A028566 -id:A028566 - cp's OEIS frontend

A158405 Triangle T(n,m) = 1+2*m of odd numbers read along rows, 0<=m

1, 1, 3, 1, 3, 5, 1, 3, 5, 7, 1, 3, 5, 7, 9, 1, 3, 5, 7, 9, 11, 1, 3, 5, 7, 9, 11, 13, 1, 3, 5, 7, 9, 11, 13, 15, 1, 3, 5, 7, 9, 11, 13, 15, 17, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23

Offset: 1

Paul Curtz, Mar 18 2009

Row sums are n^2 = A000290(n).
The triangle sums, see A180662 for their definitions, link this triangle of odd numbers with seventeen different sequences, see the crossrefs. The knight sums Kn14 - Kn110 have been added. - Johannes W. Meijer, Sep 22 2010
A208057 is the eigentriangle of A158405 such that as infinite lower triangular matrices, A158405 * A208057 shifts the latter, deleting the right border of 1's. - Gary W. Adamson, Feb 22 2012
T(n,k) = A099375(n-1,n-k), 1<=k<=n. [Reinhard Zumkeller, Mar 31 2012]

			The triangle contains the first n odd numbers in row n:
  1;
  1,3;
  1,3,5;
  1,3,5,7;
From _Seiichi Manyama_, Dec 02 2017: (Start)
    |       a(n)        |                               | A000290(n)
   -----------------------------------------------------------------
   0|                                                      (=  0)
   1|                 1 = 1/3 * ( 3)                       (=  1)
   2|             1 + 3 = 1/3 * ( 5 +  7)                  (=  4)
   3|         1 + 3 + 5 = 1/3 * ( 7 +  9 + 11)             (=  9)
   4|     1 + 3 + 5 + 7 = 1/3 * ( 9 + 11 + 13 + 15)        (= 16)
   5| 1 + 3 + 5 + 7 + 9 = 1/3 * (11 + 13 + 15 + 17 + 19)   (= 25)
(End)

Seiichi Manyama, Rows n = 1..140, flattened
Daniel Erman, The Josephus Problem, Numberphile video (2016)
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Index entries for sequences related to the Josephus Problem

Cf. A129326, A099375, A005408.
Triangle sums (see the comments): A000290 (Row1; Kn11 & Kn4 & Ca1 & Ca4 & Gi1 & Gi4); A000027 (Row2); A005563 (Kn12); A028347 (Kn13); A028560 (Kn14); A028566 (Kn15); A098603 (Kn16); A098847 (Kn17); A098848 (Kn18); A098849 (Kn19); A098850 (Kn110); A000217 (Kn21. Kn22, Kn23, Fi2, Ze2); A000384 (Kn3, Fi1, Ze3); A000212 (Ca2 & Ze4); A000567 (Ca3, Ze1); A011848 (Gi2); A001107 (Gi3). - Johannes W. Meijer, Sep 22 2010
Cf. A063656, A063657, A208057, A292610, A292611.

a158405 n k = a158405_row n !! (k-1)
a158405_row n = a158405_tabl !! (n-1)
a158405_tabl = map reverse a099375_tabl
-- Reinhard Zumkeller, Mar 31 2012

Table[2 Range[1, n] - 1, {n, 12}] // Flatten (* Michael De Vlieger, Oct 01 2015 *)

a(n) = 2*(n-floor((-1+sqrt(8*n-7))/2)*(floor((-1+sqrt(8*n-7))/2)+1)/2)-1;
vector(100, n, a(n)) \\ Altug Alkan, Oct 01 2015

a(n) = 2*i-1, where i = n-t(t+1)/2, t = floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Feb 03 2013

a(n) = 2*A002262(n-1) + 1. - Eric Werley, Sep 30 2015

Edited by R. J. Mathar, Oct 06 2009

A120071 a(n) = n*(n+20).

