Paul Curtz - cp's OEIS frontend

A383953 a(0) = 4, a(n) = 2*a(n-1) + (-1)^n.

4, 7, 15, 29, 59, 117, 235, 469, 939, 1877, 3755, 7509, 15019, 30037, 60075, 120149, 240299, 480597, 961195, 1922389, 3844779, 7689557, 15379115, 30758229, 61516459, 123032917, 246065835, 492131669, 984263339, 1968526677, 3937053355, 7874106709, 15748213419, 31496426837

Offset: 0

Paul Curtz, Aug 19 2025

Index entries for linear recurrences with constant coefficients, signature (1,2).

Cf. A005015, A052997, A340627.

Bisections give A199210 and A072261.

a[n_] := (11*2^n + (-1)^n)/3; Array[a, 34, 0] (* Amiram Eldar, Aug 20 2025 *)

a(n) = (11*2^n + (-1)^n)/3.
a(n) = A340627(n+1)/2.
a(n) = 2*A052997(n) + 1 for n >= 1.
a(n) = a(n-4) + 55*2^(n-4) for n >= 4.
G.f.: (3*x + 4)/((x + 1)*(1 - 2*x)).
E.g.f: (11*exp(2*x) + exp(-x))/3.

A385178 Triangle T(n,k) read by rows in which the n-th diagonal lists the n-th differences of A001047, 0 <= k <= n.

Original entry on oeis.org

0, 1, 1, 3, 4, 5, 7, 10, 14, 19, 15, 22, 32, 46, 65, 31, 46, 68, 100, 146, 211, 63, 94, 140, 208, 308, 454, 665, 127, 190, 284, 424, 632, 940, 1394, 2059, 255, 382, 572, 856, 1280, 1912, 2852, 4246, 6305, 511, 766, 1148, 1720, 2576, 3856, 5768, 8620, 12866, 19171

Offset: 0

Paul Curtz, Jun 20 2025

			Triangle begins:
    0;
    1,   1;
    3,   4,    5;
    7,  10,   14,   19;
   15,  22,   32,   46,   65;
   31,  46,   68,  100,  146,  211;
   63,  94,  140,  208,  308,  454,  665;
  127, 190,  284,  424,  632,  940, 1394, 2059;
  255, 382,  572,  856, 1280, 1912, 2852, 4246,  6305;
  511, 766, 1148, 1720, 2576, 3856, 5768, 8620, 12866, 19171;
  ...

Columns k=0..2: A000225, A033484, A053209 (sans 1).
Diagonals: A001047, A027649, A053581 (sans 1), A291012 (sans 2).
Cf. A028243 (row sums), A036561, A059268, A081656, A248216, A321003.

/* As triangle */ [[2^(n-k)*3^k - 2^k : k in [0..n]]: n in [0..9]]; // Vincenzo Librandi, Jun 27 2025

T:= proc(n,k) option remember;
     `if`(n=k, 3^n-2^n, T(n, k+1)-T(n-1, k))
    end:
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Jun 24 2025

t[n_, 0] := 3^n - 2^n; t[n_, k_] := t[n, k] = t[n + 1, k - 1] - t[n, k - 1]; Table[t[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Jun 20 2025 *)

T(n,n) = 3^n - 2^n = A001047(n).
T(n,k) = T(n,k+1) - T(n-1,k) for 0 <= k < n.
T(n,k) = 2^(n-k)*3^k - 2^k = A036561(n,k) - A059268(n,k).
T(2n,n) = A248216(n+1).

A380384 a(0)=0, a(n) = 2*(a(n-1) + ceiling(n/2)) - 1 for n>0.

Original entry on oeis.org

0, 1, 3, 9, 21, 47, 99, 205, 417, 843, 1695, 3401, 6813, 13639, 27291, 54597, 109209, 218435, 436887, 873793, 1747605, 3495231, 6990483, 13980989, 27962001, 55924027, 111848079, 223696185, 447392397, 894784823, 1789569675, 3579139381, 7158278793, 14316557619

Offset: 0

Paul Curtz, Jan 23 2025

Index entries for linear recurrences with constant coefficients, signature (3,-1,-3,2).

