Displaying 1-10 of 12 results found.
a(n) = n^3 + n^2 + n + 1.
+10
36
1, 4, 15, 40, 85, 156, 259, 400, 585, 820, 1111, 1464, 1885, 2380, 2955, 3616, 4369, 5220, 6175, 7240, 8421, 9724, 11155, 12720, 14425, 16276, 18279, 20440, 22765, 25260, 27931, 30784, 33825, 37060, 40495, 44136, 47989, 52060, 56355, 60880
COMMENTS
a(n) = 1111 in base n.
n^3 + n^2 + n + 1 = (n^2 + 1)*(n + 1), therefore a(n) is never prime. - Alonso del Arte, Apr 22 2014
FORMULA
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 3. - Colin Barker, Jan 02 2017
EXAMPLE
a(2) = 15 because 2^3 + 2^2 + 2 + 1 = 8 + 4 + 2 + 1 = 15.
a(3) = 40 because 3^3 + 3^2 + 3 + 1 = 27 + 9 + 3 + 1 = 40.
a(4) = 85 because 4^3 + 4^2 + 4 + 1 = 64 + 16 + 4 + 1 = 85.
The terms of the sequence are provided by the row sums of the following triangle (see the seventh formula in the previous section):
. 1;
. 3, 1;
. 9, 5, 1;
. 19, 13, 7, 1;
. 33, 25, 17, 9, 1;
. 51, 41, 31, 21, 11, 1;
. 73, 61, 49, 37, 25, 13, 1;
. 99, 85, 71, 57, 43, 29, 15, 1;
. 129, 113, 97, 81, 65, 49, 33, 17, 1;
. 163, 145, 127, 109, 91, 73, 55, 37, 19, 1;
. 201, 181, 161, 141, 121, 101, 81, 61, 41, 21, 1;
...
(End)
MATHEMATICA
FromDigits["1111", Range[0, 50]] (* Paolo Xausa, May 11 2024 *)
PROG
(PARI) Vec((1 + 5*x^2) / (1 - x)^4 + O(x^50)) \\ Colin Barker, Jan 02 2017 (Python) def a(n): return (n**3+n**2+n+1) # Torlach Rush, May 08 2024
CROSSREFS
Cf. A237627 (subset of semiprimes).
28-gonal numbers: a(n) = n*(13*n - 12).
+10
14
0, 1, 28, 81, 160, 265, 396, 553, 736, 945, 1180, 1441, 1728, 2041, 2380, 2745, 3136, 3553, 3996, 4465, 4960, 5481, 6028, 6601, 7200, 7825, 8476, 9153, 9856, 10585, 11340, 12121, 12928, 13761, 14620, 15505, 16416, 17353, 18316, 19305, 20320, 21361, 22428
COMMENTS
The defining formula can be regarded as an approximation and simplification of the expansion / propagation of native hydrophytes on the surface of stagnant waters in orthogonal directions; absence of competition / concurrence and of retrogression is assumed, mortality is taken into account. - [Translation of a comment in French sent by Pierre Gayet]
FORMULA
Product_{n>=2} (1 - 1/a(n)) = 13/14. - Amiram Eldar, Jan 22 2021
EXAMPLE
G.f. = x + 28*x^2 + 81*x^3 + 160*x^4 + 265*x^5 + 396*x^6 + 553*x^7 + ...
MATHEMATICA
lst={}; Do[a=13*n^2+14*n+1; AppendTo[lst, a], {n, 0, 5!}]; lst
Table[n*(13*n - 12), {n, 0, 100}] (* Robert Price, Oct 11 2018 *)
PROG
(Magma) [ (n+1)*(13*n+1): n in[0..50] ];
1, 17, 37, 61, 89, 121, 157, 197, 241, 289, 341, 397, 457, 521, 589, 661, 737, 817, 901, 989, 1081, 1177, 1277, 1381, 1489, 1601, 1717, 1837, 1961, 2089, 2221, 2357, 2497, 2641, 2789, 2941, 3097, 3257, 3421, 3589, 3761, 3937, 4117, 4301, 4489, 4681, 4877, 5077
COMMENTS
The defining formula can be regarded as an approximation and simplification of the expansion/propagation of native hydrophytes on the surface of stagnant waters in orthogonal directions; absence of competition/concurrence and of retrogression is assumed, mortality is taken into account. - [Translation of a comment in French sent by Pierre Gayet]
FORMULA
E.g.f.: (1 + 16*x + 2*x^2)*exp(x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)
MATHEMATICA
CoefficientList[Series[(1 + 14 x - 11 x^2) / (1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Nov 08 2014 *) Table[2n^2+14n+1, {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {1, 17, 37}, 50] (* Harvey P. Dale, Jul 14 2018 *)
PROG
(Magma) [ 2*n^2+14*n+1: n in [0..50] ];
(PARI) Vec((1+14*x-11*x^2)/(1-x)^3 + O(x^100)) \\ Colin Barker, Nov 08 2014
a(n) = 13*n^2 + 10*n + 1.
