Displaying 1-10 of 25 results found.
Partial products of successive terms of A017209; a(0)=1 .
+20
6
1, 4, 52, 1144, 35464, 1418560, 69509440, 4031547520, 270113683840, 20528639971840, 1744934397606400, 164023833375001600, 16894454837625164800, 1892178941814018457600, 228953651959496233369600, 29763974754734510338048000, 4137192490908096936988672000
FORMULA
a(n) = Sum_{k=0..n} A132393(n,k)*4^k*9^(n-k). a(n) = (-5)^n*Sum_{k=0..n} (9/5)^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012 a(n) = 9^n * Gamma(n + 4/9) / Gamma(4/9).
a(n) ~ sqrt(2*Pi) * 9^n * n^(n - 1/18) / (Gamma(4/9) * exp(n)). (End)
G.f.: hypergeometric2F0([1, 4/9], [], 9*x).
E.g.f.: (1-9*x)^(-4/9). (End)
Sum_{n>=0} 1/a(n) = 1 + (e/9^5)^(1/9)*(Gamma(4/9) - Gamma(4/9, 1/9)). - Amiram Eldar, Dec 21 2022
EXAMPLE
a(0)=1, a(1)=4, a(2)=4*13=52, a(3)=4*13*22=1144, a(4)=4*13*22*31=35464, ...
MATHEMATICA
Table[4*9^(n-1)*Pochhammer[13/9, n-1], {n, 0, 20}] (* Vaclav Kotesovec, Nov 29 2021 *)
PROG
(PARI) a(n) = (-5)^n*sum(k=0, n, (9/5)^k*stirling(n+1, n+1-k, 1)); \\ Michel Marcus, Feb 20 2015 (Magma) [n le 2 select 4^(n-1) else (9*n-14)*Self(n-1): n in [1..30]]; // G. C. Greubel, May 26 2022 (SageMath) [9^n*rising_factorial(4/9, n) for n in (0..30)] # G. C. Greubel, May 26 2022
EXTENSIONS
a(9) originally given incorrectly as 20520639971840 corrected by Peter Bala, Feb 20 2015
1, 10, 19, 28, 37, 46, 55, 64, 73, 82, 91, 100, 109, 118, 127, 136, 145, 154, 163, 172, 181, 190, 199, 208, 217, 226, 235, 244, 253, 262, 271, 280, 289, 298, 307, 316, 325, 334, 343, 352, 361, 370, 379, 388, 397, 406, 415, 424, 433, 442, 451, 460, 469, 478
COMMENTS
If A=[ A147296] 9*n^2+2*n (n>0, 11, 40, 87, ...); Y=[ A010701] 3 (3, 3, 3, ...); X=[ A017173] 9*n+1 (n>0, 10, 19, 28, ...), we have, for all terms, Pell's equation X^2 - A*Y^2 = 1. Example: 10^2 - 11*3^2 = 1; 19^2 - 40*3^2 = 1; 28^2 - 87*3^2 = 1. - Vincenzo Librandi, Aug 01 2010
FORMULA
G.f.: (1 + 8*x)/(1 - x)^2.
MATHEMATICA
LinearRecurrence[{2, -1}, {1, 10}, 60] (* Harvey P. Dale, Dec 27 2014 *)
PROG
(Sage) [i+1 for i in range(480) if gcd(i, 9) == 9] # Zerinvary Lajos, May 20 2009 (Haskell)
a017173 = (+ 1) . (* 9)
CROSSREFS
Cf. A093644 ((9, 1) Pascal, column m=1).
5, 14, 23, 32, 41, 50, 59, 68, 77, 86, 95, 104, 113, 122, 131, 140, 149, 158, 167, 176, 185, 194, 203, 212, 221, 230, 239, 248, 257, 266, 275, 284, 293, 302, 311, 320, 329, 338, 347, 356, 365, 374, 383, 392, 401, 410, 419, 428, 437, 446, 455, 464, 473, 482
FORMULA
a(n) = a(n-1) + 9, with a(0) = 5.
E.g.f.: (5 + 9*x)*exp(x). (End)
MATHEMATICA
9*Range[0, 60]+5 (* or *) LinearRecurrence[{2, -1}, {5, 14}, 60] (* Harvey P. Dale, Jul 05 2021 *)
PROG
(SageMath) [9*n+5 for n in range(51)] # G. C. Greubel, Jan 06 2023
CROSSREFS
Sequences of the form (9*n+5)^k: this sequence (k=1), A017222 (k=2), A017223 (k=3), A017224 (k=4), A017225 (k=5), A017226 (k=6), A017227 (k=7), A017228 (k=8), A017229 (k=9), A017230 (k=10), A017231 (k=11). Cf. similar sequences with closed form (2*k-1)*n+k listed in A269044.
