, 497, 631, 781, 947, 1129, 1327, 1541, 1771, 2017, 2279, 2557, 2851, 3161, 3487, 3829, 4187, 4561, 4951, 5357, 5779, 6217, 6671, 7141, 7627, 8129, 8647, 9181, 9731, 10297, 10879, 11477, 12091, 12721, 13367, 14029, 14707, 15401, 16111, 16837, 17579

Offset: 0

Paul Curtz, Feb 03 2011

Odd numbers (A005408) written clockwise as a square spiral:
.
  41--43--45--47--49--51
   |                   |
  39  13--15--17--19  53
   |   |           |   |
  37  11   1---3  21  55
   |   |       |   |   |
  35   9---7---5  23  57
   |               |   |
  33--31--29--27--25  59
                       |
  71--69--67--65--63--61
.
Walking in straight lines away from the center:
  1, 17, 49, ... = A069129(n+1) =  1 - 8*n + 8*n^2,
  1,  3, 21, ... = A033567(n)   =  1 - 6*n + 8*n^2,
  1, 15, 45, ... = A014634(n)   =  1 + 6*n + 8*n^2,
  1,  5, 25, ... = A080856(n)   =  1 - 4*n + 8*n^2,
  1, 13, 41, ... = A102083(n)   =  1 + 4*n + 8*n^2,
  1,  7, 29, ... = a(n)         =  1 - 2*n + 8*n^2,
  1, 11, 37, ... = A188135(n)   =  1 + 2*n + 8*n^2,
  1,  9, 33, ... = A081585(n)   =  1       + 8*n^2,
  5, 29, 69, ... = A108928(n+1) = -3       + 8*n^2,
  7, 31, 71, ... = A157914(n+1) = -1       + 8*n^2,
  9, 35, 77, ... = A033566(n+1) = -1 + 2*n + 8*n^2.
All are quadrisections of sequences in A181407(n) (example: A014634(n) and A033567(n) in A064038(n+1)) or of this family (?): a(n) is a quadrisection of f(n) = 1,1,1,1,2,7,11,8,11,29,37,23,28,67,79,46,... f(n) is just before A064038(n+1) (fifth vertical) in A181407(n). The companion to a(n) is A188135(n), another quadrisection of f(n). Two last quadrisections of f(n) are A054552(n) and A033951(n).
For n >= 1, bisection of A193867. - Omar E. Pol, Aug 16 2011
Also the sequence may be obtained by starting with the segment (1, 7) followed by the line from 7 in the direction 7, 29, ... in the square spiral whose vertices are the generalized hexagonal numbers (A000217). - Omar E. Pol, Aug 01 2016

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. A000217, A005408, A014634, A014635, A033566, A033567, A033951, A054552, A064038.

Cf. A069129, A080856, A081585, A102083, A108928, A157914, A181407, A188135, A193867.

[1-2*n+8*n^2: n in [0..50]]; // Vincenzo Librandi, Feb 03 2011

Table[1 - 2n + 8n^2, {n, 0, 39}] (* Alonso del Arte, Feb 03 2011 *)
CoefficientList[Series[(-1 - 4 x - 11 x^2)/(x - 1)^3, {x, 0, 47}], x] (* Michael De Vlieger, Aug 01 2016 *)

a(n)=8*n^2-2*n+1 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = a(n-1) + 16*n - 10 (n > 0).
a(n) = 2*a(n-1) - a(n-2) + 16 (n > 1).
a(n) = 3*(n-1) - 3*a(n-2) + a(n-3) (n > 2).
G.f.: (-1 - 4*x - 11*x^2)/(x-1)^3. - R. J. Mathar, Feb 03 2011
a(n) = A014635(n) + 1. - Bruno Berselli, Apr 09 2011
E.g.f.: exp(x)*(1 + 6*x + 8*x^2). - Elmo R. Oliveira, Nov 17 2024

A185871 (Even,even)-polka dot array in the natural number array A000027, by antidiagonals.

