A178218 -id:A178218 - cp's OEIS frontend

A214345 Interleaved reading of A073577 and A053755.

5, 7, 17, 23, 37, 47, 65, 79, 101, 119, 145, 167, 197, 223, 257, 287, 325, 359, 401, 439, 485, 527, 577, 623, 677, 727, 785, 839, 901, 959, 1025, 1087, 1157, 1223, 1297, 1367, 1445, 1519, 1601, 1679, 1765, 1847, 1937, 2023, 2117, 2207, 2305, 2399, 2501

Offset: 0

Yasir Karamelghani Gasmallah, Jul 13 2012

The elements of this sequence satisfy the property that for every n=2k the triple (a(2k-1)^2, a(2k)^2 , a(2k+1)^2) is an arithmetic progression, i.e., 2*a(2k)^2 = a(2k-1)^2 + a(2k+1)^2. In general a triple((x-y)^2,z^2,(x+y)^2) is an arithmetic progression if and only if x^2+y^2=z^2 : in the case of this sequence 7^2, 17^2, and 23^2 is such a triple (i.e. 15-8 =7, 17, 8+15=23, and 8^2+15^2=17^2) .

The first differences of such a sequence is always an interleaved sequence; in this case the interleaved sequence is 2,10,6,14,10,... (A142954).

			For n = 7, a(7)=2*a(6)-2*a(4)+a(3)=2*65-2*37+23=79

Guenther Schrack, Table of n, a(n) for n = 0..10001
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A053755, A073577, A178218.

First differences: A142954; 2-element moving average (a(n-1) + a(n))/2: A002378. - Guenther Schrack, Oct 25 2018

a:=[7,17];; for n in [3..50] do a[n]:=4*(n+1)+a[n-2]; od; Concatenation([5],a); # Muniru A Asiru, Oct 26 2018

I:=[5, 7, 17, 23];[n le 4 select I[n] else 2*Self(n-1)-2*Self(n-3)+Self(n-4): n in [1..75]];

seq(coeff(series((x^3-3*x^2+3*x-5)/((x-1)^3*(x+1)),x,n+1), x, n), n = 0 .. 50); # Muniru A Asiru, Oct 26 2018

LinearRecurrence[{2,0,-2,1},{5,7,17,23},50] (* Harvey P. Dale, Apr 02 2018 *)

A214345(n):=(2*n*(n+4)+3*(-1)^n+7)/2$
makelist(A214345(n),n,0,30); /* Martin Ettl, Nov 01 2012 */

a(2n+1) = A073577(n+1); a(2n) = A053755(n+1).
a(n+1)-a(n) = A142954(n+1).
a(n) = 2*a(n-1)-2*a(n-3)+a(n-4).
G.f.: (x^3-3*x^2+3*x-5)/((x-1)^3*(x+1)).
a(n) = (2*n*(n+4)+3*(-1)^n+7)/2.
2*a(2n)^2 = a(2n-1)^2 + a(2n+1)^2.
a(n) = 4*(n+1) + a(n-2) for n > 1; a(-n) = a(n-4). - Guenther Schrack, Oct 24 2018
E.g.f.: (5 + 5*x + x^2)*cosh(x) + (2 + 5*x + x^2)*sinh(x). - Stefano Spezia, Feb 22 2024

A214393 Numbers of the form (4k+3)^2+4 or (4k+5)^2-8.

Original entry on oeis.org

13, 17, 53, 73, 125, 161, 229, 281, 365, 433, 533, 617, 733, 833, 965, 1081, 1229, 1361, 1525, 1673, 1853, 2017, 2213, 2393, 2605, 2801, 3029, 3241, 3485, 3713, 3973, 4217, 4493, 4753, 5045, 5321, 5629, 5921, 6245, 6553, 6893, 7217, 7573, 7913, 8285, 8641

Offset: 0

Yasir Karamelghani Gasmallah, Jul 15 2012

For every n=2k the triple (a(2k-1)^2, a(2k)^2 , a(2k+1)^2) is an arithmetic progression, i.e., 2*a(2k)^2 = a(2k-1)^2 + a(2k+1)^2.
In general a triple((x-y)^2,z^2,(x+y)^2) is an arithmetic progression if and only if x^2+y^2=z^2, e.g., (17^2, 53^2, 73^2).
The first differences of this sequence is the interleaved sequence 4,36,20,52,36,68,52,....

			a(5) = 2*a(4) - 2*a(2) + a(1) = 2*125 - 2*53 + 17 = 161.

