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A380976 a(0) = 0, a(1) = 1. Thereafter, a(n) = a(n-1) + a(n-2) converted to base n, read in base n+1.

0, 1, 1, 2, 3, 6, 10, 18, 31, 54, 93, 172, 300, 536, 955, 1686, 2976, 5224, 9491, 17089, 30618, 54774, 97553, 172749, 305164, 554749, 982005, 1757750, 3140769, 5584153, 9924579, 17582197, 31100841, 56191241, 99509416, 177595495, 316647651, 564436376, 1002970166

Offset: 0

Kaleb Williams, Feb 10 2025

			a(5) = a(4) + a(3) = 5 = 10_5 -> 10_6 = 6.
a(13) = a(12) + a(11) = 472 = 2A4_13 -> 2A4_14 = 536.

Alois P. Heinz, Table of n, a(n) for n = 0..4302

a:= proc(n) option remember; `if`(n<2, n, (l-> add(l[i]*
     (n+1)^(i-1), i=1..nops(l)))(convert(a(n-1)+a(n-2), base, n)))
    end:
seq(a(n), n=0..43);  # Alois P. Heinz, Feb 12 2025

A380976[n_] := A380976[n] = If[n <= 1, n, FromDigits[IntegerDigits[A380976[n-1] + A380976[n-2], n], n+1]];
Array[A380976, 50, 0] (* Paolo Xausa, Feb 14 2025 *)

from itertools import count, islice
from sympy.ntheory.digits import digits
def fromdigits(digs, base): return sum(d*base**i for i, d in enumerate(digs[::-1]))
def agen(): # generator of terms
    yield from (a := [0, 1])
    for n in count(2):
        yield (an := fromdigits(digits(a[-2] + a[-1], n)[1:], n+1))
        a = [a[-1], an]
print(list(islice(agen(), 37))) # Michael S. Branicky, Feb 11 2025

A375087 Numbers added to cumulative correction term in order for prime numbers to resemble a recursive sequence.

Original entry on oeis.org

0, 1, 0, 4, 2, 4, 2, 0, 8, 2, 4, 8, 2, 0, 4, 10, 2, 4, 8, 0, 4, 4, 2, 10, 10, 2, 4, 2, -8, 14, 12, 8, -2, 10, 6, 2, 8, 4, 4, 10, -2, 10, 8, 4, -6, 2, 20, 14, 2, 0, 8, -2, 6, 10, 6, 10, 2, 4, 8, -4, -2, 20, 16, 2, -8, 12, 10, 14, 8, 0, 2, 8, 8, 8, 4, 2, 10, 4, 2, 16, 2, 10

Offset: 1

Kaleb Williams, Jul 29 2024

At n=1, prime(n+2) = prime(n+1) + prime(n) but thereafter such a form must be reduced by a "correction" amount prime(n+2) = prime(n+1) + prime(n) - A096379(n), and the present sequence is how that correction changes.

			For n = 1: a(1) = p_2 + p_1 - p_3 - (Sum_{i <= 0} a(i)) = p_2 + p_1 - p_3 ==> a(1) = 3 + 2 - 5 = 0 ==> a(1) = 0.
For n = 2: a(2) = p_3 + p_2 - p_4 - (Sum_{i <= 1} a(i)) = p_3 + p_2 - p_4 - a(1) ==> a(2) = 5 + 3 - 7 - 0 = 1 ==> a(2) = 1.
For n = 3: a(3) = p_4 + p_3 - p_5 - (Sum_{i <= 2} a(i)) = p_4 + p_3 - p_5 - (a(1) + a(2)) ==> a(3) = 7 + 5 - 11 - (0 + 1) = 0 ==> a(3) = 0.

Cf. A096379 (partial sums), A066495 (indices of 0's).

lista(nn) = my(va = vector(nn)); for (n=1, nn, va[n] = prime(n+1) + prime(n) - prime(n+2) - sum(i=1, n-1, va[i]);); va; \\ Michel Marcus, Jul 30 2024

a(n) = 2*prime(n+1) - prime(n+2) - prime(n-1), for n>=2.
a(n) = A096379(n) - A096379(n-1), for n>=2.
prime(n+2) = prime(n+1) + prime(n) - Sum_{i=1..n} a(i)
a(n) = prime(n+1) + prime(n) - prime(n+2) - Sum_{i=0..n-1} a(i).

A374610 Smallest result of n in "base-p" using digits 0,1,2 and interpreted as base-3.

