A097581 -id:A097581 - cp's OEIS frontend

A201630 a(n) = a(n-1) + 2*a(n-2) with n>1, a(0)=2, a(1)=7.

2, 7, 11, 25, 47, 97, 191, 385, 767, 1537, 3071, 6145, 12287, 24577, 49151, 98305, 196607, 393217, 786431, 1572865, 3145727, 6291457, 12582911, 25165825, 50331647, 100663297, 201326591, 402653185, 805306367, 1610612737, 3221225471, 6442450945, 12884901887

Offset: 0

Bruno Berselli, Dec 03 2011

B. Satyanarayana and K. S. Prasad, Discrete Mathematics and Graph Theory, PHI Learning Pvt. Ltd. (Eastern Economy Edition), 2009, p. 73 (problem 3.3).

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2).

Cf. A001045, A005010, A042948, A097581.

[n le 2 select 5*n-3 else Self(n-1)+2*Self(n-2): n in [1..33]];

Mathematica
```
LinearRecurrence[{1, 2}, {2,7}, 33]
```

a[0]:2$ a[1]:7$ a[n]:=a[n-1]+2*a[n-2]$ makelist(a[n], n, 0, 32);

v=vector(33); v[1]=2; v[2]=7; for(i=3, #v, v[i]=v[i-1]+2*v[i-2]); v

def A201630(n): return 3*2**n - (-1)**n
print([A201630(n) for n in range(31)]) # G. C. Greubel, Feb 07 2025

G.f.: (2+5*x)/((1+x)*(1-2*x)).
a(n) = 3*2^n - (-1)^n.
a(n) = 7 + 2*Sum_{i=0..n-2} a(i), for n>0.
a(n) = A097581(A042948(n+1)).
a(n+2) - a(n) = a(n+1) + a(n) = A005010(n).
E.g.f.: 3*exp(2*x) - exp(-x). - G. C. Greubel, Feb 07 2025

A358935 a(n) is the least k > 0 such that fusc(n) = fusc(n + k) or fusc(n) = fusc(n - k) (provided that n - k >= 0), where "fusc" is Stern's diatomic series (A002487).

Original entry on oeis.org

1, 1, 3, 2, 2, 3, 2, 4, 6, 3, 2, 6, 2, 4, 3, 8, 4, 3, 4, 6, 6, 4, 2, 12, 2, 4, 6, 8, 4, 6, 3, 16, 30, 3, 12, 6, 4, 8, 18, 12, 4, 12, 10, 8, 6, 4, 2, 24, 2, 4, 6, 8, 10, 12, 4, 16, 18, 7, 4, 12, 9, 6, 3, 32, 7, 3, 7, 6, 12, 9, 8, 12, 46, 7, 12, 11, 12, 21, 7

Offset: 1

Rémy Sigrist, Dec 07 2022

nonn

Every positive integer appears infinitely many times in A002487, hence the sequence is well defined.

			The first terms, alongside fusc(n) and the direction where to find the same value, are:
  n   a(n)  fusc(n)  dir
  --  ----  -------  ---
   1     1        1  +
   2     1        1  -
   3     3        2  +
   4     2        1  -
   5     2        3  +
   6     3        2  -
   7     2        3  -
   8     4        1  -
   9     6        4  +
  10     3        3  -
  11     2        5  +
  12     6        2  -

Rémy Sigrist, Table of n, a(n) for n = 1..8192
Rémy Sigrist, PARI program
Index entries for sequences related to Stern's sequences

Cf. A002487, A097581.

PARI
```
See Links section.
```

a(2^k) = 2^(k-1) for any k > 0.

a(n) = 2 iff n belongs to A097581 \ {2}.

cp's OEIS Frontend

A201630 a(n) = a(n-1) + 2*a(n-2) with n>1, a(0)=2, a(1)=7.

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