Paolo Xausa, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A178218, A214345.

I:=[13, 17, 53, 73]; [n le 4 select I[n] else 2*Self(n-1)-2*Self(n-3)+Self(n-4): n in [1..75]];

A214393[n_] := 4*n*(n+3) + 6*(-1)^n + 7; Array[A214393, 50, 0] (* or *)
LinearRecurrence[{2, 0, -2, 1}, {13, 17, 53, 73}, 50] (* Paolo Xausa, Feb 22 2024 *)

A214393(n):=4*n*(n+3)+6*(-1)^n+7$
makelist(A214393(n),n,0,30); /* Martin Ettl, Nov 01 2012 */

a(n) = 2*a(n-1)-2*a(n-3)+a(n-4).
G.f.: (13-9*x+19*x^2-7*x^3)/((1+x)*(1-x)^3).
a(n) = 4*n*(n+3)+6*(-1)^n+7.
2*a(2n)^2 = a(2n-1)^2 + a(2n+1)^2.

A214405 Numbers of the form (4k+3)^2-8 or (4k+5)^2+4.

Original entry on oeis.org

1, 29, 41, 85, 113, 173, 217, 293, 353, 445, 521, 629, 721, 845, 953, 1093, 1217, 1373, 1513, 1685, 1841, 2029, 2201, 2405, 2593, 2813, 3017, 3253, 3473, 3725, 3961, 4229, 4481, 4765, 5033, 5333, 5617, 5933, 6233, 6565, 6881, 7229, 7561, 7925, 8273, 8653

Offset: 1

Yasir Karamelghani Gasmallah, Jul 15 2012

For every odd n the triple (a(n-1)^2, a(n)^2 , a(n+1)^2) is an arithmetic progression, i.e., 2*a(n)^2 = a(n-1)^2 + a(n+1)^2.
In general a triple((x-y)^2,z^2,(x+y)^2) is an arithmetic progression if and only if x^2+y^2=z^2.
The first differences of this sequence is the interleaved sequence 28,12,44,28,60,44....

			a(4) = 2*a(3) - 2*a(1) + a(0) = 2*85 - 2*29 + 1 = 113.

Cf. A178218, A214345.

I:=[1, 29, 41, 85]; [n le 4 select I[n] else 2*Self(n-1)-2*Self(n-3)+Self(n-4): n in [1..75]];

A214405(n):=4*n*(n+3)-6*(-1)^n+7$
makelist(A214405(n),n,0,30); /* Martin Ettl, Nov 01 2012 */

a(n) = 2*a(n-1)-2*a(n-3)+a(n-4).
O.G.f.: (1+27*x-17*x^2+5*x^3)/((1+x)*(1-x)^3).
a(n) = 4*n*(n+3)-6*(-1)^n+7.
2*a(2n+1)^2 = a(2n)^2 + a(2n+2)^2.

A216876 20k^2-20k-5 interleaved with 20k^2+5 for k=>0.

Original entry on oeis.org

-5, 5, -5, 25, 35, 85, 115, 185, 235, 325, 395, 505, 595, 725, 835, 985, 1115, 1285, 1435, 1625, 1795, 2005, 2195, 2425, 2635, 2885, 3115, 3385, 3635, 3925, 4195, 4505, 4795, 5125, 5435, 5785, 6115, 6485, 6835, 7225, 7595, 8005, 8395, 8825, 9235, 9685

Offset: 0

Eddie Gutierrez, Sep 18 2012

The sequence (the second in the family) is present as a family of single interleaved sequence of which are separated or factored out of the larger sequence to give individual sequences. The larger sequence produces two smaller interleaved sequences where one of them has the formula above and a first interleaved sequence. There are a total of two sequences in this family.

