| Displaying 1-7 of 7 results found.
page
1
a(n) = n*(n^8 + 1)*(n^4 + 1)*(n^2 + 1)*(n + 1) + 1.
+10
14
1, 17, 131071, 64570081, 5726623061, 190734863281, 3385331888947, 38771752331201, 321685687669321, 2084647712458321, 11111111111111111, 50544702849929377, 201691918794585181, 720867993281778161, 2345488209948553531, 7037580381120954241
COMMENTS
a(n) = Phi_17(n) where Phi_k(x) is the k-th cyclotomic polynomial.
LINKS
Index entries for linear recurrences with constant coefficients, signature (17, -136, 680, -2380, 6188, -12376, 19448, -24310, 24310, -19448, 12376, -6188, 2380, -680, 136, -17, 1).
FORMULA
G.f.: (1 +130918*x^2 +62343506*x^3 +4646748160*x^4 +102074708252*x^5 +878064150546*x^6 +3419813860214*x^7 +6502752956958*x^8 +6232856389160*x^9 +3004612851498*x^10 +701875014878*x^11 +73106078368*x^12 +2893069436*x^13 +31542430*x^14 +43674*x^15 +x^16)/(1 - x)^17.
Sum_{n>=0} 1/a(n) = 1.05883117453...
MATHEMATICA
Table[Cyclotomic[17, n], {n, 0, 15}]
PROG
(Magma) [n*(n^8+1)*(n^4+1)*(n^2+1)*(n+1)+1: n in [0..20]]; // Vincenzo Librandi, Feb 27 2016 (Sage) [n*(n^8+1)*(n^4+1)*(n^2+1)*(n+1)+1 for n in (0..20)] # G. C. Greubel, Apr 24 2019 (GAP) List([0..20], n-> n*(n^8+1)*(n^4+1)*(n^2+1)*(n+1)+1) # G. C. Greubel, Apr 24 2019
CROSSREFS
Cf. similar sequences of the type Phi_k(n), where Phi_k is the k-th cyclotomic polynomial: A000012 (k=0), A023443 (k=1), A000027 (k=3), A002522 (k=4), A053699 (k=5), A002061 (k=6), A053716 (k=7), A002523 (k=8), A060883 (k=9), A060884 (k=10), A060885 (k=11), A060886 (k=12), A060887 (k=13), A060888 (k=14), A060889 (k=15), A060890 (k=16), this sequence (k=17), A060891 (k=18), A269446 (k=19).
a(n) = n^3 - n^2 + n - 1 = (n-1) * (n^2 + 1).
+10
13
-1, 0, 5, 20, 51, 104, 185, 300, 455, 656, 909, 1220, 1595, 2040, 2561, 3164, 3855, 4640, 5525, 6516, 7619, 8840, 10185, 11660, 13271, 15024, 16925, 18980, 21195, 23576, 26129, 28860, 31775, 34880, 38181, 41684, 45395, 49320, 53465, 57836, 62439, 67280, 72365, 77700, 83291, 89144, 95265, 101660
COMMENTS
Number of walks of length 4 between any two distinct vertices of the complete graph K_{n+1} (n >= 1). Example: a(2) = 5 because in the complete graph ABC we have the following walks of length 4 between A and B: ABACB, ABCAB, ACACB, ACBAB and ACBCB. - Emeric Deutsch, Apr 01 2004 1/a(n) for n >= 2, is in base n given by 0.repeat(0,0,1,1), due to (1/n^3 + 1/n^4)*(1/(1-1/n^4)) = 1/((n-1)*(n^2+1)). - Wolfdieter Lang, Jun 20 2014 For odd n, a(n) * (n+1) / 2 + 1 also represents the first integer in a sum of n^4 consecutive integers that equals n^8. - Patrick J. McNab, Dec 26 2016
FORMULA
a(n) = round(n^4/(n+1)) for n >= 2.
G.f.: (4*x-1)*(1+x^2)/(1-x)^4 (for the signed sequence). - Emeric Deutsch, Apr 01 2004 a(n) = floor(n^5/(n^2+n)) for n > 0. - Gary Detlefs, May 27 2010
EXAMPLE
a(4) = 4^3 - 4^2 + 4 - 1 = 64 - 16 + 4 - 1 = 51.
