cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A330513 a(n) = a(n-1) + a(floor(n/4)), a(1)=a(2)=a(3) = 1.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 17, 19, 21, 24, 27, 30, 33, 37, 41, 45, 49, 54, 59, 64, 69, 75, 81, 87, 93, 100, 107, 114, 121, 129, 137, 145, 153, 162, 171, 180, 189, 199, 209, 219, 229, 240, 251, 262, 273, 285, 297, 309, 321, 334, 347
Offset: 1

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Author

Jeffrey Shallit, Dec 16 2019

Keywords

Comments

Also the number of finite sequences b(1..r) satisfying the conditions b(1) = 1, b(i+1) >= 4 b(i) for 1 <= i < r, and b(r) <= n.

Links

Crossrefs

Cf. A033485, A330500.

Programs

  • Maple
    a:= proc(n) option remember;
     `if`(n<4, signum(n), a(n-1)+a(iquo(n, 4)))
    end:
seq(a(n), n=1..75);  # Alois P. Heinz, Dec 16 2019
  • Mathematica
    Nest[Append[#1, #1[[-1]] + #1[[Floor[#2/4] ]] ] & @@ {#, Length@ # + 1} &, {1, 1, 1}, 58] (* Michael De Vlieger, Mar 04 2020 *)

A338380 Replace every term a(n) by the pair [a(n), a(n)] to form a new sequence S: S is the succession of the absolute differences of the starting sequence.

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 13, 18, 23, 16, 9, 19, 29, 42, 55, 73, 91, 68, 45, 61, 77, 86, 95, 114, 133, 104, 75, 117, 159, 214, 269, 342, 415, 324, 233, 165, 97, 142, 187, 248, 309, 232, 155, 241, 327, 422, 517, 631, 745, 612, 479, 375, 271, 196, 121, 238, 355, 514, 673, 887, 1101, 832, 563, 905, 1247, 1662, 2077
Offset: 1

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Author

Eric Angelini and Carole Dubois, Nov 05 2020

Keywords

Comments

This is the lexicographically earliest sequence of distinct positive terms with this property.

Examples

			The successive absolute differences between two successive terms are 1, 1, 2, 2, 3, 3, 5, 5, 7, 7, 10, 10, 13, 13, 18, 18, 23, 23, 16, 16, 9, 9,... which is the sequence itself with every term duplicated.

Links

Crossrefs

Cf. A033485 (same sequence, but strictly monotonically increasing).

A346912 a(0) = 1; a(n) = a(n-1) + a(floor(n/2)) + 1.

Original entry on oeis.org

1, 3, 7, 11, 19, 27, 39, 51, 71, 91, 119, 147, 187, 227, 279, 331, 403, 475, 567, 659, 779, 899, 1047, 1195, 1383, 1571, 1799, 2027, 2307, 2587, 2919, 3251, 3655, 4059, 4535, 5011, 5579, 6147, 6807, 7467, 8247, 9027, 9927, 10827, 11875, 12923, 14119, 15315
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 11 2021

Keywords

Links

Crossrefs

Cf. A000123, A018819, A022907, A033485, A102378, A102379.

Programs

  • Maple
    f:= proc(n) option remember; procname(n-1) + procname(floor(n/2)) + 1 end proc;
f(0):= 1:
map(f, [$1..50]); # Robert Israel, May 04 2025
  • Mathematica
    a[0] = 1; a[n_] := a[n] = a[n - 1] + a[Floor[n/2]] + 1; Table[a[n], {n, 0, 47}]
nmax = 47; CoefficientList[Series[(1/(1 - x)) (-1 + 2 Product[1/(1 - x^(2^k)), {k, 0, Floor[Log[2, nmax]]}]), {x, 0, nmax}], x]
  • Python
    from itertools import islice
from collections import deque
def A346912_gen(): # generator of terms
    aqueue, f, b, a = deque([2]), True, 1, 2
    yield from (1, 3, 7)
    while True:
        a += b
        yield 4*a - 1
        aqueue.append(a)
        if f: b = aqueue.popleft()
        f = not f
A346912_list = list(islice(A346912_gen(),40)) # Chai Wah Wu, Jun 08 2022

Formula

G.f.: (1/(1 - x)) * (-1 + 2 * Product_{k>=0} 1/(1 - x^(2^k))).
a(n) = n + 1 + Sum_{k=1..n} a(floor(k/2)).
a(n) = 2 * A000123(n) - 1.
a(n) = 4 * A033485(n) - 1 for n > 0. - Hugo Pfoertner, Aug 12 2021
From Michael Tulskikh, Aug 12 2021: (Start)
2*a(2n) = a(2n-1) + a(2n+1).
a(2n) = a(2n-2) + a(n-1) + a(n) + 2.
a(2n) = 2*(Sum_{i=0..n} a(i)) - a(n) + 2n. (End)

A347027 a(1) = 1; a(n) = a(n-1) + 2 * a(floor(n/2)).

Original entry on oeis.org

1, 3, 5, 11, 17, 27, 37, 59, 81, 115, 149, 203, 257, 331, 405, 523, 641, 803, 965, 1195, 1425, 1723, 2021, 2427, 2833, 3347, 3861, 4523, 5185, 5995, 6805, 7851, 8897, 10179, 11461, 13067, 14673, 16603, 18533, 20923, 23313, 26163, 29013, 32459, 35905, 39947, 43989
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 11 2021

Keywords

Crossrefs

Cf. A033485, A033489, A058039.
Partial sums of A039722.

Programs

  • Mathematica
    a[1] = 1; a[n_] := a[n] = a[n - 1] + 2 a[Floor[n/2]]; Table[a[n], {n, 1, 47}]
nmax = 47; A[] = 0; Do[A[x] = (x + 2 (1 + x) A[x^2])/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
  • Python
    from collections import deque
from itertools import islice
def A347027_gen(): # generator of terms
    aqueue, f, b, a = deque([3]), True, 1, 3
    yield from (1, 3)
    while True:
        a += 2*b
        yield a
        aqueue.append(a)
        if f: b = aqueue.popleft()
        f = not f
A347027_list = list(islice(A347027_gen(),40)) # Chai Wah Wu, Jun 08 2022

Formula

G.f. A(x) satisfies: A(x) = (x + 2 * (1 + x) * A(x^2)) / (1 - x).
a(n) = 1 + 2 * Sum_{k=2..n} a(floor(k/2)).

A351620 a(1) = 1; a(n) = a(n-1) + Sum_{k=2..n} a(floor(n/k)).

Original entry on oeis.org

1, 2, 4, 8, 13, 22, 32, 48, 67, 93, 120, 164, 209, 266, 331, 418, 506, 626, 747, 905, 1076, 1276, 1477, 1755, 2039, 2370, 2723, 3149, 3576, 4112, 4649, 5295, 5971, 6737, 7519, 8501, 9484, 10590, 11744, 13109, 14475, 16092, 17710, 19561, 21504, 23650, 25797, 28386, 30986, 33903
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 20 2022

Keywords

Comments

Partial sums of A320225.

Crossrefs

Cf. A001519, A022825, A025523, A033485, A078346, A320225, A351621.

Programs

  • Mathematica
    a[1] = 1; a[n_] := a[n] = a[n - 1] + Sum[a[Floor[n/k]], {k, 2, n}]; Table[a[n], {n, 1, 50}]

Formula

G.f. A(x) satisfies: A(x) = x/(1 - x) + (1/(1 - x)^2) * Sum_{k>=2} (1 - x^k) * A(x^k).
Previous Showing 31-35 of 35 results.

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