Eddie Gutierrez New Interleaved Sequences Part E on oddwheel.com, Section B1 Line No. 25 (square_sequencesV.html), Part E.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A178218, A214345, A214493, A214393, A214405, A216871.

&cat[[20*k^2-20*k-5, 20*k^2+5]: k in [0..22]]; // Bruno Berselli, Sep 27 2012

Flatten[Table[{20*n^2 - 20*n - 5, 20*n^2 + 5}, {n, 0, 30}]] (* T. D. Noe, Sep 26 2012 *)

A216876(n):=(5/2)*(2*n*(n-2)-3*(-1)^n+1)$
makelist(A216876(n),n,0,30); /* Martin Ettl, Nov 01 2012 */

vector(60,n,k=(n-1)\2;if(n%2,20*k^2-20*k-5,20*k^2+5)) \\ Charles R Greathouse IV, Sep 27 2012

Contribution from Bruno Berselli, Sep 27 2012: (Start)
G.f.: -5*(1-3*x+3*x^2-5*x^3)/((1+x)*(1-x)^3).
a(n) = (5/2)*(2*n*(n-2)-3*(-1)^n+1).
a(n) = 5*A214345(n-3) with A214345(-3)=-1, A214345(-2)=1, A214345(-1)=-1. (End)

More terms from T. D. Noe, Sep 26 2012

Definition rewritten by Bruno Berselli, Oct 25 2012

A273182 a(n) is the second number in a triple consisting of 3 numbers, which when squared are part of a right diagonal of a magic square of squares.

Original entry on oeis.org

14, 84, 490, 2856, 16646, 97020, 565474, 3295824, 19209470, 111960996, 652556506, 3803378040, 22167711734, 129202892364, 753049642450, 4389094962336, 25581520131566, 149100025827060, 869018634830794, 5065011783157704, 29521052064115430, 172061300601534876

Offset: 0

Eddie Gutierrez, May 17 2016

The multiplying factor 6 appears to come from the ratio of a(1)/a(0) of the sequence. Each of the lines of tables (V vs VII) or (VI vs VIII) in oddwheel.com/ImaginaryB.html generates this factor.

			a(2) = 84*6 -14 = 490; a(3) = 490*6 - 84 = 2856; a(4) = 2856*6 - 490 = 16646.

Colin Barker, Table of n, a(n) for n = 0..1000
Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
E. Gutierrez, Recursion Methods to Generate New Integer Sequences (Part VIF)
E. Gutierrez, Table of Tuples and Use of Magic Ratio for Tuple Conversion (Part IB)
E. Gutierrez, Table of Tuples for Square of Squares (Part IC)
Index entries for linear recurrences with constant coefficients, signature (6,-1).

Cf. A178218, A273187, A273189.

CoefficientList[Series[14/(1 - 6 x + x^2), {x, 0, 21}], x] (* Michael De Vlieger, May 18 2016 *)

Vec(14/(1-6*x+x^2) + O(x^50)) \\ Colin Barker, May 18 2016

a(0)=14, a(1)= 84, a(n+1)= a(n)*6 - a(n-1).
G.f.: 14 / (1-6*x+x^2). - Colin Barker, May 18 2016
E.g.f.: 7*(3*sqrt(2)*sinh(2*sqrt(2)*x) + 4*cosh(2*sqrt(2)*x))*exp(3*x)/2. - Ilya Gutkovskiy, May 18 2016

A214493 Numbers of the form ((6k+5)^2+9)/2 or 2(3k+4)^2-9.