Original entry on oeis.org

0, 21, 44, 69, 96, 125, 156, 189, 224, 261, 300, 341, 384, 429, 476, 525, 576, 629, 684, 741, 800, 861, 924, 989, 1056, 1125, 1196, 1269, 1344, 1421, 1500, 1581, 1664, 1749, 1836, 1925, 2016, 2109, 2204, 2301, 2400, 2501, 2604, 2709, 2816, 2925, 3036, 3149, 3264

Offset: 0

Wolfdieter Lang, Jul 20 2006

Felix P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, Preprint on ResearchGate, March 2014.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

a(n-10), n >= 11, tenth column (used for the n=10 series of the hydrogen atom) of triangle A120070.
For n*(n+18) see A098850.
Cf. A001008, A001477, A005563, A028347, A028552, A028557, A028560, A028563, A028566, A028569, A098603, A098847, A098848, A098849, A102928, A120061, A132759, A132760, A132761, A132762, A132765.

seq(sum(4*k-n, k=7..n), n=6..51); # Zerinvary Lajos, Feb 17 2008

s=0;lst={};Do[s+=n;AppendTo[lst,s],{n,21,6!,2}];lst (* Vladimir Joseph Stephan Orlovsky, Feb 26 2009 *)

a(n)=n*(n+20) \\ Charles R Greathouse IV, Jan 21 2015

a(n) = (n+10)^2 - 10^2 = n*(n+20), n >= 0.
G.f.: x*(21-19*x)/(1-x)^3.
a(n) = 2*n + a(n-1) + 19 (with a(0)=0). - Vincenzo Librandi, Nov 13 2010
From Amiram Eldar, Jan 16 2021: (Start)
Sum_{n>=1} 1/a(n) = H(20)/20 = A001008(20)/A102928(20) = 11167027/62078016, where H(k) is the k-th harmonic number.
Sum_{n>=1} (-1)^(n+1)/a(n) = 155685007/4655851200. (End)
From Elmo R. Oliveira, Jan 12 2025: (Start)
E.g.f.: exp(x)*x*(21 + x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3. (End)

A132765 a(n) = n*(n + 23).

Original entry on oeis.org

0, 24, 50, 78, 108, 140, 174, 210, 248, 288, 330, 374, 420, 468, 518, 570, 624, 680, 738, 798, 860, 924, 990, 1058, 1128, 1200, 1274, 1350, 1428, 1508, 1590, 1674, 1760, 1848, 1938, 2030, 2124, 2220, 2318, 2418, 2520, 2624, 2730, 2838, 2948, 3060, 3174, 3290, 3408

Offset: 0

Omar E. Pol, Aug 28 2007

G. C. Greubel, Table of n, a(n) for n = 0..5000
Felix P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, ResearchGate, 2014.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001008, A001477, A002378, A005563, A028347, A028552, A028557, A028560, A028563, A028566, A028569, A056126, A098603, A098847, A098848, A098849, A098850, A102928, A120071, A132759, A132760, A132761, A132762, A132763, A132764, A132765, A132766.

Table[n (n + 23), {n, 0, 50}] (* Bruno Berselli, Sep 03 2018 *)

a(n)=n*(n+23) \\ Charles R Greathouse IV, Jun 17 2017

[n*(n+23) for n in (0..50)] # G. C. Greubel, Mar 14 2022

a(n) = n*(n + 23).
a(n) = 2*n + a(n-1) + 22 for n>0, a(0)=0. - Vincenzo Librandi, Aug 03 2010
From Chai Wah Wu, Dec 17 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.
G.f.: 2*x*(12 - 11*x)/(1-x)^3. (End)
From Amiram Eldar, Jan 16 2021: (Start)
Sum_{n>=1} 1/a(n) = H(23)/23 = A001008(23)/A102928(23) = 444316699/2736605872, where H(k) is the k-th harmonic number.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2)/23 - 3825136961/123147264240. (End)
E.g.f.: x*(24 + x)*exp(x). - G. C. Greubel, Mar 14 2022

A094728 Triangle read by rows: T(n,k) = n^2 - k^2, 0 <= k < n.