Numbers with digital root 1, 4, 7 or 9.
+10
20
1, 4, 7, 9, 10, 13, 16, 18, 19, 22, 25, 27, 28, 31, 34, 36, 37, 40, 43, 45, 46, 49, 52, 54, 55, 58, 61, 63, 64, 67, 70, 72, 73, 76, 79, 81, 82, 85, 88, 90, 91, 94, 97, 99, 100, 103, 106, 108, 109, 112, 115, 117, 118, 121, 124, 126, 127, 130, 133, 135, 136, 139, 142
COMMENTS
All squares are members (see A070433). May also be defined as: possible sums of digits of squares. - Zak Seidov, Feb 11 2008 First differences are periodic: 3, 3, 2, 1, 3, 3, 2, 1, 3, 3, 2, 1, 3, 3, 2, 1, 3, 3, 2, 1, 3, 3, 2, 1, 3, 3, 2, 1, ... - Zak Seidov, Feb 11 2008 Minimal n with corresponding sum-of-digits(n^2) are: 1, 2, 4, 3, 8, 7, 13, 24, 17, 43, 67, 63, 134, 83, 167, 264, 314, 313, 707, 1374, 836, 1667, 2236, 3114, 4472, 6833, 8167, 8937, 16667, 21886, 29614, 60663, 41833, 74833, 89437, 94863, 134164, 191833.
a(n) is the set of all m such that 9k+m can be a perfect square (quadratic residues of 9 including the trivial case of 0). - Gary Detlefs, Mar 19 2010 The sum of digits of any term belongs to the sequence. Also the products of any terms belong to the sequence.
Positive integers of the forms x^2 + (2*m+1)*x*y + (m^2+m-2)*y^2, for integers m.
This sequence is closed under multiplication. (End)
FORMULA
O.g.f.: x*(2x+1)*(x^2+x+1)/((-1+x)^2*(x+1)*(x^2+1)).
a(n) = a(n-4) + 9. (End)
a(n) = 3*(n - floor(n/4)) - (3 - i^n - (-i)^n - (-1)^n)/2, where i = sqrt(-1). - Gary Detlefs, Mar 19 2010 a(n) = 3*n - floor(n/4) - 2*floor((n+3)/4). - Ridouane Oudra, Jan 21 2024 E.g.f.: (cos(x) + (9*x - 1)*cosh(x) - 3*sin(x) + (9*x - 2)*sinh(x))/4. - Stefano Spezia, Feb 21 2024
MAPLE
seq( 3*(n-floor(n/4)) - (3-I^n-(-I)^n-(-1)^n)/2, n=1..63); # Gary Detlefs, Mar 19 2010
MATHEMATICA
LinearRecurrence[{1, 0, 0, 1, -1}, {1, 4, 7, 9, 10}, 70] (* Harvey P. Dale, Aug 29 2015 *)
a(0) = 1; a(n) = (5*3^(n-1) - 1)/2 for n > 0.
+10
20
1, 2, 7, 22, 67, 202, 607, 1822, 5467, 16402, 49207, 147622, 442867, 1328602, 3985807, 11957422, 35872267, 107616802, 322850407, 968551222, 2905653667, 8716961002, 26150883007, 78452649022, 235357947067, 706073841202
COMMENTS
From Erich Friedman's math magic page 2nd paragraph under "Answers" section.
Let A be the Hessenberg matrix of order n, defined by: A[1,j] = 1, A[i,i] = 2,(i>1), A[i,i-1] = -1, and A[i,j] = 0 otherwise. Then, for n >= 1, a(n) = (-1)^n*charpoly(A,-1). - Milan Janjic, Jan 26 2010 If n > 0 and H = hex number ( A003215), then 9*H(a(n)) - 2 = H(a(n+1)), for example 9*H(2) - 2 = 9*19 - 2 = 169 = H(7). For n > 2, this is a subsequence of A017209, see formula. - Klaus Purath, Mar 31 2021
FORMULA
The following is a summary of formulas added over the past 18 years.
a(n) = 3*a(n-1) + 1; with a(0)=1, a(1)=2. - Jason Earls, Apr 29 2001 a(n) = 4*a(n-1) - 3*a(n-2) for n > 2.
