Displaying 1-10 of 20 results found.
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
FORMULA
G.f.: x*(9-8*x)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
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 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)
PROG
(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
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
COMMENTS
a(n) is the first Zagreb index of the helm graph H[n] (n>=3). - Emeric Deutsch, Nov 05 2016 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)
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
Eric Weisstein's World of Mathematics, Helm Graph.
FORMULA
a(n) = n*(n + 17).
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 *)
Triangle T(n,m) = 1+2*m of odd numbers read along rows, 0<=m<n.
+10
19
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
COMMENTS
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
FORMULA
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
EXAMPLE
The triangle contains the first n odd numbers in row n:
1;
1,3;
1,3,5;
1,3,5,7;
-----------------------------------------------------------------
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)
PROG
(Haskell)
a158405 n k = a158405_row n !! (k-1)
a158405_row n = a158405_tabl !! (n-1)
a158405_tabl = map reverse a099375_tabl
(PARI) a(n) = 2*(n-floor((-1+sqrt(8*n-7))/2)*(floor((-1+sqrt(8*n-7))/2)+1)/2)-1;
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
FORMULA
a(n) = (n+10)^2 - 10^2 = n*(n+20), n>=0.
G.f.: x*(21-19*x)/(1-x)^3.
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)
CROSSREFS
a(n-10), n>=11, tenth column (used for the n=10 series of the hydrogen atom) of triangle A120070. Cf. A001008, A001477, A098849, A102928, A120061, A132765, A132760, A132761, A132762, A005563, A028552, A028347, A028557, A028560, A028563, A028566, A028569, A098603, A098847, A132759, A098848.
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
FORMULA
a(n) = n*(n + 15).
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
Table[n(n+15), {n, 0, 60}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 16, 34}, 60] (* Harvey P. Dale, Jan 20 2019 *)
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
FORMULA
a(n) = n*(n + 23).
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)
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)
CROSSREFS
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.
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
FORMULA
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)
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)
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {0, 20, 42}, 60] (* Harvey P. Dale, Jun 03 2021 *)
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.
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
FORMULA
a(n) = n*(n + 22).
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 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)
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 *)
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.
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
FORMULA
a(n) = n*(n + 21).
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 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)
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 *)
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.
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
FORMULA
a(n) = n*(n + 24).
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 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)
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 *)
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.
Search completed in 0.018 seconds
| 