MAPLE
a:=n->sum(1+sum(n, k=1..n), k=2..n):seq(a(n), n=0...43); # Zerinvary Lajos, Aug 24 2008
PROG
(PARI) { for (n=0, 1000, write("b062158.txt", n, " ", n*(n*(n - 1) + 1) - 1) ) } \\ Harry J. Smith, Aug 02 2009
Square array T(n,k) = (n^k - (-1)^k)/(n+1), n >= 0, k >= 0, read by falling antidiagonals.
+10
8
0, 1, 0, -1, 1, 0, 1, 0, 1, 0, -1, 1, 1, 1, 0, 1, 0, 3, 2, 1, 0, -1, 1, 5, 7, 3, 1, 0, 1, 0, 11, 20, 13, 4, 1, 0, -1, 1, 21, 61, 51, 21, 5, 1, 0, 1, 0, 43, 182, 205, 104, 31, 6, 1, 0, -1, 1, 85, 547, 819, 521, 185, 43, 7, 1, 0, 1, 0, 171, 1640, 3277, 2604, 1111, 300, 57, 8, 1, 0, -1, 1, 341, 4921, 13107, 13021, 6665, 2101, 455, 73, 9, 1, 0
COMMENTS
For n >= 1, T(n, k) equals the number of walks of length k between any two distinct vertices of the complete graph K_(n+1). - Peter Bala, May 30 2024
FORMULA
T(n, k) = n^(k-1) - n^(k-2) + n^(k-3) - ... + (-1)^(k-1) = n^(k-1) - T(n, k-1) = n*T(n, k-1) - (-1)^k = (n - 1)*T(n, k-1) + n*T(n, k-2) = round[n^k/(n+1)] for n > 1.
T(n, k) = (-1)^(k+1) * resultant( n*x + 1, (x^k-1)/(x-1) ). - Max Alekseyev, Sep 28 2021 E.g.f. of row n: (exp(n*x) - exp(-x))/(n+1). - Stefano Spezia, Feb 20 2024 Binomial transform of the m-th row: Sum_{k = 0..n} binomial(n, k)*T(m, k) = (m + 1)^(n-1) for n >= 1.
Let R(m, x) denote the g.f. of the m-th row of the square array. Then R(m_1, x) o R(m_2, x) = R(m_1 + m_2 + m_1*m_2, x), where o denotes the black diamond product of power series as defined by Dukes and White. Cf. A109502. T(m_1 + m_2 + m_1*m_2, k) = Sum_{i = 0..k} Sum_{j = i..k} binomial(k, i)* binomial(k-i, j-i)*T(m_1, j)*T(m_2, k-i). (End)
EXAMPLE
Square array begins:
0, 1, -1, 1, -1, 1, -1, 1, ...
0, 1, 0, 1, 0, 1, 0, 1, ...
0, 1, 1, 3, 5, 11, 21, 43, ...
0, 1, 2, 7, 20, 61, 182, 547, ...
0, 1, 3, 13, 51, 205, 819, 3277, ...
0, 1, 4, 21, 104, 521, 2604, 13021, ...
0, 1, 5, 31, 185, 1111, 6665, 39991, ...
0, 1, 6, 43, 300, 2101, 14706, 102943, ... (End)
MAPLE
seq(print(seq((n^k - (-1)^k)/(n+1), k = 0..10)), n = 0..10); # Peter Bala, May 31 2024
MATHEMATICA
T[n_, k_]:=(n^k - (-1)^k)/(n+1); Join[{0}, Table[Reverse[Table[T[n-k, k], {k, 0, n}]], {n, 12}]]//Flatten (* Stefano Spezia, Feb 20 2024 *)
CROSSREFS
Rows include A062157, A000035, A001045, A015518, A015521, A015531, A015540, A015552, A015565, A015577, A015585, A015592, A015609. Related to repunits in negative bases (cf. A055129 for positive bases).
a(n) = n^5 - n^4 + n^3 - n^2 + n - 1.