G.f.: (1-2*x+2*x^2)/((1-x)*(1-3*x)). (End)
For n > 0, a(n) = floor(3^n*5/6). - M. F. Hasler, Apr 06 2019 a(n) = 2 + Sum_{i = 0..n-2} A005030(i). a(n+1)*a(n+2) = a(n)*a(n+3) + 20*3^n, n > 1.
MATHEMATICA
LinearRecurrence[{4, -3}, {1, 2, 7}, 30] (* Harvey P. Dale, Nov 15 2022 *)
PROG
(PARI) { for (n=0, 200, if (n>1, a1=a=3*a1 + 1, if (n==0, a=1, a1=a=2)); write("b060816.txt", n, " ", a); ) } \\ Harry J. Smith, Jul 13 2009
6, 15, 24, 33, 42, 51, 60, 69, 78, 87, 96, 105, 114, 123, 132, 141, 150, 159, 168, 177, 186, 195, 204, 213, 222, 231, 240, 249, 258, 267, 276, 285, 294, 303, 312, 321, 330, 339, 348, 357, 366, 375, 384, 393, 402, 411, 420, 429, 438, 447, 456, 465, 474, 483
COMMENTS
Numbers whose digital root is 6; that is, A010888(a(n)) = 6. (Ball essentially says that Iamblichus (circa 350) announced that a number equal to the sum of three integers 3n, 3n - 1, and 3n - 2 has 6 as what is now called the number's digital root.) - Rick L. Shepherd, Apr 01 2014
REFERENCES
W. W. R. Ball, A Short Account of the History of Mathematics, Sterling Publishing Company, Inc., 2001 (Facsimile Edition) [orig. pub. 1912], pages 110-111.
FORMULA
Sum_{n>=0} (-1)^n/a(n) = sqrt(3)*Pi/27 - log(2)/9. - Amiram Eldar, Dec 12 2021
MATHEMATICA
LinearRecurrence[{2, -1}, {6, 15}, 60] (* Harvey P. Dale, Feb 01 2014 *)
AUTHOR
David J. Horn and Laura Krebs Gordon (lkg615(AT)verizon.net), 1985
Triangle read by rows: T(n,k) = 2*n*k + n + k (1 <= k <= n).
+10
9
4, 7, 12, 10, 17, 24, 13, 22, 31, 40, 16, 27, 38, 49, 60, 19, 32, 45, 58, 71, 84, 22, 37, 52, 67, 82, 97, 112, 25, 42, 59, 76, 93, 110, 127, 144, 28, 47, 66, 85, 104, 123, 142, 161, 180, 31, 52, 73, 94, 115, 136, 157, 178, 199, 220, 34, 57, 80, 103, 126, 149, 172, 195, 218, 241, 264
COMMENTS
T(n,k) gives number of edges (of unit length) in a k X n grid.
The values 2*T(n,k)+1 = (2*n+1)*(2*k+1) are nonprime and therefore in A047845.
FORMULA
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = A182868((n+1)/2) if n is odd otherwise A182868(n/2) + 1. (End)
EXAMPLE
Triangle begins:
4;
7, 12;
10, 17, 24;
13, 22, 31, 40;
16, 27, 38, 49, 60;
19, 32, 45, 58, 71, 84;
22, 37, 52, 67, 82, 97, 112;
25, 42, 59, 76, 93, 110, 127, 144;
28, 47, 66, 85, 104, 123, 142, 161, 180;
MATHEMATICA
T[n_, k_]:= 2 n k + n + k; Table[T[n, k], {n, 10}, {k, n}]//Flatten (* Vincenzo Librandi, Jun 01 2014 *)
PROG
(Magma) [(2*n*k + n + k): k in [1..n], n in [1..11]]; // Vincenzo Librandi, Jun 01 2014 (Python)
def T(r, c): return 2*r*c + r + c
a = [T(r, c) for r in range(12) for c in range(1, r+1)]
(SageMath) flatten([[2*n*k +n +k for k in range(1, n+1)] for n in range(1, 14)]) # G. C. Greubel, Oct 17 2023
AUTHOR
Artemario Tadeu Medeiros da Silva (artemario(AT)uol.com.br), Jun 09 2003
a(n) = 3^(n+1) + (3^n-1)/2.
