A098832 -id:A098832 - cp's OEIS frontend

A098603 a(n) = n*(n+10).

0, 11, 24, 39, 56, 75, 96, 119, 144, 171, 200, 231, 264, 299, 336, 375, 416, 459, 504, 551, 600, 651, 704, 759, 816, 875, 936, 999, 1064, 1131, 1200, 1271, 1344, 1419, 1496, 1575, 1656, 1739, 1824, 1911, 2000, 2091, 2184, 2279, 2376, 2475, 2576, 2679, 2784

Offset: 0

Eugene McDonnell (eemcd(AT)mac.com), Nov 04 2004

These are the only positive integer values of t for which the Binet-de Moivre formula for the recurrence b(n) = 10*b(n-1)+t*b(n-2) with b(0)=0 and b(1)=1 has a root which is a square. In particular, sqrt(10^2+4*t) is a positive integer since 10^2+4*t = 10^2+4*a(m) = (2*m+10)^2. Thus the characteristic roots are r1=10+m and r2 = -m. - Felix P. Muga II, Mar 28 2014

Shawn A. Broyles, Table of n, a(n) for n = 0..1000
Felix P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, Preprint on ResearchGate, March 2014.
Wikipedia, Hydrogen spectral series.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A098832.

a(n-5), n>=6, fifth column (used for the Pfund series of the hydrogen atom) of triangle A120070.

[n*(n+10): n in [0..50]]; // G. C. Greubel, Jul 31 2022

seq(n*(n+10), n=0..53); # Emeric Deutsch, Mar 11 2005

Table[n(n+10),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,11,24},50] (* Harvey P. Dale, Jul 26 2014 *)

a(n)=n*(n+10) \\ Charles R Greathouse IV, Jun 16 2017

[n*(n+10) for n in (0..50)] # G. C. Greubel, Jul 31 2022

a(n) = (n+5)^2 - 5^2 = n*(n+10), n>=0.
G.f.: x*(11-9*x)/(1-x)^3.
a(n) = a(n-1) + 2*n + 9, (with a(0)=0). - Vincenzo Librandi, Nov 17 2010
Sum_{n>=1} 1/a(n) = 7381/25200 via sum_{n>=0} 1/((n+x)*(n+y)) = (psi(x)-psi(y))/(x-y). - R. J. Mathar, Jul 14 2012
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), with a(0)=0, a(1)=11, a(2)=24. - Harvey P. Dale, Jul 26 2014
Sum_{n>=1} (-1)^(n+1)/a(n) = 1627/25200. - Amiram Eldar, Jan 15 2021
E.g.f.: x*(11 + x)*exp(x). - G. C. Greubel, Jul 31 2022
From Amiram Eldar, Feb 12 2024: (Start)
Product_{n>=1} (1 - 1/a(n)) = -18144*sqrt(2/13)*sin(sqrt(26)*Pi)/(935*Pi).
Product_{n>=1} (1 + 1/a(n)) = 126*sqrt(6)*sin(2*sqrt(6)*Pi)/(23*Pi). (End)

More terms from Emeric Deutsch, Mar 11 2005

A098849 a(n) = n*(n + 16).

Original entry on oeis.org

0, 17, 36, 57, 80, 105, 132, 161, 192, 225, 260, 297, 336, 377, 420, 465, 512, 561, 612, 665, 720, 777, 836, 897, 960, 1025, 1092, 1161, 1232, 1305, 1380, 1457, 1536, 1617, 1700, 1785, 1872, 1961, 2052, 2145, 2240, 2337, 2436, 2537, 2640, 2745, 2852, 2961

Offset: 0

Eugene McDonnell (eemcd(AT)mac.com), Nov 04 2004

G. C. Greubel, Table of n, a(n) for n = 0..1000
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, A098832, A001477, A056126, A102928, A120071, A132760, A132761, A132765.

a(n-8), n>=9, eighth column (used for the n=8 series of the hydrogen atom) of triangle A120070.