Original entry on oeis.org

1, 4, 3, 9, 8, 5, 16, 15, 12, 7, 25, 24, 21, 16, 9, 36, 35, 32, 27, 20, 11, 49, 48, 45, 40, 33, 24, 13, 64, 63, 60, 55, 48, 39, 28, 15, 81, 80, 77, 72, 65, 56, 45, 32, 17, 100, 99, 96, 91, 84, 75, 64, 51, 36, 19, 121, 120, 117, 112, 105, 96, 85, 72, 57, 40, 21

Offset: 1

Reinhard Zumkeller, May 24 2004

(T(n,k) mod 4) <> 2, see A042965, A016825.
All numbers m occur A034178(m) times.
The row polynomials T(n,x) appear in the calculation of the column g.f.s of triangle A120070 (used to find the frequencies of the spectral lines of the hydrogen atom).

			n=3: T(3,x) = 9+8*x+5*x^2.
Triangle begins:
   1;
   4,  3;
   9,  8,  5;
  16, 15, 12,  7;
  25, 24, 21, 16,  9;
  36, 35, 32, 27, 20, 11;
  49, 48, 45, 40, 33, 24, 13;
  64, 63, 60, 55, 48, 39, 28, 15;
  81, 80, 77, 72, 65, 56, 45, 32, 17;
  ... etc. - _Philippe Deléham_, Mar 07 2013

G. C. Greubel, Table of n, a(n) for n = 1..5050

Cf. A000290, A000567, A004736, A005408, A005563, A008586, A008590.
Cf. A008594, A008598, A016825, A016945, A017329, A028347, A028560.
Cf. A028566, A034178, A042965, A094727, A120070, A128624.
Cf. A000384 (alternating row sums), A002412 (row sums).

[n^2-k^2: k in [0..n-1], n in [1..15]]; // G. C. Greubel, Mar 12 2024

Table[n^2 - k^2, {n,12}, {k,0,n-1}]//Flatten (* Michael De Vlieger, Nov 25 2015 *)

flatten([[n^2-k^2 for k in range(n)] for n in range(1,16)]) # G. C. Greubel, Mar 12 2024

Row polynomials: T(n,x) = n^2*Sum_{m=0..n} x^m - Sum_{m=0..n} m^2*x^m = Sum_{k=0..n-1} T(n,k)*x^k, n >= 1.
T(n, k) = A004736(n,k)*A094727(n,k).
T(n, 0) = A000290(n).
T(n, 1) = A005563(n-1) for n>1.
T(n, 2) = A028347(n) for n>2.
T(n, 3) = A028560(n-3) for n>3.
T(n, 4) = A028566(n-4) for n>4.
T(n, n-1) = A005408(n).
T(n, n-2) = A008586(n-1) for n>1.
T(n, n-3) = A016945(n-2) for n>2.
T(n, n-4) = A008590(n-2) for n>3.
T(n, n-5) = A017329(n-3) for n>4.
T(n, n-6) = A008594(n-3) for n>5.
T(n, n-8) = A008598(n-2) for n>7.
T(A005408(k), k) = A000567(k).
Sum_{k=0..n} T(n, k) = A002412(n) (row sums).
From G. C. Greubel, Mar 12 2024: (Start)
Sum_{k=0..n-1} (-1)^k * T(n, k) = A000384(floor((n+1)/2)).
Sum_{k=0..floor((n-1)/2)} T(n-k, k) = A128624(n).
Sum_{k=0..floor((n-1)/2)} (-1)^k*T(n-k, k) = (1/2)*n*(n+1 - (-1)^n*cos(n*Pi/2)). (End)
G.f.: x*(1 - 3*x^2*y + x*(1 + y))/((1 - x)^3*(1 - x*y)^2). - Stefano Spezia, Aug 04 2025

A132762 a(n) = n*(n + 19).

Original entry on oeis.org

0, 20, 42, 66, 92, 120, 150, 182, 216, 252, 290, 330, 372, 416, 462, 510, 560, 612, 666, 722, 780, 840, 902, 966, 1032, 1100, 1170, 1242, 1316, 1392, 1470, 1550, 1632, 1716, 1802, 1890, 1980, 2072, 2166, 2262, 2360, 2460, 2562, 2666, 2772, 2880, 2990, 3102, 3216

Offset: 0

Omar E. Pol, Aug 28 2007

G. C. Greubel, Table of n, a(n) for n = 0..5000
Felix P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, Preprint on ResearchGate, March 2014.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001008, A002378, A005563, A028347, A028552, A028557, A028560, A028563, A028566, A028569, A051942, A098603, A098847, A098848, A098849, A098850, A102928, A120071, A132759, A132760, A132761, A132763, A132764, A132765, A132766, A132767.