+10
7
1, 24, 73, 148, 249, 376, 529, 708, 913, 1144, 1401, 1684, 1993, 2328, 2689, 3076, 3489, 3928, 4393, 4884, 5401, 5944, 6513, 7108, 7729, 8376, 9049, 9748, 10473, 11224, 12001, 12804, 13633, 14488, 15369, 16276, 17209, 18168, 19153, 20164
COMMENTS
The defining formula can be regarded as an approximation and simplification of the expansion / propagation of native hydrophytes on the surface of stagnant waters in orthogonal directions; absence of competition / concurrence and of retrogression is assumed, mortality is taken into account. - [Translation of a comment in French sent by Pierre Gayet]
FORMULA
G.f.: (1 + 21*x + 4*x^2)/(1-x)^3.
a(n-1) = A202141(n) - 1 with a(-1)=4. (End)
MATHEMATICA
Table[13n^2+10n+1, {n, 0, 40}] (* or *) LinearRecurrence[{3, -3, 1}, {1, 24, 73}, 40] (* Harvey P. Dale, Nov 06 2014 *)
PROG
(Magma) [ 13*n^2+10*n+1: n in [0..50] ];
1, 29, 73, 133, 209, 301, 409, 533, 673, 829, 1001, 1189, 1393, 1613, 1849, 2101, 2369, 2653, 2953, 3269, 3601, 3949, 4313, 4693, 5089, 5501, 5929, 6373, 6833, 7309, 7801, 8309, 8833, 9373, 9929, 10501, 11089, 11693, 12313, 12949, 13601, 14269, 14953, 15653, 16369
COMMENTS
The defining formula can be regarded as an approximation and simplification of the expansion / propagation of native hydrophytes on the surface of stagnant waters in orthogonal directions; absence of competition / concurrence and of retrogression is assumed, mortality is taken into account. - (Translation of a comment in French sent by P. Gayet)
FORMULA
G.f.: (1 + 26*x - 11*x^2)/(1 - x)^3.
E.g.f.: (1 + 28*x + 8*x^2)*exp(x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)
PROG
(Magma) [ 8*n^2+20*n+1: n in [0..50] ];
1, 26, 61, 106, 161, 226, 301, 386, 481, 586, 701, 826, 961, 1106, 1261, 1426, 1601, 1786, 1981, 2186, 2401, 2626, 2861, 3106, 3361, 3626, 3901, 4186, 4481, 4786, 5101, 5426, 5761, 6106, 6461, 6826, 7201, 7586, 7981, 8386, 8801, 9226, 9661, 10106, 10561, 11026
COMMENTS
The defining formula can be regarded as an approximation and simplification of the expansion / propagation of native hydrophytes on the surface of stagnant waters in orthogonal directions; absence of competition / concurrence and of retrogression is assumed, mortality is taken into account. - [Translation of a comment in French sent by Pierre Gayet]
FORMULA
E.g.f.: (5*x^2 + 25*x + 1)*exp(x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)
MATHEMATICA
lst={}; Do[a=5*n^2+20*n+1; AppendTo[lst, a], {n, 0, 5!}]; lst
Table[5n^2+20n+1, {n, 0, 40}] (* or *) LinearRecurrence[{3, -3, 1}, {1, 26, 61}, 40] (* or *) CoefficientList[Series[(14x^2-23x-1)/(x-1)^3, {x, 0, 40}], x] (* Harvey P. Dale, May 07 2023 *)
PROG
(Magma) [ 5*n^2+20*n+1: n in [0..50] ];
0, 6, 21, 48, 90, 150, 231, 336, 468, 630, 825, 1056, 1326, 1638, 1995, 2400, 2856, 3366, 3933, 4560, 5250, 6006, 6831, 7728, 8700, 9750, 10881, 12096, 13398, 14790, 16275, 17856, 19536, 21318, 23205, 25200, 27306, 29526, 31863, 34320, 36900, 39606, 42441, 45408, 48510
COMMENTS
After 0, this sequence is the third column of the array in A185874. Sequence is related to A051744 by A051744(n) = n*a(n)/3 - Sum_{i=0..n-1} a(i) for n>0.