+10
4
-1, 0, 21, 182, 819, 2604, 6665, 14706, 29127, 53144, 90909, 147630, 229691, 344772, 501969, 711914, 986895, 1340976, 1790117, 2352294, 3047619, 3898460, 4929561, 6168162, 7644119, 9390024, 11441325, 13836446, 16616907, 19827444, 23516129, 27734490, 32537631, 37984352, 44137269
COMMENTS
Number of walks of length 6 between any two distinct nodes of the complete graph K_{n+1} (n>=1). - Emeric Deutsch, Apr 01 2004 For odd n, a(n) * (n+1) / 2 + 1 also represents the first integer in a sum of n^6 consecutive integers that equals n^12. - Patrick J. McNab, Dec 26 2016
FORMULA
a(n) = round(n^6/(n+1)) for n>2 = A062160(n,6). G.f.: (76x^3 + 6x^2 + 27x^4 + 6x^5 + 6x - 1)/(1-x)^6 (for the signed sequence). - Emeric Deutsch, Apr 01 2004 a(0)=-1, a(1)=0, a(2)=21, a(3)=182, a(4)=819, a(5)=2604, a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6). - Harvey P. Dale, Dec 20 2015 E.g.f.: exp(x)*(x^5 + 9*x^4 + 20*x^3 + 10*x^2 + x - 1). - Stefano Spezia, Apr 22 2023
EXAMPLE
a(4) = 4^5 - 4^4 + 4^3 - 4^2 + 4 - 1 = 1024 - 256 + 64 - 16 + 4 - 1 = 819.
MATHEMATICA
LinearRecurrence[{6, -15, 20, -15, 6, -1}, {-1, 0, 21, 182, 819, 2604}, 40] (* Harvey P. Dale, Dec 20 2015 *)
PROG
(PARI) { for (n=0, 1000, write("b062159.txt", n, " ", n*(n*(n*(n*(n - 1) + 1) - 1) + 1) - 1) ) } \\ Harry J. Smith, Aug 02 2009
Square array read by antidiagonals: T(m, n) = Phi_m(n), the m-th cyclotomic polynomial at x=n.
+10
3
1, 1, -1, 1, 0, 1, 1, 1, 2, 1, 1, 2, 3, 3, 1, 1, 3, 4, 7, 2, 1, 1, 4, 5, 13, 5, 5, 1, 1, 5, 6, 21, 10, 31, 1, 1, 1, 6, 7, 31, 17, 121, 3, 7, 1, 1, 7, 8, 43, 26, 341, 7, 127, 2, 1, 1, 8, 9, 57, 37, 781, 13, 1093, 17, 3, 1, 1, 9, 10, 73, 50, 1555, 21, 5461, 82, 73, 1, 1, 1, 10, 11, 91, 65, 2801, 31, 19531, 257, 757, 11, 11, 1, 1, 11, 12, 111, 82, 4681, 43, 55987, 626, 4161, 61, 2047, 1, 1
COMMENTS
Outside of rows 0, 1, 2 and columns 0, 1, only terms of A206942 occur. Conjecture: There are infinitely many primes in every row (except row 0) and every column (except column 0), the indices of the first prime in n-th row and n-th column are listed in A117544 and A117545. (See A206864 for all the primes apart from row 0, 1, 2 and column 0, 1.) Another conjecture: Except row 0, 1, 2 and column 0, 1, the only perfect powers in this table are 121 (=Phi_5(3)) and 343 (=Phi_3(18)=Phi_6(19)).
EXAMPLE
Read by antidiagonals:
m\n 0 1 2 3 4 5 6 7 8 9 10 11 12
------------------------------------------------------
0 1 1 1 1 1 1 1 1 1 1 1 1 1
1 -1 0 1 2 3 4 5 6 7 8 9 10 11
2 1 2 3 4 5 6 7 8 9 10 11 12 13
3 1 3 7 13 21 31 43 57 73 91 111 133 157
4 1 2 5 10 17 26 37 50 65 82 101 122 145
5 1 5 31 121 341 781 ... ... ... ... ... ... ...
6 1 1 3 7 13 21 31 43 57 73 91 111 133
etc.
The cyclotomic polynomials are:
n n-th cyclotomic polynomial
0 1
1 x-1
2 x+1
3 x^2+x+1
4 x^2+1
5 x^4+x^3+x^2+x+1
6 x^2-x+1
...
