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Search: a132765 -id:a132765
Displaying 1-10 of 16 results found.
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A056126
a(n) = n*(n + 17)/2.
+10
21
0, 9, 19, 30, 42, 55, 69, 84, 100, 117, 135, 154, 174, 195, 217, 240, 264, 289, 315, 342, 370, 399, 429, 460, 492, 525, 559, 594, 630, 667, 705, 744, 784, 825, 867, 910, 954, 999, 1045, 1092, 1140, 1189, 1239, 1290, 1342, 1395, 1449, 1504, 1560, 1617, 1675
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
G.f.: x*(9-8*x)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = A126890(n,8) for n>7. - Reinhard Zumkeller, Dec 30 2006
If we define f(n,i,a) = Sum_{k=0..n-i} binomial(n,k)*stirling1(n-k,i)* Product_{j=0..k-1} (-a-j), then a(n) = -f(n,n-1,9), for n>=1. - Milan Janjic, Dec 20 2008
a(n) = a(n-1) + n + 8 (with a(0)=0). - Vincenzo Librandi, Aug 07 2010
a(n) = 9*n - floor(n/2) + floor(n^2/2). - Wesley Ivan Hurt, Jun 15 2013
E.g.f.: x*(18 + x)*exp(x)/2. - G. C. Greubel, Jan 19 2020
From Amiram Eldar, Jan 10 2021: (Start)
Sum_{n>=1} 1/a(n) = 2*A001008(17)/(17*A002805(17)) = 42142223/104144040.
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/17 - 1768477/20828808. (End)
MAPLE
seq( n*(n+17)/2, n=0..50); # G. C. Greubel, Jan 19 2020
MATHEMATICA
Table[n(n+17)/2, {n, 0, 50}] (* Harvey P. Dale, Apr 25 2011 *)
PROG
(PARI) a(n)=n*(n+17)/2 \\ Charles R Greathouse IV, Sep 24 2015
(Magma) [n*(n+17)/2: n in [0..50]]; // G. C. Greubel, Jan 19 2020
(Sage) [n*(n+17)/2 for n in (0..50)] # G. C. Greubel, Jan 19 2020
(GAP) List([0..50], n-> n*(n+17)/2 ); # G. C. Greubel, Jan 19 2020
CROSSREFS
Cf. A000096, A001477, A051942, A056000, A056121.
Cf. A001008, A002805, A098849, A120071, A132760, A132761, A132765.
KEYWORD
easy,nonn
AUTHOR
Barry E. Williams, Jul 07 2000
STATUS
approved
A098849
a(n) = n*(n + 16).
+10
21
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
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).
FORMULA
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)
MAPLE
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
MATHEMATICA
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 *)
PROG
(PARI) a(n)=n*(n+16) \\ Charles R Greathouse IV, Jul 30 2016
CROSSREFS
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.
KEYWORD
nonn,easy
AUTHOR
Eugene McDonnell (eemcd(AT)mac.com), Nov 04 2004
EXTENSIONS
More terms from Emeric Deutsch, Mar 26 2005
STATUS
approved
A132761
a(n) = n*(n+17).
+10
19
0, 18, 38, 60, 84, 110, 138, 168, 200, 234, 270, 308, 348, 390, 434, 480, 528, 578, 630, 684, 740, 798, 858, 920, 984, 1050, 1118, 1188, 1260, 1334, 1410, 1488, 1568, 1650, 1734, 1820, 1908, 1998, 2090, 2184, 2280, 2378, 2478, 2580, 2684, 2790, 2898, 3008, 3120
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
a(n) is the first Zagreb index of the helm graph H[n] (n>=3). - Emeric Deutsch, Nov 05 2016
From Emeric Deutsch, Nov 07 2016: (Start)
a(n) is the first Zagreb index of the graph obtained by joining one vertex of the cycle graph C[n] with each vertex of a second cycle graph C[n].
The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices. Alternately, it is the sum of the degree sums d(i)+d(j) over all edges ij of the graph. (End)
From Emeric Deutsch, May 11 2018: (Start)
The M-polynomial of the Helm graph H[n] is M(H[n];x,y) = n*x*y^4 + n*x^4*y^4 + n*x^4*y^n.
The helm graph H[n] is the graph obtained from an n-wheel graph by adjoining a pendant edge at each node of the cycle. (End)
LINKS
Muniru A Asiru, Table of n, a(n) for n = 0..5000
Emeric Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, Vol. 6, No. 2 (2015), pp. 93-102.
Ivan Gutman and Kinkar Ch. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50 (2004), pp. 83-92.
Felix P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, Preprint on ResearchGate, March 2014.
Eric Weisstein's World of Mathematics, Helm Graph.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
a(n) = n*(n + 17).
a(n) = A132760(n) + 2*n = A132765(n) - 6*n = A098849(n) + 1*n = A120071(n) - 3*n. - Zerinvary Lajos, Feb 17 2008
a(n) = 2*n + a(n-1) + 16 for n>0, a(0)=0. - Vincenzo Librandi, Aug 03 2010
G.f.: 2*x*(9 - 8*x)/(1 - x)^3. - Emeric Deutsch, Nov 07 2016
From Amiram Eldar, Jan 16 2021: (Start)
Sum_{n>=1} 1/a(n) = H(17)/17 = A001008(17)/A102928(17) = 42142223/208288080, where H(k) is the k-th harmonic number.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2)/17 - 1768477/41657616. (End)
MATHEMATICA
Table[n(n+17), {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 18, 38}, 50] (* Harvey P. Dale, Sep 12 2020 *)
PROG
(PARI) a(n)=n*(n+17) \\ Charles R Greathouse IV, Nov 07 2016
(GAP) List([0..50], n->n*(n+17)); # Muniru A Asiru, May 11 2018
CROSSREFS
Cf. A001008, A002378, A098849, A102928, A120071, A132760, A132765.