Table[n (n + 19), {n, 0, 50}] (* Bruno Berselli, Sep 05 2018 *)
LinearRecurrence[{3,-3,1},{0,20,42},60] (* Harvey P. Dale, Jun 03 2021 *)

a(n)=n*(n+19) \\ Charles R Greathouse IV, Jun 17 2017

[n*(n+19) for n in (0..50)] # G. C. Greubel, Mar 14 2022

a(n) = 2*n + a(n-1) + 18 for n > 0, a(0) = 0. - Vincenzo Librandi, Aug 03 2010
From Chai Wah Wu, Dec 17 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.
G.f.: 2*x*(10 - 9*x)/(1-x)^3. (End)
a(n) = 2*A051942(n+9). - R. J. Mathar, Sep 05 2018
From Amiram Eldar, Jan 16 2021: (Start)
Sum_{n>=1} 1/a(n) = H(19)/19 = A001008(19)/A102928(19) = 275295799/1474352880, where H(k) is the k-th harmonic number.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2)/19 - 33464927/884611728. (End)
E.g.f.: x*(20 + x)*exp(x). - G. C. Greubel, Mar 14 2022

A132764 a(n) = n*(n+22).

Original entry on oeis.org

0, 23, 48, 75, 104, 135, 168, 203, 240, 279, 320, 363, 408, 455, 504, 555, 608, 663, 720, 779, 840, 903, 968, 1035, 1104, 1175, 1248, 1323, 1400, 1479, 1560, 1643, 1728, 1815, 1904, 1995, 2088, 2183, 2280, 2379, 2480, 2583, 2688, 2795, 2904, 3015, 3128, 3243, 3360

Offset: 0

Omar E. Pol, Aug 28 2007

			a(1)=2*1+0+21=23; a(2)=2*2+23+21=48; a(3)=2*3+48+21=75. - _Vincenzo Librandi_, Aug 03 2010

G. C. Greubel, Table of n, a(n) for n = 0..5000
Felix P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, Preprint on ResearchGate, March 2014.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001008, A002378, A005563, A028347, A028552, A028557, A028560, A028563, A028566, A028569, A098603, A098847, A098848, A098849, A098850, A102928, A120071, A132759, A132760, A132761, A132762, A132763, A132765, A132766, A132767.

Table[n(n+22),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,23,48},50] (* Harvey P. Dale, May 02 2012 *)

a(n)=n*(n+22) \\ Charles R Greathouse IV, Oct 07 2015

[n*(n+22) for n in (0..50)] # G. C. Greubel, Mar 14 2022

a(n) = n*(n + 22).
a(n) = 2*n + a(n-1) + 21 (with a(0)=0). - Vincenzo Librandi, Aug 03 2010
a(0)=0, a(1)=23, a(2)=48, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, May 02 2012
From Amiram Eldar, Jan 16 2021: (Start)
Sum_{n>=1} 1/a(n) = H(22)/22 = A001008(22)/A102928(22) = 19093197/113809696, where H(k) is the k-th harmonic number.
Sum_{n>=1} (-1)^(n+1)/a(n) = 156188887/5121436320. (End)
From G. C. Greubel, Mar 14 2022: (Start)
G.f.: x*(23 - 21*x)/(1-x)^3.
E.g.f.: x*(23 + x)*exp(x). (End)

A132763 a(n) = n*(n+21).

Original entry on oeis.org

0, 22, 46, 72, 100, 130, 162, 196, 232, 270, 310, 352, 396, 442, 490, 540, 592, 646, 702, 760, 820, 882, 946, 1012, 1080, 1150, 1222, 1296, 1372, 1450, 1530, 1612, 1696, 1782, 1870, 1960, 2052, 2146, 2242, 2340, 2440, 2542, 2646, 2752, 2860, 2970, 3082, 3196, 3312

Offset: 0

Omar E. Pol, Aug 28 2007

G. C. Greubel, Table of n, a(n) for n = 0..5000
Felix P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, Preprint on ResearchGate, March 2014.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001008, A002378, A005563, A028347, A028552, A028557, A028560, A028563, A028566, A028569, A098603, A098847, A098848, A098849, A098850, A102928, A120071, A132759, A132760, A132761, A132762, A132764, A132765, A132766, A132767.