Original entry on oeis.org

5, 12, 14, 23, 25, 27, 38, 40, 42, 44, 57, 59, 61, 63, 65, 80, 82, 84, 86, 88, 90, 107, 109, 111, 113, 115, 117, 119, 138, 140, 142, 144, 146, 148, 150, 152, 173, 175, 177, 179, 181, 183, 185, 187, 189, 212, 214, 216, 218, 220, 222, 224, 226, 228, 230, 255, 257, 259, 261, 263, 265, 267, 269, 271, 273, 275, 302, 304, 306, 308, 310, 312, 314, 316, 318, 320, 322, 324, 353, 355, 357, 359, 361, 363, 365, 367, 369, 371, 373, 375, 377, 408, 410, 412, 414, 416, 418, 420, 422, 424, 426, 428, 430, 432, 434

Offset: 1

Clark Kimberling, Feb 05 2011

This is the fourth of four polka dot arrays in the natural number array A000027.  See A185868.
row 1: A096376
col 1: A014106
col 2: A071355
diag (5,25,...): A080856
diag (12,40,...): A033586
antidiagonal sums: A048395 (sums of consecutive squares)

			Northwest corner:
  5....12...23...38...57
  14...25...40...59...82
  27...42...61...84...111
  44...63...86...113..144

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A000027 (as an array), A185868, A185869, A185870.

f[n_,k_]:=2n+(n+k-1)(2n+2k-1);
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]]
Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten

from math import comb, isqrt
def A185871(n):
    a = (m:=isqrt(k:=n<<1))+(k>m*(m+1))
    x = n-comb(a,2)
    y = a-x+1
    return y*((y+(c:=x<<1)<<1)-3)+x*(c-1)+1 # Chai Wah Wu, Jun 18 2025

T(n,k) = 2*n + (n+k-1)*(2*n+2*k-1), k>=1, n>=1.

A060820 a(n) = (2n-1)^2 + (2n)^2.

Original entry on oeis.org

5, 25, 61, 113, 181, 265, 365, 481, 613, 761, 925, 1105, 1301, 1513, 1741, 1985, 2245, 2521, 2813, 3121, 3445, 3785, 4141, 4513, 4901, 5305, 5725, 6161, 6613, 7081, 7565, 8065, 8581, 9113, 9661, 10225, 10805, 11401, 12013, 12641, 13285, 13945, 14621, 15313

Offset: 1

Jason Earls, May 05 2001

			a(1)=5 because 1^2+2^2=5. a(2)=25 because 3^2+4^2=25.

Marilyn vos Savant and Leonore Fleischer, Brain Building in Just 12 Weeks, Bantam Books, New York, NY, 1991, pp. 104-105, 119.

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

Essentially identical to A080856, which is the main entry for this sequence. - N. J. A. Sloane, Jul 13 2025

Table[(2*n - 1)^2 + (2*n)^2, {n, 300}] (* Vladimir Joseph Stephan Orlovsky, Feb 16 2012 *)
LinearRecurrence[{3,-3,1},{5,25,61},60] (* Harvey P. Dale, Oct 13 2020 *)

a(n) = (2*n - 1)^2 + (2*n)^2 \\ Harry J. Smith, Jul 12 2009

a(n) = 3*a(n-1)-3*a(n-2)+a(n-3). G.f.: x*(5+10*x+x^2)/(1-x)^3. - Colin Barker, Apr 22 2012

A341896 a(n) is the number of words of length n over the alphabet {a,b,c} with an even number of appearances of the letter 'a' and the sum of appearances of the letters 'b' and 'c' add up to at most 3.

Original entry on oeis.org

1, 2, 5, 14, 25, 90, 61, 294, 113, 690, 181, 1342, 265, 2314, 365, 3670, 481, 5474, 613, 7790, 761, 10682, 925, 14214, 1105, 18450, 1301, 23454, 1513, 29290, 1741, 36022, 1985, 43714, 2245, 52430, 2521, 62234, 2813, 73190, 3121, 85362, 3445, 98814, 3785, 113610

Offset: 0

Luis Mantilla, Feb 28 2021

			a(0) = 1 : the empty word.
a(1) = 2 : {b, c}.
a(2) = 5 : {aa, bb, cc, bc, cb}.
a(3) = 14 : {aab, aac, aba, aca, baa, bbb, bbc, bcb, bcc, caa, cbb, cbc, ccb, bbb}.
a(4) = 25 : {aaaa, aabb, aabc, aacb, aacc, abab, abac, abba, abca, acab, acac, acba, baab, baac, baba, baca, bbaa, bcaa, caab, caac, caba, caca, cbaa, ccaa, acca}.