Original entry on oeis.org

1, 3, 2, 4, 6, 5, 7, 11, 8, 15, 14, 16, 20, 17, 24, 23, 25, 41, 26, 47, 44, 51, 50, 52, 68, 53, 74, 71, 78, 77, 79, 125, 80, 131, 133, 149, 134, 155, 152, 159, 158, 160, 206, 161, 212, 214, 230, 215, 236, 233, 240, 239, 241, 401, 242, 449, 404, 455, 457, 473

Offset: 2

Kaleb Williams, Jul 13 2024

nonn

A "base-p" representation of n is n = Sum_{i} d_i * prime(i+1) with i >= 0 and with "digits" 0 <= d_i <= 2.
Among possible representations, a(n) is the smallest a(n) = Sum_{i} d_i * 3^i.
The effect is to the lazy representation of n as the sum of up to 2 of each prime, with lazy meaning work from largest to smallest prime and use the fewest of each as long as smaller primes will be sufficient to complete the sum.

			For n = 3, 3 = 1*3 + 0*2 = 10_p => 10_3 = 1*3 + 0*1 = 3_10 = a(3).
For n = 5, 5 = 1*3 + 1*2 = 11_p => 11_3 = 1*3 + 1*1 = 4_10 = a(5).
For n = 19, 19 = 1*7 + 1*5 + 1*3 + 2*2 = 1112_p => 1112_3 = 1*27 + 1*9 + 1*3 + 2*1 = 41_10 = a(19).

A374490 Greatest common divisor of sums of n consecutive cubes.

Original entry on oeis.org

1, 1, 9, 4, 5, 9, 7, 8, 27, 5, 11, 36, 13, 7, 45, 16, 17, 27, 19, 20, 63, 11, 23, 72, 25, 13, 81, 28, 29, 45, 31, 32, 99, 17, 35, 108, 37, 19, 117, 40, 41, 63, 43, 44, 135, 23, 47, 144, 49, 25, 153, 52, 53, 81, 55, 56, 171, 29, 59, 180, 61

Offset: 1

Kaleb Williams, Jul 09 2024

A quasipolynomial of order 12 and degree 2. - Charles R Greathouse IV, Jul 11 2024

			For n=3, the sum of 3 consecutive cubes is S(x) = x^3 + (x+1)^3 + (x+2)^3 which has S(0) = 9 and thereafter remains a multiple of 9 since S(x) - S(x-1) = 9*(x^2 + x + 1), so that the GCD of all S(x) is a(3) = 9.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Wikipedia, Integer-valued polynomial
Kaleb Williams, Proof that 3*n^3 + 15*n is divisible 9 by induction
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,-1).

Cf. A026741 (for consecutive integers), A060789 (for consecutive squares).

Cf. A359380.

f(n,x='x)=n*x^3 + (3/2*n^2 - 3/2*n)*x^2 + (n^3 - 3/2*n^2 + 1/2*n)*x + (1/4*n^4 - 1/2*n^3 + 1/4*n^2)
Polya(P)=my(x=variable(P),D=poldegree(P),f=D!,t=0); forstep(d=D,0,-1, my(c=polcoef(P,d,x)*d!); P-=c*binomial(x,d); t=gcd(t,c); f/=max(d,1)); t
a(n)=Polya(f(n)) \\ Charles R Greathouse IV, Jul 09 2024

From Stefano Spezia, Jul 10 2024: (Start)
G.f.: x*(1 + x + 9*x^2 + 4*x^3 + 5*x^4 + 9*x^5 + 7*x^6 + 8*x^7 + 27*x^8 + 5*x^9 + 11*x^10 + 36*x^11 + 11*x^12 + 5*x^13 + 27*x^14 + 8*x^15 + 7*x^16 + 9*x^17 + 5*x^18 + 4*x^19 + 9*x^20 + x^21 + x^22)/((1 - x)^2*(1 + x)^2*(1 + x^2)^2*(1 - x + x^2)^2*(1 + x + x^2)^2*(1 - x^2 + x^4)^2).
a(A359380(n)) = A359380(n). (End)
a(n) = n/2 if n is 2 or 10 mod 12; a(n) = 3n if n is 0, 3, or 9 mod 12; a(n) = 3n/2 if n = 6 mod 12; and a(n) = n otherwise (if n is 1, 4, 5, 7, 8, or 11 mod 12). In particular, n/2 <= a(n) <= 3n. - Charles R Greathouse IV, Jul 11 2024

a(41)-a(61) from Charles R Greathouse IV, Jul 09 2024

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User: Kaleb Williams

A380976 a(0) = 0, a(1) = 1. Thereafter, a(n) = a(n-1) + a(n-2) converted to base n, read in base n+1.

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A375087 Numbers added to cumulative correction term in order for prime numbers to resemble a recursive sequence.

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A374610 Smallest result of n in "base-p" using digits 0,1,2 and interpreted as base-3.

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A374490 Greatest common divisor of sums of n consecutive cubes.

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