Original entry on oeis.org

17, 23, 65, 89, 149, 191, 269, 329, 425, 503, 617, 713, 845, 959, 1109, 1241, 1409, 1559, 1745, 1913, 2117, 2303, 2525, 2729, 2969, 3191, 3449, 3689, 3965, 4223, 4517, 4793, 5105, 5399, 5729, 6041, 6389, 6719, 7085, 7433, 7817, 8183, 8585, 8969, 9389, 9791, 10229, 10649, 11105, 11543, 12017, 12473, 12965, 13439, 13949

Offset: 0

Yasir Karamelghani Gasmallah, Jul 19 2012

For every n=2k the triple (a(2k-1)^2, a(2k)^2 , a(2k+1)^2) is an arithmetic progression, i.e., 2*a(2k)^2 = a(2k-1)^2 + a(2k+1)^2.
In general a triple((x-y)^2,z^2,(x+y)^2) is an arithmetic progression if and only if x^2+y^2=z^2.
The first differences of this sequence is the interleaved sequence 6,42,24,60,42,78.... = 9*n*(39-27*(-1)^n)/2.

			For n = 7, a(7)=2*a(6)-2*a(4)+a(3)=2*269-2*149+89=329.

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A178218, A214345.

I:=[17, 23, 65, 89]; [n le 4 select I[n] else 2*Self(n-1)-2*Self(n-3)+Self(n-4): n in [1..75]];

LinearRecurrence[{2,0,-2,1},{17,23,65,89},60] (* Harvey P. Dale, Aug 07 2015 *)

a(n) = 2*a(n-1)-2*a(n-3)+a(n-4).
G.f.: (17-11*x+19*x^2-7*x^3)/((1+x)*(1-x)^3).
a(n) = (6*n*(3*n+10)+27*(-1)^n+41)/4.
2*a(2n)^2 = a(2n-1)^2 + a(2n+1)^2.

A215098 a(0)=0, a(1)=1, a(n) = n*(n-1) - a(n-2).

Original entry on oeis.org

0, 1, 2, 5, 10, 15, 20, 27, 36, 45, 54, 65, 78, 91, 104, 119, 136, 153, 170, 189, 210, 231, 252, 275, 300, 325, 350, 377, 406, 435, 464, 495, 528, 561, 594, 629, 666, 703, 740, 779, 820, 861, 902, 945, 990, 1035, 1080, 1127, 1176, 1225, 1274, 1325, 1378, 1431

Offset: 0

Alex Ratushnyak, Aug 03 2012

Same seed, b(n) = n*(n+1) - b(n-2) : 0, 1, 6, 11, 14, 19, 28, 37, 44, 53, 66, 79, 90, 103, 120, 137, 152, 169, 190, 211, 230, 251, 276, 301, 324, 349, 378, 407, 434, 463, 496, 529, 560, 593, ...

b(n) = a(n+1) - 1 if (n mod 4) < 2, otherwise b(n) = a(n+1) + 1.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-4,4,-3,1).

Cf. A007590 (a(0)=0, a(n) = n*(n-1) - a(n-1)).

Cf. A178218 (a(1)=1, a(n) = n*(n+1) - a(n-1)).

[n le 2 select n-1 else  2*Binomial(n-1,2) -Self(n-2): n in [1..81]]; // G. C. Greubel, Nov 25 2022

CoefficientList[Series[(x -x^2 +3x^3 -x^4)/(1 -3x +4x^2 -4x^3 +3x^4 -x^5), {x, 0, 70}], x] (* Vincenzo Librandi, Jul 18 2013 *)
RecurrenceTable[{a[0]==0,a[1]==1,a[n]==n(n-1)-a[n-2]},a,{n,60}] (* or *) LinearRecurrence[{3,-4,4,-3,1},{0,1,2,5,10},60] (* Harvey P. Dale, May 15 2016 *)

prpr = 0
prev = 1
for n in range(2,77):
    print(prpr, end=', ')
    curr = n*(n-1) - prpr
    prpr = prev
    prev = curr

def A215098(n):
    if (n<2): return n
    else: return 2*binomial(n,2) - A215098(n-2)
[A215098(n) for n in range(81)] # G. C. Greubel, Nov 25 2022

G.f.: x*(1-x+3*x^2-x^3)/(1-3*x+4*x^2-4*x^3+3*x^4-x^5). - David Scambler, Aug 06 2012

a(n) = (n^2 +n -1 +cos(pi*n/2) +sin(pi*n/2))/2. - Vaclav Kotesovec, Aug 11 2012

A273189 a(n) is the third number in a triple consisting of 3 numbers, which when squared are part of a right diagonal of a magic square of squares.