Rodrigo de Castro, Teoría de la computación, 2004, unilibros.

Luis Mantilla, Table of n, a(n) for n = 0..46
Luis Mantilla, demonstration
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-6,0,4,0,-1).

Bisection gives: A080856 (even part).

a(n) = 4*a(n-2) - 6*a(n-4) + 4*a(n-6) - a(n-8).
G.f.: (10*x^7-13*x^6+46*x^5+11*x^4+6*x^3+x^2+2*x+1)/((x-1)^4*(x+1)^4).
a(n) = 2*n + 8*C(n,3) if n is odd, a(n) = 1 + 4*C(n,2) if n is even. - Alois P. Heinz, Mar 01 2021

A381552 Triangle read by rows T(n,k) is the number of diamond coverings for a specific number of diamonds covering an odd length row of triangles.

Original entry on oeis.org

3, 4, 4, 5, 12, 4, 6, 25, 20, 4, 7, 44, 61, 28, 4, 8, 70, 146, 113, 36, 4, 9, 104, 301, 344, 181, 44, 4, 10, 147, 560, 876, 670, 265, 52, 4, 11, 200, 966, 1968, 2035, 1156, 365, 60, 4, 12, 264, 1572, 4026, 5368, 4082, 1834, 481, 68, 4, 13, 340, 2442, 7656, 12727, 12376, 7385, 2736, 613, 76, 4, 14, 429, 3652, 13728, 27742, 33397, 25312, 12376, 3894, 761, 84, 4

Offset: 0

Craig Knecht, Feb 27 2025

The total number of ways the diamond can cover a single row of length(n) triangles is the Fibonacci series. This total can be subdivided into categories based on the number of covering diamonds. The number of categories increase with the length of the row providing the structure of the triangle (see illustrations in the link below).

A381555 provides additional graphics explaining the diamond coverings.

			Triangle begins:
  3;
  4,  4;
  5, 12,   4;
  6, 25,  20,   4;
  7, 44,  61,  28,  4;
  8, 70, 146, 113, 36, 4;
  ...

Craig Knecht, Splitting the Fibonacci numbers.
Craig Knecht, Fibonacci Expansion.

Cf. A000297, A080856, A381555.

cp's OEIS Frontend

A386478 Array read by upward antidiagonals: T(k,n) = 1 (k = 0, n >= 0), T(k,n) = k^2*n^2/2 - (3*k-4)*n/2 + 1 (k >= 1, n >= 0).

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A049061 Triangle a(n,k) (1<=k<=n) of "signed Eulerian numbers" read by rows.

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A190816 a(n) = 5*n^2 - 4*n + 1.

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A360418 Numbers k such that, in a listing of all congruence classes of positive integers, the k-th congruence class contains k. Here the class r mod m (with r in {1,...,m}) precedes the class a' mod b' (with r' in {1,...,m'}) iff m < m' or r > r'.

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A185438 a(n) = 8*n^2 - 2*n + 1.

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A185871 (Even,even)-polka dot array in the natural number array A000027, by antidiagonals.

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A060820 a(n) = (2*n-1)^2 + (2*n)^2.

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A386478 Array read by upward antidiagonals: T(k,n) = 1 (k = 0, n >= 0), T(k,n) = k^2n^2/2 - (3k-4)*n/2 + 1 (k >= 1, n >= 0).

A190816 a(n) = 5n^2 - 4n + 1.

A185438 a(n) = 8n^2 - 2n + 1.

A060820 a(n) = (2n-1)^2 + (2n)^2.