FORMULA
O.g.f.: 3*x*(2 - x)/(1 - x)^4.
E.g.f.: x*(12 + 9*x + x^2)*exp(x)/2.
a(n) = n*(n + 1)*(n + 5)/2.
a(n) = Sum_{i=0..n} n*(n - i) + 5*i, that is: a(n) = A002411(n) + A028895(n). More generally, Sum_{i=0..n} n*(n - i) + k*i = n*(n + 1)*(n + k)/2. a(n+1) - 3*a(n) + 3*a(n-1) = 3* A105163(n) for n>0. Sum_{n>=1} 1/a(n) = 163/600.
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/5 - 253/600. (End)
EXAMPLE
The sequence is also provided by the row sums of the following triangle (see the fourth formula above):
. 0;
. 1, 5;
. 4, 7, 10;
. 9, 11, 13, 15;
. 16, 17, 18, 19, 20;
. 25, 25, 25, 25, 25, 25;
. 36, 35, 34, 33, 32, 31, 30;
. 49, 47, 45, 43, 41, 39, 37, 35;
. 64, 61, 58, 55, 52, 49, 46, 43, 40;
. 81, 77, 73, 69, 65, 61, 57, 53, 49, 45, etc.
Third column is included in A189834. Diagonal 1, 11, 25, 43, 65, 91, 121, ... is A161532.
MATHEMATICA
Table[n (n + 1) (n + 5)/2, {n, 0, 50}]
LinearRecurrence[{4, -6, 4, -1}, {0, 6, 21, 48}, 50] (* Harvey P. Dale, Jul 18 2019 *)
PROG
(PARI) vector(50, n, n--; n*(n+1)*(n+5)/2)
(Sage) [n*(n+1)*(n+5)/2 for n in (0..50)]
(Magma) [n*(n+1)*(n+5)/2: n in [0..50]];
CROSSREFS
Cf. similar sequences of the type n*(n+1)*(n+k)/2: A002411 (k=0), A006002 (k=1), A027480 (k=2), A077414 (k=3, with offset 1), A212343 (k=4, without the initial 0), this sequence (k=5).
2, 4, 10, 20, 34, 52, 74, 100, 130, 164, 202, 244, 290, 340, 394, 452, 514, 580, 650, 724, 802, 884, 970, 1060, 1154, 1252, 1354, 1460, 1570, 1684, 1802, 1924, 2050, 2180, 2314, 2452, 2594, 2740, 2890, 3044, 3202, 3364, 3530, 3700, 3874, 4052, 4234, 4420, 4610, 4804, 5002, 5204, 5410, 5620
COMMENTS
Numbers n such that 2*n - 4 is a perfect square.
For n > 2, the number of square a(n)-gonal numbers is finite. - Muniru A Asiru, Oct 16 2016
FORMULA
Sum_{n>=1} 1/a(n) = (1 + Pi*coth(Pi))/4 = 1.038337023734290587067... . - Vaclav Kotesovec, Apr 11 2016 E.g.f.: 2*(exp(x)*(x^2 - x + 2) - 2).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)
EXAMPLE
a(1) = 2*1^2 - 4*1 + 4 = 2.
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {2, 4, 10}, 60] (* Harvey P. Dale, Jul 18 2023 *)
PROG
(Magma) [ 2*n^2 - 4*n + 4: n in [1..60]];
(Magma) [ n: n in [1..6000] | IsSquare(2*n-4)];
(PARI) x='x+O('x^99); Vec(2*x*(1-x+2*x^2)/(1-x)^3) \\ Altug Alkan, Apr 11 2016
CROSSREFS
Cf. A002522, numbers n such that 2*n + k is a perfect square: no sequence (k = -9), A255843 (k = -8), A271649 (k = -7), A093328 (k = -6), A097080 (k = -5), this sequence (k = -4), A051890 (k = -3), A058331 (k = -2), A001844 (k = -1), A001105 (k = 0), A046092 (k = 1), A056222 (k = 2), A142463 (k = 3), A054000 (k = 4), A090288 (k = 5), A268581 (k = 6), A059993 (k = 7), (-1)* A147973 (k = 8), A139570 (k = 9), A271625 (k = 10), A222182 (k = 11), A152811 (k = 12), A181510 (k = 13), A161532 (k = 14), no sequence (k = 15).