Original entry on oeis.org

51, 401, 2451, 14401, 84051, 490001, 2856051, 16646401, 97022451, 565488401, 3295908051, 19209960001, 111963852051, 652573152401, 3803475062451, 22168277222401, 129206188272051, 753068852410001, 4389206926188051, 25582172704718401, 149103829302122451

Offset: 0

Eddie Gutierrez, May 17 2016

The multiplying factor 6 (in the recursion formulas below) appears to come from the ratio of b(1)/b(0) of the sequence. Each of the lines of tables (V vs VII) or (VI vs VIII) in oddwheel.com/ImaginaryB.html generates this factor.
k is obtained from the difference of the offsets of two relate sequences. this one, (II), starting at 51 and a second, (I), at 99 (to be submitted separately). Thus, k =[Ic(n)- IIc(n)]*2. When n=0, Ic(0)=99 and IIc(0)=51 giving the value for k of (99-51)*2=96. Furthermore, k is the same constant number for any value of n.
The differences between number in the sequence are identical in both of the related sequences.

			a(2)= 401*6 - (51 - 96)= 2451;
a(3)= 2451*6 - (401 - 96)= 14401;
a(4)= 14401*6 - (2451 - 96)= 84051.

Colin Barker, Table of n, a(n) for n = 0..1000
E. Gutierrez, Recursion Methods to Generate New Integer Sequences (Part VIF)
E. Gutierrez, Table of Tuples and Use of Magic Ratio for Tuple Conversion (Part IB)
E. Gutierrez, Table of Tuples for Square of Squares (Part IC)
Index entries for linear recurrences with constant coefficients, signature (7,-7,1).

Cf. A178218, A273182, A273187.

CoefficientList[Series[(51 + 44 x + x^2)/((1 - x) (1 - 6 x + x^2)), {x, 0, 20}], x] (* Michael De Vlieger, May 18 2016 *)
LinearRecurrence[{7,-7,1},{51,401,2451},30] (* Harvey P. Dale, Feb 21 2020 *)

Vec((51+44*x+x^2)/((1-x)*(1-6*x+x^2)) + O(x^50)) \\ Colin Barker, May 18 2016

a(0)= 51, a(1)= 401, a(n+1)= a(n)*6 - a(n-1) + k where k=96.
From Colin Barker, May 18 2016: (Start)
a(n) = (-24+25/2*(3-2*sqrt(2))^(1+n)+25/2*(3+2*sqrt(2))^(1+n)).
a(n) = 7*a(n-1)-7*a(n-2)+a(n-3) for n>2.
G.f.: (51+44*x+x^2) / ((1-x)*(1-6*x+x^2)).
(End)

More terms from Colin Barker, May 18 2016

A014112 a(n) = n(n-1) + (n-2)(n-3) + ... + 10 + 1 for n odd; otherwise, a(n) = n(n-1) + (n-2)(n-3) + ... + 21.

Original entry on oeis.org

1, 2, 7, 14, 27, 44, 69, 100, 141, 190, 251, 322, 407, 504, 617, 744, 889, 1050, 1231, 1430, 1651, 1892, 2157, 2444, 2757, 3094, 3459, 3850, 4271, 4720, 5201, 5712, 6257, 6834, 7447, 8094, 8779, 9500, 10261, 11060, 11901, 12782, 13707, 14674, 15687

Offset: 1

Jon Wild, Jul 14 1997

			From _Bruno Berselli_, Mar 12 2018: (Start)
n=1: 1;
n=2: 1*2;
n=3: 1 + 0*1 + 2*3 = 7;
n=4: 1*2 + 3*4 = 14;
n=5: 1 + 0*1 + 2*3 + 4*5 = 27;
n=6: 1*2 + 3*4 + 5*6 = 44;
n=7: 1 + 0*1 + 2*3 + 4*5 + 6*7 = 69, etc.
(End)

Delbert L. Johnson, Table of n, a(n) for n = 1..20000
Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Cf. A064999, A178218 (first differences).

