A055010 -id:A055010 - cp's OEIS frontend

A295149 Numbers n such that gcd(A002487(n), A002487(n+2)) > 1.

5, 11, 23, 33, 45, 47, 49, 61, 73, 85, 95, 105, 117, 153, 163, 165, 187, 191, 195, 217, 219, 229, 257, 259, 269, 271, 273, 283, 285, 313, 325, 339, 353, 363, 365, 367, 369, 381, 383, 385, 397, 399, 401, 403, 413, 427, 441, 453, 481, 483, 493, 495, 497, 507

Offset: 1

Rémy Sigrist, Nov 15 2017

nonn

All terms are odd (as for any k > 0, gcd(A002487(2*k), A002487(2*k+2)) = gcd(A002487(k), A002487(k+1)) = 1).
This sequence is infinite as it contains A055010(n) for any n > 1.
For any n > 1, gcd(A002487(A055010(n)), A002487(A055010(n)+2)) = 2*n-1.
For any n > 0, gcd(A002487(a(n)), A002487(a(n)+2)) is odd (as A002487(k) is even iff k is divisible by 3).

Cf. A002487, A055010.

fusc(n)=local(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); b \\ after Charles R Greathouse IV at A002487
for (n=1, 507, if (gcd(fusc(n),fusc(n+2))>1, print1 (n", ")))

A336337 Total number of records over all length n ternary words (words on alphabet {0,1,2}).

Original entry on oeis.org

0, 3, 12, 41, 132, 413, 1272, 3881, 11772, 35573, 107232, 322721, 970212, 2914733, 8752392, 26273561, 78853452, 236625893, 710008752, 2130288401, 6391389492, 19175217053, 57527748312, 172587439241, 517770706332, 1553328896213, 4660020243072, 13980127838081

Offset: 0

Geoffrey Critzer, Jul 18 2020

A record in a word a_1,a_2,...,a_n is a letter a_j that is larger than all the preceding letters. That is, a_j>a_i for all i

Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Cf. A055010, A285852.

nn = 25; Range[0, nn]!; CoefficientList[Series[D[Product[1 + v z/(1 - j z), {j, 1, 3}], v] /. v -> 1, {z, 0, nn}], z]

O.g.f.: x*(-3 + 6*x - 2*x^2)/(-1 + 6*x - 11*x^2 + 6*x^3) = d/dy A(x,y)|y=1 where A(x,y) is the o.g.f. for A285852.
a(n) = Sum_{k=0..3} A285852(n,k)*k.
a(n) = 11/2*3^(n-1)-2^n-1/2, n>0. - R. J. Mathar, Aug 19 2022

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cp's OEIS Frontend

A295149 Numbers n such that gcd(A002487(n), A002487(n+2)) > 1.

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