KEYWORD
nonn,easy
AUTHOR
Omar E. Pol, Aug 28 2007
STATUS
approved
A120071
a(n) = n*(n+20).
+10
18
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Table of n, a(n) for n=0..48.
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).
FORMULA
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)
MAPLE
seq(sum(4*k-n, k=7..n), n=6..51); # Zerinvary Lajos, Feb 17 2008
MATHEMATICA
s=0; lst={}; Do[s+=n; AppendTo[lst, s], {n, 21, 6!, 2}]; lst (* Vladimir Joseph Stephan Orlovsky, Feb 26 2009 *)
PROG
(PARI) a(n)=n*(n+20) \\ Charles R Greathouse IV, Jan 21 2015
CROSSREFS
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, A098849, A102928, A120061, A132765, A132760, A132761, A132762, A005563, A028552, A028347, A028557, A028560, A028563, A028566, A028569, A098603, A098847, A132759, A098848.
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Jul 20 2006
STATUS
approved
A132760
a(n) = n*(n+15).
+10
17
0, 16, 34, 54, 76, 100, 126, 154, 184, 216, 250, 286, 324, 364, 406, 450, 496, 544, 594, 646, 700, 756, 814, 874, 936, 1000, 1066, 1134, 1204, 1276, 1350, 1426, 1504, 1584, 1666, 1750, 1836, 1924, 2014, 2106, 2200, 2296, 2394, 2494
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Table of n, a(n) for n=0..43.
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).
FORMULA
a(n) = n*(n + 15).
a(n) = 2*A056121(n). - Reinhard Zumkeller, Mar 20 2009
a(n) = 2*n + a(n-1) + 14 (with a(0)=0). - Vincenzo Librandi, Aug 03 2010
G.f.: 2*x*(-8+7*x) / (x-1)^3 . - R. J. Mathar, Jul 14 2012
Sum_{n>=1} 1/a(n) = 1195757/5405400 = 0.22121526621... - R. J. Mathar, Jul 14 2012
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2)/15 - 52279/1081080. - Amiram Eldar, Jan 15 2021
MATHEMATICA
s=0; lst={}; Do[s+=n; AppendTo[lst, s], {n, 16, 6!, 2}]; lst (* Vladimir Joseph Stephan Orlovsky, Feb 26 2009 *)
Table[n(n+15), {n, 0, 60}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 16, 34}, 60] (* Harvey P. Dale, Jan 20 2019 *)
PROG
(PARI) a(n)=n*(n+15) \\ Charles R Greathouse IV, Sep 24 2015
CROSSREFS
Cf. A002378, A001477, A056121, A056126, A098849, A120071, A132761, A132765.
KEYWORD
easy,nonn
AUTHOR
Omar E. Pol, Aug 28 2007
STATUS
approved
A132762
a(n) = n*(n + 19).
+10
13
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
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).
FORMULA
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
MATHEMATICA
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 *)
PROG
(PARI) a(n)=n*(n+19) \\ Charles R Greathouse IV, Jun 17 2017
(Sage) [n*(n+19) for n in (0..50)] # G. C. Greubel, Mar 14 2022
CROSSREFS
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.
KEYWORD
nonn,easy
AUTHOR
Omar E. Pol, Aug 28 2007
STATUS
approved
A132764
a(n) = n*(n+22).
+10
13
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
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).
FORMULA
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)
EXAMPLE
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
MATHEMATICA
Table[n(n+22), {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 23, 48}, 50] (* Harvey P. Dale, May 02 2012 *)
PROG
(PARI) a(n)=n*(n+22) \\ Charles R Greathouse IV, Oct 07 2015
(Sage) [n*(n+22) for n in (0..50)] # G. C. Greubel, Mar 14 2022
CROSSREFS
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.
KEYWORD
easy,nonn
AUTHOR
Omar E. Pol, Aug 28 2007
STATUS
approved
A132763
a(n) = n*(n+21).
+10
12
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
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).
FORMULA
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)
MATHEMATICA
Table[n(n+21), {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 22, 46}, 50] (* Harvey P. Dale, May 25 2014 *)
PROG
(PARI) a(n)=n*(n+21) \\ Charles R Greathouse IV, Oct 07 2015
(Sage) [n*(n+21) for n in (0..50)] # G. C. Greubel, Mar 14 2022
CROSSREFS
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.
KEYWORD
easy,nonn
AUTHOR
Omar E. Pol, Aug 28 2007
STATUS
approved
A132766
a(n) = n*(n+24).
+10
12
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
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).
FORMULA
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)
MATHEMATICA
Table[n (n + 24), {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 25, 52}, 50] (* Harvey P. Dale, Feb 11 2016 *)
PROG
(PARI) a(n)=n*(n+24) \\ Charles R Greathouse IV, Jun 17 2017
(Sage) [n*(n+24) for n in (0..50)] # G. C. Greubel, Mar 14 2022
CROSSREFS
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.
KEYWORD
easy,nonn
AUTHOR
Omar E. Pol, Aug 28 2007
STATUS
approved
A132767
a(n) = n*(n + 25).
+10
11
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
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
REFERENCES
Douglas B. West, Introduction to Graph Theory, 2nd ed., Prentice-Hall, NJ, 2001.
LINKS
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).
FORMULA
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
MATHEMATICA
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 *)
PROG
(PARI) a(n)=n*(n+25) \\ Charles R Greathouse IV, Jun 17 2017
(Sage) [n*(n+25) for n in (0..50)] # G. C. Greubel, Mar 13 2022
CROSSREFS
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.
KEYWORD
nonn,easy
AUTHOR
Omar E. Pol, Aug 28 2007
STATUS
approved
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