[n le 2 select n else Self(n-2)+n*(n-1):n in [1..50]]; // Vincenzo Librandi, Feb 28 2016
(C#) public BigInteger a(BigInteger n) => (n * (n + 2) * (2 * n - 1) + 9) / 12; // Delbert L. Johnson, Mar 19 2023

LinearRecurrence[{3, -2, -2, 3, -1}, {1, 2, 7, 14, 27}, 50] (* Vincenzo Librandi, Feb 28 2016 *)

a(n) = a(n-2) + n*(n-1) for n > 2, a(1)=1, a(2)=2.
G.f.: x*(1 - x^3 + 3*x^2 - x)/((x + 1)*(x - 1)^4). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 10 2009
a(n) + a(n+1) = A064999(n). - R. J. Mathar, Feb 27 2016
a(n) = n*(n + 2)*(2*n - 1)/12 + 3*(1 - (-1)^n)/8. - Bruno Berselli, Mar 12 2018

More terms from Erich Friedman

A216844 4k^2-8k+2 interleaved with 4k^2-4k+2 for k>=0.

Original entry on oeis.org

2, 2, -2, 2, 2, 10, 14, 26, 34, 50, 62, 82, 98, 122, 142, 170, 194, 226, 254, 290, 322, 362, 398, 442, 482, 530, 574, 626, 674, 730, 782, 842, 898, 962, 1022, 1090, 1154, 1226, 1294, 1370, 1442, 1522, 1598, 1682, 1762, 1850, 1934, 2026, 2114, 2210, 2302, 2402

Offset: 0

Eddie Gutierrez, Sep 17 2012

The sequence is present as a family of single interleaved sequence of which there are many which are separated or factored out to give individual sequences. The larger sequence produces two smaller interleaved sequences where one of them has the formulas above and the other interleaved sequence has the formulas (4n^2 + 4n -1) and (4n^2+1). The latter interleaved sequence is A214345.

Eddie Gutierrez New Interleaved Sequences Part A on oddwheel.com, Section B1 Line No. 21 (square_sequencesI.html) Part A
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A178218, A214345, A214393, A214405, A216576.

&cat[[4*k^2-8*k+2, 4*k^2-4*k+2]: k in [0..25]]; // Bruno Berselli, Sep 30 2012

Flatten[Table[{4 n^2 - 8 n + 2, 4 n^2 - 4 n + 2}, {n, 0, 25}]] (* Bruno Berselli, Sep 30 2012 *)
LinearRecurrence[{2,0,-2,1},{2,2,-2,2},60] (* Harvey P. Dale, Jul 18 2020 *)

G.f.: 2*(1-x-3*x^2+5*x^3)/((1+x)*(1-x)^3). [Bruno Berselli, Sep 30 2012]
a(n) = (1/2)*(2*n*(n-4)-3*(-1)^n+7). [Bruno Berselli, Sep 30 2012]
a(n) = 2*A178218(n-3) with A178218(-3)=1, A178218(-2)=1, A178218(-1)=-1, A178218(0)=1. [Bruno Berselli, Oct 01 2012]

Definition rewritten by Bruno Berselli, Oct 25 2012

cp's OEIS Frontend

A214345 Interleaved reading of A073577 and A053755.

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A214393 Numbers of the form (4k+3)^2+4 or (4k+5)^2-8.

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A214405 Numbers of the form (4k+3)^2-8 or (4k+5)^2+4.

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A216876 20k^2-20k-5 interleaved with 20k^2+5 for k=>0.

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A273182 a(n) is the second number in a triple consisting of 3 numbers, which when squared are part of a right diagonal of a magic square of squares.

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A214493 Numbers of the form ((6k+5)^2+9)/2 or 2(3k+4)^2-9.

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A215098 a(0)=0, a(1)=1, a(n) = n*(n-1) - a(n-2).

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A014112 a(n) = n(n-1) + (n-2)(n-3) + ... + 10 + 1 for n odd; otherwise, a(n) = n(n-1) + (n-2)(n-3) + ... + 21.