Table[n(n+21),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,22,46},50] (* Harvey P. Dale, May 25 2014 *)

a(n)=n*(n+21) \\ Charles R Greathouse IV, Oct 07 2015

[n*(n+21) for n in (0..50)] # G. C. Greubel, Mar 14 2022

a(n) = n*(n + 21).
a(n) = 2*n + a(n-1) + 20 (with a(0)=0). - Vincenzo Librandi, Aug 03 2010
a(0)=0, a(1)=22, a(2)=46, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, May 25 2014
From Amiram Eldar, Jan 16 2021: (Start)
Sum_{n>=1} 1/a(n) = H(21)/21 = A001008(21)/A102928(21) = 18858053/108636528, where H(k) is the k-th harmonic number.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2)/21 - 166770367/4888643760. (End)
From Stefano Spezia, Jan 30 2021: (Start)
O.g.f.: 2*x*(11 - 10*x)/(1 - x)^3.
E.g.f.: x*(22 + x)*exp(x). (End)

A132766 a(n) = n*(n+24).

Original entry on oeis.org

0, 25, 52, 81, 112, 145, 180, 217, 256, 297, 340, 385, 432, 481, 532, 585, 640, 697, 756, 817, 880, 945, 1012, 1081, 1152, 1225, 1300, 1377, 1456, 1537, 1620, 1705, 1792, 1881, 1972, 2065, 2160, 2257, 2356, 2457, 2560, 2665, 2772, 2881, 2992, 3105, 3220, 3337

Offset: 0

Omar E. Pol, Aug 28 2007

G. C. Greubel, Table of n, a(n) for n = 0..5000
Felix P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, Preprint on ResearchGate, March 2014.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001008, A002378, A005563, A028347, A028552, A028557, A028560, A028563, A028566, A028569, A098603, A098849, A098850, A098847, A098848, A102928, A120071, A132759, A132760, A132761, A132762, A132763, A132764, A132765.

Table[n (n + 24), {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 25, 52}, 50] (* Harvey P. Dale, Feb 11 2016 *)

a(n)=n*(n+24) \\ Charles R Greathouse IV, Jun 17 2017

[n*(n+24) for n in (0..50)] # G. C. Greubel, Mar 14 2022

a(n) = n*(n + 24).
a(n) = 2*n + a(n-1) + 23 (with a(0)=0). - Vincenzo Librandi, Aug 03 2010
a(0)=0, a(1)=25, a(2)=52; for n>2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Feb 11 2016
From Amiram Eldar, Jan 16 2021: (Start)
Sum_{n>=1} 1/a(n) = H(24)/24 = A001008(24)/A102928(24) = 1347822955/8566766208, where H(k) is the k-th harmonic number.
Sum_{n>=1} (-1)^(n+1)/a(n) = 3602044091/128501493120. (End)
From G. C. Greubel, Mar 14 2022: (Start)
G.f.: 2*x*(13 - 12*x)/(1-x)^3.
E.g.f.: x*(26 + x)*exp(x). (End)

A132767 a(n) = n*(n + 25).

Original entry on oeis.org

0, 26, 54, 84, 116, 150, 186, 224, 264, 306, 350, 396, 444, 494, 546, 600, 656, 714, 774, 836, 900, 966, 1034, 1104, 1176, 1250, 1326, 1404, 1484, 1566, 1650, 1736, 1824, 1914, 2006, 2100, 2196, 2294, 2394, 2496, 2600, 2706, 2814, 2924, 3036, 3150, 3266, 3384

Offset: 0

Omar E. Pol, Aug 28 2007

a(n) is the Zagreb 1 index of the Mycielskian of the cycle graph C[n]. See p. 205 of the D. B. West reference. - Emeric Deutsch, Nov 04 2016

Douglas B. West, Introduction to Graph Theory, 2nd ed., Prentice-Hall, NJ, 2001.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Felix P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, Preprint on ResearchGate, March 2014.
Wikipedia, Mycielskian.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001008, A002378, A005563, A028347, A028552, A028557, A028560, A028563, A028566, A028569, A098603, A098847, A098848, A098849, A098850, A102928, A120071, A132759, A132760, A132761, A132762, A132763, A132764, A132765, A132766.