seq(n*(n+16),n=0..55); # Emeric Deutsch, Mar 26 2005
a:=n->sum(n, j=17..n): seq(a(n), n=16..63); # Zerinvary Lajos, Feb 17 2008

s=0;lst={};Do[s+=n;AppendTo[lst,s],{n,17,6!,2}];lst (* Vladimir Joseph Stephan Orlovsky, Feb 26 2009 *)
LinearRecurrence[{3, -3, 1}, {0, 17, 36}, 50] (* G. C. Greubel, Jul 29 2016 *)
Table[n(n+16),{n,0,50}] (* Harvey P. Dale, Jul 18 2024 *)

a(n)=n*(n+16) \\ Charles R Greathouse IV, Jul 30 2016

a(n) = (n+8)^2 - 8^2 = n*(n + 16), n>=0.
G.f.: x*(17 - 15*x)/(1-x)^3.
a(n) = a(n-1) + 2*n + 15 (with a(0)=0). - Vincenzo Librandi, Nov 17 2010
From G. C. Greubel, Jul 29 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
E.g.f.: x*(17 + x)*exp(x). (End)
From Amiram Eldar, Jan 15 2021: (Start)
Sum_{n>=1} 1/a(n) = H(16)/16 = A001008(16)/A102928(16) = 2436559/11531520, where H(k) is the k-th harmonic number.
Sum_{n>=1} (-1)^(n+1)/a(n) = 95549/2306304. (End)

More terms from Emeric Deutsch, Mar 26 2005

A098847 a(n) = n*(n + 12).

Original entry on oeis.org

0, 13, 28, 45, 64, 85, 108, 133, 160, 189, 220, 253, 288, 325, 364, 405, 448, 493, 540, 589, 640, 693, 748, 805, 864, 925, 988, 1053, 1120, 1189, 1260, 1333, 1408, 1485, 1564, 1645, 1728, 1813, 1900, 1989, 2080, 2173, 2268, 2365, 2464, 2565, 2668, 2773

Offset: 0

Eugene McDonnell (eemcd(AT)mac.com), Nov 04 2004

G. C. Greubel, Table of n, a(n) for n = 0..1000
Felix P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, Preprint on ResearchGate, March 2014.
Wikipedia, Hydrogen spectral series, Humphreys series.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A098832, A063930.

a(n-6), n>=7, sixth column (used for the n=6 series of the hydrogen atom) of triangle A120070.

Table[ n(n + 12), {n, 0, 50}] (* Robert G. Wilson v, Jul 14 2005 *)
LinearRecurrence[{3,-3,1},{0,13,28},50] (* Harvey P. Dale, May 24 2012 *)

a(n)=n*(n+12) \\ Charles R Greathouse IV, Sep 24 2015

a(n) = (n+6)^2 - 6^2 = n*(n + 12), n>=0.
G.f.: x*(13 - 11*x)/(1-x)^3.
a(n) = 2*n + a(n-1) + 11 (with a(0)=0). - Vincenzo Librandi, Nov 17 2010
a(0)=0, a(1)=13, a(2)=28, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, May 24 2012
Sum_{n>=1} 1/a(n) = 86021/332640 = 0.258600... via Sum_{n>=0} 1/((n+x)(n+y)) = (psi(x)-psi(y))/(x-y). - R. J. Mathar, Jul 14 2012
E.g.f.: x*(13 + x)*exp(x). - G. C. Greubel, Jul 29 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = 18107/332640. - Amiram Eldar, Jan 15 2021

More terms from Robert G. Wilson v, Jul 14 2005

A098848 a(n) = n*(n + 14).

Original entry on oeis.org

0, 15, 32, 51, 72, 95, 120, 147, 176, 207, 240, 275, 312, 351, 392, 435, 480, 527, 576, 627, 680, 735, 792, 851, 912, 975, 1040, 1107, 1176, 1247, 1320, 1395, 1472, 1551, 1632, 1715, 1800, 1887, 1976, 2067, 2160, 2255, 2352, 2451, 2552, 2655, 2760, 2867

Offset: 0

Eugene McDonnell (eemcd(AT)mac.com), Nov 04 2004

G. C. Greubel, Table of n, a(n) for n = 0..1000
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. A098832.

a(n-7), n>=8, seventh column (used for the n=7 series of the hydrogen atom) of triangle A120070.

Table[ n(n + 14), {n, 0, 50}] (* Robert G. Wilson v, Jul 14 2005 *)
LinearRecurrence[{3, -3, 1}, {0, 15, 32}, 50] (* G. C. Greubel, Jul 29 2016 *)

a(n)=n*(n+14) \\ Charles R Greathouse IV, Sep 24 2015

a(n) = (n+7)^2 - 7^2 = n*(n + 14), n>=0.
G.f.: x*(15 - 13*x)/(1-x)^3.
a(n) = 2*n + a(n-1) + 13 (with a(0)=0). - Vincenzo Librandi, Nov 16 2010
Sum_{n>=1} 1/a(n) = 1171733/5045040 = 0.2322544518... via Sum_{n>=0} 1/((n+x)(n+y)) = (psi(x)-psi(y))/(x-y). - R. J. Mathar, Jul 14 2012
From G. C. Greubel, Jul 29 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
E.g.f.: x*(15 + x)*exp(x). (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = 237371/5045040. - Amiram Eldar, Jan 15 2021

More terms from Robert G. Wilson v, Jul 14 2005

A098850 a(n) = n*(n + 18).