MATHEMATICA
Table[Cyclotomic[m, k-m], {k, 0, 49}, {m, 0, k}]
PROG
(PARI) t1(n)=n-binomial(floor(1/2+sqrt(2+2*n)), 2)
t2(n)=binomial(floor(3/2+sqrt(2+2*n)), 2)-(n+1)
T(m, n) = if(m==0, 1, polcyclo(m, n))
a(n) = T(t1(n), t2(n))
CROSSREFS
Rows 0-16 are A000012, A023443, A000027, A002061, A002522, A053699, A002061, A053716, A002523, A060883, A060884, A060885, A060886, A060887, A060888, A060889, A060890. Columns 0-13 are A158388, A020500, A019320, A019321, A019322, A019323, A019324, A019325, A019326, A019327, A019328, A019329, A019330, A019331. Indices of primes in n-th row for n = 1-20 are A008864, A006093, A002384, A005574, A049409, A055494, A100330, A000068, A153439, A246392, A162862, A246397, A217070, A250174, A250175, A006314, A217071, A164989, A217072, A250176. Indices of primes in main diagonal is A070519. Cf. A117544 (indices of first prime in n-th row), A085398 (indices of first prime in n-th row apart from column 1), A117545 (indices of first prime in n-th column). Cf. A206942 (all terms (sorted) for rows>2 and columns>1). Cf. A206864 (all primes (sorted) for rows>2 and columns>1).
Square array A(n,k) = (n^(2*k + 1) + 1)/(n + 1), n >= 0, k >= 0, read by antidiagonals.
+10
2
1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 11, 7, 1, 1, 1, 43, 61, 13, 1, 1, 1, 171, 547, 205, 21, 1, 1, 1, 683, 4921, 3277, 521, 31, 1, 1, 1, 2731, 44287, 52429, 13021, 1111, 43, 1, 1, 1, 10923, 398581, 838861, 325521, 39991, 2101, 57, 1, 1, 1, 43691, 3587227, 13421773, 8138021, 1439671
FORMULA
A(n,k) = Sum_{j=0..2*k} (-n)^j.
EXAMPLE
Array begins:
=====================================================================
n/k | 0 1 2 3 4 5 6 ...
----+----------------------------------------------------------------
0 | 1 1 1 1 1 1 1 ...
1 | 1 1 1 1 1 1 1 ...
2 | 1 3 11 43 171 683 2731 ...
3 | 1 7 61 547 4921 44287 398581 ...
4 | 1 13 205 3277 52429 838861 13421773 ...
5 | 1 21 521 13021 325521 8138021 203450521 ...
6 | 1 31 1111 39991 1439671 51828151 1865813431 ...
...
PROG
(PARI) A(n, k) = (n^(2*k + 1) + 1)/(n + 1) \\ Andrew Howroyd, May 03 2023 (Magma) /* as array */ [[&+[(-n)^j: j in [0..2*k]]: k in [0..6]]: n in [0..6]]; // Juri-Stepan Gerasimov, May 06 2023
1, 3, 262657, 387440173, 68719738881, 3814699218751, 101559966746113, 1628413638264057, 18014398643699713, 150094635684419611, 1000000001000000001, 5559917315850179173, 26623333286045024257, 112455406962561892503, 426878854231297789441, 1477891880073843750001
COMMENTS
a(n) = Phi_27(n) where Phi_k(x) is the k-th cyclotomic polynomial.
MATHEMATICA
Table[Cyclotomic[27, n], {n, 0, 17}]
CROSSREFS
Sequences of the type Phi_k(n), where Phi_k is the k-th cyclotomic polynomial: A000012 (k=0), A023443 (k=1), A000027 (k=2), A002061 (k=3), A002522 (k=4), A053699 (k=5), A002061 (k=6), A053716 (k=7), A002523 (k=8), A060883 (k=9), A060884 (k=10), A060885 (k=11), A060886 (k=12), A060887 (k=13), A060888 (k=14), A060889 (k=15), A060890 (k=16), A269442 (k=17), A060891 (k=18), A269446 (k=19), A060892 (k=20), A269483 (k=21), A269486 (k=22), A060893 (k=24), A269527 (k=25), A266229 (k=26), this sequence (k=27), A270204 (k=28), A060894 (k=30), A060895 (k=32), A060896 (k=36). Cf. A153440 (indices of prime terms).
Search completed in 0.013 seconds
|