Table[n (n + 25), {n, 0, 50}] (* Bruno Berselli, Aug 22 2018 *)
LinearRecurrence[{3,-3,1},{0,26,54},60] (* Harvey P. Dale, Feb 20 2023 *)

a(n)=n*(n+25) \\ Charles R Greathouse IV, Jun 17 2017

[n*(n+25) for n in (0..50)] # G. C. Greubel, Mar 13 2022

a(n) = 2*n + a(n-1) + 24 (with a(0)=0). - Vincenzo Librandi, Aug 03 2010
a(n) = n^2 + 25*n. - Omar E. Pol, Nov 04 2016
From Chai Wah Wu, Dec 17 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.
G.f.: 2*x*(13 - 12*x)/(1-x)^3. (End)
From Amiram Eldar, Jan 16 2021: (Start)
Sum_{n>=1} 1/a(n) = H(25)/25 = A001008(25)/A102928(25) = 34052522467/223092870000, where H(k) is the k-th harmonic number.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2)/25 - 19081066231/669278610000. (End)
E.g.f.: x*(26 + x)*exp(x). - G. C. Greubel, Mar 13 2022

A100345 Triangle read by rows: T(n,k) = n*(n+k), 0 <= k <= n.

Original entry on oeis.org

0, 1, 2, 4, 6, 8, 9, 12, 15, 18, 16, 20, 24, 28, 32, 25, 30, 35, 40, 45, 50, 36, 42, 48, 54, 60, 66, 72, 49, 56, 63, 70, 77, 84, 91, 98, 64, 72, 80, 88, 96, 104, 112, 120, 128, 81, 90, 99, 108, 117, 126, 135, 144, 153, 162, 100, 110, 120, 130, 140, 150, 160, 170, 180, 190

Offset: 0

Reinhard Zumkeller, Nov 18 2004

Distinct members (except 0) are in A071562. Numbers occurring at least twice are in A175040. - Franklin T. Adams-Watters, Apr 04 2010

			Triangle begins:
   0
   1   2
   4   6   8
   9  12  15  18
  16  20  24  28  32
  25  30  35  40  45  50
  36  42  48  54  60  66  72
  49  56  63  70  77  84  91  98
  64  72  80  88  96 104 112 120 128

Cf. A000290, A000384, A001105, A002378, A005563, A014107, A028347, A028552, A028557, A028560, A028563, A028566, A028569, A046092, A054000, A071355, A071562, A085789, A098603, A175040.

Table[n(n+k),{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, Aug 16 2018 *)

row(n) = vector(n+1, k, n*(n+k-1)); \\ Amiram Eldar, May 09 2025

T(n,0) = A000290(n).
T(n,1) = A002378(n) for n > 0.
T(n,2) = A005563(n) for n > 1.
T(n,3) = A028552(n) for n > 2.
T(n,4) = A028347(n+2) for n > 3.
T(n,5) = A028557(n) for n > 4.
T(n,6) = A028560(n) for n > 5.
T(n,7) = A028563(n) for n > 6.
T(n,8) = A028566(n) for n > 7.
T(n,9) = A028569(n) for n > 8.
T(n,10) = A098603(n) for n > 9.
T(n,n-5) = A071355(n-4) for n > 4.
T(n,n-4) = A054000(n-1) for n > 3.
T(n,n-3) = A014107(n) for n > 2.
T(n,n-2) = A046092(n-1) for n > 1.
T(n,n-1) = A000384(n) for n > 0.
T(n,n) = A001105(n).
Row sums give A085789 for n > 0.
G.f.: x*(1 + 2*y + 6*x^3*y^2 - 3*x^2*y*(1 + 2*y) + x*(1 - 3*y + 2*y^2))/((1 - x)^3*(1 - x*y)^3). - Stefano Spezia, Jul 03 2025

cp's OEIS Frontend

A158405 Triangle T(n,m) = 1+2*m of odd numbers read along rows, 0<=m

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A120071 a(n) = n*(n+20).

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A132765 a(n) = n*(n + 23).

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A094728 Triangle read by rows: T(n,k) = n^2 - k^2, 0 <= k < n.

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A132762 a(n) = n*(n + 19).

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A132764 a(n) = n*(n+22).

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A132763 a(n) = n*(n+21).

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