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0, 13, 26, 39, 52, 65, 78, 91, 104, 117, 130, 143, 156, 169, 182, 195, 208, 221, 234, 247, 260, 273, 286, 299, 312, 325, 338, 351, 364, 377, 390, 403, 416, 429, 442, 455, 468, 481, 494, 507, 520, 533, 546, 559, 572, 585, 598, 611, 624, 637, 650, 663, 676
1, 2, 4, 7, 8, 11, 13, 14, 16, 17, 19, 22, 23, 26, 28, 29, 31, 32, 34, 37, 38, 41, 43, 44, 46, 47, 49, 52, 53, 56, 58, 59, 61, 62, 64, 67, 68, 71, 73, 74, 76, 77, 79, 82, 83, 86, 88, 89, 91, 92, 94, 97, 98, 101, 103, 104, 106, 107, 109, 112, 113, 116, 118, 119
COMMENTS
a(n) - 15*floor((n-1)/8) - 2*((n-1) mod 8) has period 8, repeating [1,0,0,1,0,1,1,0].
Numbers whose odd part is 7-rough: products of terms of A007775 and powers of 2 (terms of A000079). - Peter Munn, Aug 04 2020 The asymptotic density of this sequence is 8/15. - Amiram Eldar, Oct 18 2020
FORMULA
a(n+8) = a(n) + 15.
a(n) = 15*floor((n-1)/8) +2*f(n) +floor(2*phi*(f(n+1)+2)) -2*floor(phi*(f(n+1)+2)), where f(n) = (n-1) mod 8 and phi=(1+sqrt(5))/2.
a(n) = 15*floor((n-1)/8) +2*f(n) +floor((2*f(n)+5)/5) -floor((f(n)+2)/3), where f(n) = (n-1) mod 8.
G.f.: x*(1 +x +2*x^2 +3*x^3 +x^4 +3*x^5 +2*x^6 +x^7 +x^8) / ((1-x)^2*(1+x)*(1+x^2)*(1+x^4)). -
a(n) = a(n-1) +a(n-8) -a(n-9) for n>9. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*sqrt(7 + sqrt(5) - sqrt(6*(5 + sqrt(5))))*Pi/15. - Amiram Eldar, Dec 13 2021
MAPLE
for n from 1 to 500 do if n mod 3<>0 and n mod 5<>0 then print(n) fi od
MATHEMATICA
Select[Range[120], GCD[#, 15] == 1 &] (* or *) t = 70; CoefficientList[Series[(1 + x + 2 x^2 + 3 x^3 + x^4 + 3 x^5 + 2 x^6 + x^7 + x^8)/((1 - x)^2 (1 + x) (1 + x^2) (1 + x^4)) , {x, 0, t}], x] (* Bruno Berselli, Oct 01 2013 *) Select[Range[120], CoprimeQ[#, 15]&] (* Harvey P. Dale, Oct 31 2013 *)
PROG
(Magma) [n: n in [1..120] | IsOne(GCD(n, 15))]; // Bruno Berselli, Oct 01 2013 (Sage) [i for i in range(120) if gcd(i, 15) == 1] # Bruno Berselli, Oct 01 2013
0, 4095, 531440, 16777215, 244140624, 2176782335, 13841287200, 68719476735, 282429536480, 999999999999, 3138428376720, 8916100448255, 23298085122480, 56693912375295, 129746337890624, 281474976710655, 582622237229760, 1156831381426175, 2213314919066160
COMMENTS
a(n) mod 13 = 0 iff n mod 13 > 0; a( A008595(n)) = 12; a( A113763(n)) = 0.
LINKS
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).
FORMULA
a(n) = 13*a(n-1) - 78*a(n-2) + 286*a(n-3) - 715*a(n-4) + 1287*a(n-5) - 1716*a(n-6) + 1716*a(n-7) - 1287*a(n-8) + 715*a(n-9) - 286*a(n-10) + 78*a(n-11) - 13*a(n-12) + a(n-13) for n > 12.
G.f.: x*(4095 + 478205*x + 10187905*x^2 + 66317979*x^3 + 162513078*x^4 + 162511362*x^5 + 66319266*x^6 + 10187190*x^7 + 478491*x^8 + 4017*x^9 + 13*x^10 - x^11)/(1 - x)^13. (End)
Maximal run length in base-12 representation of n.
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1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1
COMMENTS
For all numbers n in A023806, a(n)=1, but this holds also for others, e.g., n=145, not in A023806. The same relation holds for the terms of A113763 less than n=144, but becomes wrong from then on. - M. F. Hasler, Jul 24 2013
PROG
(PARI) A043286(n, b=12){=my(m, c=1); while(n>0, n%b==(n\=b)%b&&c++&&next; m=max(m, c); c=1); m} \\ M. F. Hasler, Jul 23 2013 - typos fixed by Antti Karttunen, Dec 06 2017
Fibonacci sequence beginning 2, 21.
+10
3
2, 21, 23, 44, 67, 111, 178, 289, 467, 756, 1223, 1979, 3202, 5181, 8383, 13564, 21947, 35511, 57458, 92969, 150427, 243396, 393823, 637219, 1031042, 1668261, 2699303, 4367564, 7066867, 11434431, 18501298, 29935729, 48437027, 78372756, 126809783, 205182539, 331992322, 537174861
REFERENCES
Steven Vajda, Fibonacci and Lucas Numbers, and the Golden Section: Theory and Applications, Dover Publications (2008), page 24 (formula 8).
FORMULA
G.f.: (2 + 19*x)/(1 - x - x^2).
a(n) = a(n-1) + a(n-2).
Let g(r,s;n) be the n-th generalized Fibonacci number with initial values r, s. We have:
a(n) = Lucas(n) + g(0,20;n), see A022354; a(n) = Fibonacci(n) + g(2,20;n), see A022372; a(n) = 2*g(1,21;n) - g(0,21;n);
a(n) = g(1,k;n) + g(1,21-k;n) for all k in Z.
a(h+k) = a(h)*Fibonacci(k-1) + a(h+1)*Fibonacci(k) for all h, k in Z (see S. Vajda in References section). For h=0 and k=n:
a(n) = 2*Fibonacci(n-1) + 21*Fibonacci(n).
Sum_{j=0..n} a(j) = a(n+2) - 21.
a(n) = (2^(-n)*((1-sqrt(5))^n*(-20+sqrt(5)) + (1+sqrt(5))^n*(20+sqrt(5)))) / sqrt(5). - Colin Barker, Oct 25 2017
MATHEMATICA
LinearRecurrence[{1, 1}, {2, 21}, 40]
PROG
(Magma) a0:=2; a1:=21; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..40]];
(PARI) Vec((2 + 19*x)/(1 - x - x^2) + O(x^40)) \\ Colin Barker, Oct 25 2017 (Sage)
a = BinaryRecurrenceSequence(1, 1, 2, 21)
CROSSREFS
Sequences of the type g(2,k;n): A118658 (k=0), A000032 (k=1), 2* A000045 (k=2,4), A020695 (k=3), A001060 (k=5), A022112 (k=6), A022113 (k=7), A294157 (k=8), A022114 (k=9), A022367 (k=10), A022115 (k=11), A022368 (k=12), A022116 (k=13), A022369 (k=14), A022117 (k=15), A022370 (k=16), A022118 (k=17), A022371 (k=18), A022119 (k=19), A022372 (k=20), this sequence (k=21), A022373 (k=22); A022374 (k=24); A022375 (k=26); A022376 (k=28), A190994 (k=29), A022377 (k=30); A022378 (k=32).
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