A028557 -id:A028557 - cp's OEIS frontend

A062190 Coefficient triangle of certain polynomials N(5; m,x).

1, 1, 6, 1, 14, 21, 1, 24, 84, 56, 1, 36, 216, 336, 126, 1, 50, 450, 1200, 1050, 252, 1, 66, 825, 3300, 4950, 2772, 462, 1, 84, 1386, 7700, 17325, 16632, 6468, 792, 1, 104, 2184, 16016, 50050, 72072, 48048, 13728, 1287, 1, 126, 3276, 30576, 126126, 252252, 252252, 123552, 27027, 2002

Offset: 0

Wolfdieter Lang, Jun 19 2001

The e.g.f. of the m-th (unsigned) column sequence without leading zeros of the generalized (a=5) Laguerre triangle L(5; n+m,m)= A062138(n+m,m), n >= 0, is N(5; m,x)/(1-x)^(2*(m+3)), with the row polynomials N(5; m,x) := Sum_{k=0..m} a(m,k)*x^k.

			Triangle begins as:
  1;
  1,   6;
  1,  14,   21;
  1,  24,   84,    56;
  1,  36,  216,   336,    126;
  1,  50,  450,  1200,   1050,    252;
  1,  66,  825,  3300,   4950,   2772,     462;
  1,  84, 1386,  7700,  17325,  16632,    6468,    792;
  1, 104, 2184, 16016,  50050,  72072,   48048,  13728,   1287;
  1, 126, 3276, 30576, 126126, 252252,  252252, 123552,  27027,  2002;
  1, 150, 4725, 54600, 286650, 756756, 1051050, 772200, 289575, 50050, 3003;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Family of polynomials (see A062145): A008459 (c=1), A132813 (c=2), A062196 (c=3), A062145 (c=4), A062264 (c=5), this sequence (c=6).
Columns k: A028557 (k=1), A104676 (k=2), A104677 (k=3), A104678 (k=4), A104679 (k=5), A104680 (k=6).
Diagonals: A000389 (k=n), A027818 (k=n-1), A104670 (k=n-2), A104671 (k=n-3), A104672 (k=n-4), A104673 (k=n-5), A104674 (k=n-6).
Cf. A003516 (row sums), A113894 (main diagonal).

A062190:= func< n,k | Binomial(n,k)*Binomial(n+5,k) >;
[A062190(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 28 2025

A062190 := proc(m,k)
    add( (binomial(m, j)*(2*m+5-j)!/((m+5)!*(m-j)!))*(x^(m-j))*(1-x)^j,j=0..m) ;
    coeftayl(%,x=0,k) ;
end proc: # R. J. Mathar, Nov 29 2015

NN[5, m_, x_] := x^m*(2*m+5)!*Hypergeometric2F1[-m, -m, -2*m-5, (x-1)/x]/((m+5)!*m!); Table[CoefficientList[NN[5, m, x], x], {m, 0, 8}] // Flatten (* Jean-François Alcover, Sep 18 2013 *)
A062190[n_,k_]:= Binomial[n,k]*Binomial[n+5,k];
Table[A062190[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 28 2025 *)

def A062190(n,k): return binomial(n,k)*binomial(n+5,k)
print(flatten([[A062190(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Feb 28 2025

T(m, k) = [x^k]N(5; m, x), with N(5; m, x) = ((1-x)^(2*(m+3)))*(d^m/dx^m)(x^m/(m!*(1-x)^(m+6))).
N(5; m, x) = Sum_{j=0..m} ((binomial(m, j)*(2*m+5-j)!/((m+5)!*(m-j)!))*(x^(m-j))*(1-x)^j).
N(5; m, x)= x^m*(2*m+5)! * 2F1(-m, -m; -2*m-5; (x-1)/x)/((m+5)!*m!). - Jean-François Alcover, Sep 18 2013
T(n, k) = binomial(n, k)*binomial(n+5, k). - G. C. Greubel, Feb 28 2025

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

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

A080096 a(1) = a(2) = 1, a(3) = 2, thereafter a(n) = abs(a(n-1) - a(n-2) - a(n-3)).

Original entry on oeis.org

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

Offset: 1

Benoit Cloitre, Jan 28 2003

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A028347, A028557, A077623, A077653, A079623, A079624, A088226.

a080096 n = a080096_list !! (n-1)
a080096_list = 1 : 1 : 2 : zipWith3 (\u v w -> abs (w - v - u))
               a080096_list (tail a080096_list) (drop 2 a080096_list)
-- Reinhard Zumkeller, Oct 11 2014

m:=120;
A080096:=[n le 3 select Floor((n+1)/2) else Abs(Self(n-1) - Self(n-2) - Self(n-3)): n in [1..m+5]];
[A080096[n]: n in [1..m]]; // G. C. Greubel, Sep 11 2024

nxt[{a_,b_,c_}]:={b,c,Abs[c-b-a]}; NestList[nxt,{1,1,2},110][[All,1]] (* Harvey P. Dale, Nov 14 2021 *)

a(n)=local(k,m); if(n<1,0,k=sqrtint(n+4); m=n+4-k^2; if(m%2,m\2+1,k-m\2))

@CachedFunction
def a(n): # a = A080096
    if n<4: return int((n+1)//2)
    else: return abs(a(n-1)-a(n-2)-a(n-3))
[a(n) for n in range(1,101)] # G. C. Greubel, Sep 11 2024

For n>=3 Max( a(k) : 1<=k<=n ) = floor ( sqrt(n+4)).
Special cases: a(n^2 + 4*n - 1) = 0 and a(n^2 - 4) = n.
a(A028557(n)) = a(A028557(n+1)).
Sum_{k=(n-1)^2 .. n^2} a(k) = n^2.

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