+10
9
3, 10, 31, 94, 283, 850, 2551, 7654, 22963, 68890, 206671, 620014, 1860043, 5580130, 16740391, 50221174, 150663523, 451990570, 1355971711, 4067915134, 12203745403, 36611236210, 109833708631, 329501125894, 988503377683, 2965510133050, 8896530399151
COMMENTS
a(n-1) agrees with the graph radius of the n-Sierpinski carpet graph for n = 2 to at least n = 5. See A100774 for the graph diameter of the n-Sierpinski carpet graph. The inverse binomial transform gives 3, 7, 14, 28, 56, ... i.e., A005009 with a leading 3. - R. J. Mathar, Jan 08 2020 First differences of A108765. The digital root of a(n) for n > 1 is always 4. a(n) is never divisible by 7 or by 12. a(n) == 10 (mod 84) for odd n. a(n) == 31 (mod 84) for even n > 0. Conjecture: This sequence contains no prime factors p == {11, 13, 23, 61 71, 73} (mod 84). - Klaus Purath, Apr 13 2020
FORMULA
G.f.: (3-2*x)/((1-x)*(1-3*x)).
a(n) = 3*a(n-1) + 1 for n > 0, a(0)=3. (Note that if a(0) were 1 in this recurrence we would get A003462, if it were 2 we would get A060816. - N. J. A. Sloane, Dec 06 2019) a(n) = 4*a(n-1) - 3*a(n-2) for n > 1, a(0)=3, a(1)=10.
a(n) = 2*a(n-1) + 3*a(n-2) + 2 for n > 1.
EXAMPLE
Ternary....................Decimal
10...............................3
101.............................10
1011............................31
10111...........................94
101111.........................283
1011111........................850
10111111......................2551
101111111.....................7654, etc.
MATHEMATICA
Table[(7 3^n - 1)/2, {n, 0, 20}]
(7 3^Range[0, 20] - 1)/2
LinearRecurrence[{4, -3}, {10, 31}, {0, 20}]
CoefficientList[Series[(3 - 2 x)/((x - 1) (3 x - 1)), {x, 0, 20}], x]
(* End *)
PROG
(PARI) Vec((3 - 2*x) / ((1 - x)*(1 - 3*x)) + O(x^30)) \\ Colin Barker, Nov 27 2019
Coins left when packing boomerangs into n X n coins.
+10
8
4, 3, 7, 13, 6, 13, 22, 9, 19, 31, 12, 25, 40, 15, 31, 49, 18, 37, 58, 21, 43, 67, 24, 49, 76, 27, 55, 85, 30, 61, 94, 33, 67, 103, 36, 73, 112, 39, 79, 121, 42, 85, 130, 45, 91, 139, 48, 97, 148, 51, 103, 157, 54, 109, 166, 57
COMMENTS
The coins left after packing boomerangs into n X n coins using the same rule as A229593. See illustration in links.
FORMULA
G.f. -x^2*(-4-3*x-7*x^2-5*x^3+x^5) / ( (x-1)^2*(1+x+x^2)^2 ). - R. J. Mathar, Oct 16 2013
MATHEMATICA
LinearRecurrence[{0, 0, 2, 0, 0, -1}, {4, 3, 7, 13, 6, 13}, 80] (* Harvey P. Dale, Jan 19 2019 *)
PROG
(Small Basic)
u[2]=4
d[3]=-1
d[4]=4
d[5]=6
For n=2 To 100
If n+1 >=6 Then
If Math.Remainder(n+1, 3)=0 Then
d[n+1]=d[n-2]-6
Else
d[n+1]=d[n-2]+3
EndIf
EndIf
u[n+1]=u[n]+d[n+1]
TextWindow.Write(u[n]+", ")
EndFor
Odd composite numbers congruent to 4 modulo 9.
+10
8
49, 85, 121, 175, 247, 265, 301, 319, 355, 391, 427, 445, 481, 517, 535, 553, 589, 625, 679, 697, 715, 805, 841, 895, 913, 931, 949, 985, 1003, 1057, 1075, 1111, 1147, 1165, 1183, 1219, 1255, 1273, 1309, 1345, 1363, 1417, 1435, 1507, 1525, 1561, 1615, 1633
MATHEMATICA
Select[18Range[125] + 13, Not[PrimeQ[#]] &] (* Alonso del Arte, Sep 24 2014 *) Select[Range[13, 1700, 18], CompositeQ] (* Harvey P. Dale, Aug 21 2024 *)
PROG
(PARI) lista(nn) = {forcomposite(n=1, nn, if ((n % 2) && ((n % 9) == 4), print1(n, ", ")); ); } \\ Michel Marcus, Sep 22 2014
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