3, 13, 27, 45, 67, 93, 123, 157, 195, 237, 283, 333, 387, 445, 507, 573, 643, 717, 795, 877, 963, 1053, 1147, 1245, 1347, 1453, 1563, 1677, 1795, 1917, 2043, 2173, 2307, 2445, 2587, 2733, 2883, 3037, 3195, 3357, 3523, 3693, 3867, 4045, 4227, 4413, 4603, 4797, 4995, 5197, 5403, 5613, 5827
COMMENTS
Numbers n such that 2*n + 10 is a perfect square.
FORMULA
Sum_{n>=1} 1/a(n) = 13/30 - Pi*cot(sqrt(5/2)*Pi)/(2*sqrt(10)) = 0.5627678459924... . - Vaclav Kotesovec, Apr 11 2016 E.g.f.: exp(x)*(2*x^2 + 6*x - 3) + 3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)
EXAMPLE
a(1) = 2*1^2 + 4*1 - 3 = 3.
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {3, 13, 27}, 60] (* Harvey P. Dale, Jun 08 2023 *)
PROG
(Magma) [ 2*n^2 + 4*n - 3: n in [1..60]];
(Magma) [ n: n in [1..6000] | IsSquare(2*n+10)];
(PARI) x='x+O('x^99); Vec(x*(3+4*x-3*x^2)/(1-x)^3) \\ Altug Alkan, Apr 11 2016
CROSSREFS
Numbers h such that 2*h + k is a perfect square: A294774 (k=-9), A255843 (k=-8), A271649 (k=-7), A093328 (k=-6), A097080 (k=-5), A271624 (k=-4), A051890 (k=-3), A058331 (k=-2), A001844 (k=-1), A001105 (k=0), A046092 (k=1), A056222 (k=2), A142463 (k=3), A054000 (k=4), A090288 (k=5), A268581 (k=6), A059993 (k=7), (-1)* A147973 (k=8), A139570 (k=9), this sequence (k=10), A222182 (k=11), A152811 (k=12), A181510 (k=13), A161532 (k=14), no sequence (k=15).
4, 8, 16, 28, 44, 64, 88, 116, 148, 184, 224, 268, 316, 368, 424, 484, 548, 616, 688, 764, 844, 928, 1016, 1108, 1204, 1304, 1408, 1516, 1628, 1744, 1864, 1988, 2116, 2248, 2384, 2524, 2668, 2816, 2968, 3124, 3284, 3448, 3616, 3788, 3964, 4144, 4328, 4516, 4708, 4904, 5104, 5308, 5516
COMMENTS
Numbers n such that 2n - 7 is a perfect square.
Galois numbers for three-dimensional vector space, defined as the total number of subspaces in a three-dimensional vector space over GF(n-1), when n-1 is a power of a prime. - Artur Jasinski, Aug 31 2016, corrected by Robert Israel, Sep 23 2016
FORMULA
Sum_{n>=1} 1/a(n) = tanh(sqrt(7)*Pi/2)*Pi/(2*sqrt(7)). - Amiram Eldar, Jul 30 2024
EXAMPLE
a(1) = 2*(1^2 - 1 + 2) = 4.
MATHEMATICA
Table[2 (n^2 - n + 2), {n, 53}] (* or *)
Select[Range@ 5516, IntegerQ@ Sqrt[2 # - 7] &] (* or *)
Table[SeriesCoefficient[(-4 (1 - x + x^2))/(-1 + x)^3, {x, 0, n}], {n, 0, 52}] (* Michael De Vlieger, Apr 11 2016 *) LinearRecurrence[{3, -3, 1}, {4, 8, 16}, 60] (* Harvey P. Dale, Jun 14 2022 *)
PROG
(Magma) [ 2*n^2 - 2*n + 4: n in [1..60]];
(Magma) [ n: n in [1..6000] | IsSquare(2*n-7)];
CROSSREFS
Numbers h such that 2*h + k is a perfect square: no sequence (k=-9), A255843 (k=-8), this sequence (k=-7), A093328 (k=-6), A097080 (k=-5), A271624 (k=-4), A051890 (k=-3), A058331 (k=-2), A001844 (k=-1), A001105 (k=0), A046092 (k=1), A056222 (k=2), A142463 (k=3), A054000 (k=4), A090288 (k=5), A268581 (k=6), A059993 (k=7), (-1)* A147973 (k=8), A139570 (k=9), A271625 (k=10), A222182 (k=11), A152811 (k=12), A181510 (k=13), A161532 (k=14), no sequence (k=15).
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