Original entry on oeis.org

0, 19, 40, 63, 88, 115, 144, 175, 208, 243, 280, 319, 360, 403, 448, 495, 544, 595, 648, 703, 760, 819, 880, 943, 1008, 1075, 1144, 1215, 1288, 1363, 1440, 1519, 1600, 1683, 1768, 1855, 1944, 2035, 2128, 2223, 2320, 2419, 2520, 2623, 2728, 2835, 2944, 3055

Offset: 0

Eugene McDonnell (eemcd(AT)mac.com), Nov 04 2004

G. C. Greubel, Table of n, a(n) for n = 0..1000
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, A098832, A102928.

a(n-9), n>=10, ninth column (used for the n=9 series of the hydrogen atom) of triangle A120070.

seq(n*(n+18),n=0..52); # Emeric Deutsch, Mar 06 2005

LinearRecurrence[{3, -3, 1}, {0, 19, 40}, 25] (* G. C. Greubel, Jul 29 2016 *)

a(n)=n*(n+18) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = (n+9)^2 - 9^2 = n*(n + 18), n>=0.
G.f.: x*(19 - 17*x)/(1-x)^3.
a(n) = 2*n + a(n-1) + 17 (with a(0)=0). - Vincenzo Librandi, Nov 17 2010
From G. C. Greubel, Jul 29 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
E.g.f.: x*(19 + x)*exp(x). (End)
From Amiram Eldar, Jan 16 2021: (Start)
Sum_{n>=1} 1/a(n) = H(18)/18 = A001008(18)/A102928(18) = 14274301/73513440, where H(k) is the k-th harmonic number.
Sum_{n>=1} (-1)^(n+1)/a(n) = 1632341/44108064. (End)

More terms from Emeric Deutsch, Mar 06 2005

A165351 Numerator of 3*n/2.

Original entry on oeis.org

0, 3, 3, 9, 6, 15, 9, 21, 12, 27, 15, 33, 18, 39, 21, 45, 24, 51, 27, 57, 30, 63, 33, 69, 36, 75, 39, 81, 42, 87, 45, 93, 48, 99, 51, 105, 54, 111, 57, 117, 60, 123, 63, 129, 66, 135, 69, 141, 72, 147, 75, 153, 78, 159, 81, 165, 84, 171, 87, 177, 90, 183, 93, 189, 96, 195

Offset: 0

Paul Curtz, Sep 16 2009

First trisection of A026741. The other trisections are A165355 and A165367.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A000034 (denominator).

Cf. A008585, A016945, A026741, A098832, A165355, A165367.

[Numerator(3*n/2): n in [0..100]]; // Vincenzo Librandi, Mar 03 2014

A165351:=n->numer(3*n/2); seq(A165351(k), k=0..100); # Wesley Ivan Hurt, Oct 11 2013

Table[Numerator[3n/2], {n, 0, 100}] (* Wesley Ivan Hurt, Oct 11 2013 *)
CoefficientList[Series[3*x*(1+x+x^2)/(1-x^2)^2, {x, 0, 70}], x] (* Vincenzo Librandi, Mar 03 2014 *)
LinearRecurrence[{0,2,0,-1},{0,3,3,9},70] (* Harvey P. Dale, Jun 20 2021 *)

[3*n*(3-(-1)^n)/4 for n in (0..100)] # G. C. Greubel, Jul 31 2022

a(n) = A026741(3*n) = 3*A026741(n).
a(2n) = A008585(n).
a(2n+1) = A016945(n).
G.f.: 3*x*(1+x+x^2)/((1-x)^2 * (1+x)^2).
a(n) = numerator(3n/2). - Wesley Ivan Hurt, Oct 11 2013
a(n) = 3*n / (1 + ((n+1) mod 2)). - Wesley Ivan Hurt, Feb 25 2014
From G. C. Greubel, Jul 31 2022: (Start)
a(n) = 3*n*(3 - (-1)^n)/4.
E.g.f.: (3*x/2)*( 2*cosh(x) + sinh(x) ). (End)

Edited and extended by R. J. Mathar, Sep 26 2009

New name from Wesley Ivan Hurt, Oct 13 2013

cp's OEIS Frontend

A098603 a(n) = n*(n+10).

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A098849 a(n) = n*(n + 16).

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A098847 a(n) = n*(n + 12).

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A098848 a(n) = n*(n + 14).

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A098850 a(n) = n*(n + 18).

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A165351 Numerator of 3*n/2.

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