Cf. A081254, A109613, A250777.

LinearRecurrence[{3, -1, -3, 2}, {0, 1, 3, 9}, 40] (* Amiram Eldar, Jan 24 2025 *)

G.f.: x*(x^2+1)/((x+1)*(1-2*x)*(x-1)^2).
a(n) = floor(2^n*5/3) - n - 1.
a(n) = A081254(n+1) - n - 1.
a(n) = a(n-4) + A250777(n-3).

A379530 a(n) = (A135318(3n) + A135318(3n+1) + A135318(3*n+2))/3.

Original entry on oeis.org

1, 3, 8, 23, 64, 185, 512, 1479, 4096, 11833, 32768, 94663, 262144, 757305, 2097152, 6058439, 16777216, 48467513, 134217728, 387740103, 1073741824, 3101920825, 8589934592, 24815366599, 68719476736, 198522932793, 549755813888, 1588183462343, 4398046511104, 12705467698745

Offset: 0

Paul Curtz, Dec 24 2024

Index entries for linear recurrences with constant coefficients, signature (0,7,0,8).

Cf. A001018, A013730, A015565, A135318.

LinearRecurrence[{0, 7, 0, 8}, {1, 3, 8, 23}, 30] (* Amiram Eldar, Dec 31 2024 *)

a(n) = 7*a(n-2) + 8*a(n-4) with a(0)=1, a(1)=3, a(2)=8, a(3)=23 for n >= 4.
a(2*n) = A001018(n).
a(2*n+1) = A015565(n+1) + A013730(n).

A378120 a(n) = (A000217(n) + A005132(n))/2.

Original entry on oeis.org

0, 1, 3, 6, 6, 11, 17, 24, 24, 33, 33, 44, 44, 57, 57, 72, 72, 89, 107, 126, 126, 147, 147, 147, 171, 171, 197, 197, 225, 225, 255, 255, 287, 320, 354, 354, 390, 390, 390, 429, 429, 470, 470, 513, 513, 558, 558, 605, 605, 654, 654, 705, 705, 758, 758, 813, 813

Offset: 0

Paul Curtz, Nov 17 2024

nonn

a(n) is the sum of the up steps in the first n terms of Recaman's sequence A005132.

Index entries for sequences related to Recamán's sequence

Cf. A000217, A005132, A377748.

a(n) = A005132(n) + A377748(n).

A377748 a(n) = (A000217(n) - A005132(n))/2.

Original entry on oeis.org

0, 0, 0, 0, 4, 4, 4, 4, 12, 12, 22, 22, 34, 34, 48, 48, 64, 64, 64, 64, 84, 84, 106, 129, 129, 154, 154, 181, 181, 210, 210, 241, 241, 241, 241, 276, 276, 313, 351, 351, 391, 391, 433, 433, 477, 477, 523, 523, 571, 571, 621, 621, 673, 673, 727, 727, 783, 783, 841, 841

Offset: 0

Paul Curtz, Nov 06 2024

nonn

a(n) is the sum of the down steps in the first n terms of Recamán's sequence A005132.

			a(0) = (0-0)/2, a(1) = (1-1)/2, a(2) = (3-3)/2, a(3) = (6-6)/2, a(4) = (10-2)/2, a(5) = (15-7)/2 ... .

Index entries for sequences related to Recamán's sequence

Cf. A000217, A005132, A160356.

a(n) = Sum_{k=1..n} max(0, -A160356(k)).

A375966 Powers of 3 alternating with powers of 4.

Original entry on oeis.org

1, 1, 3, 4, 9, 16, 27, 64, 81, 256, 243, 1024, 729, 4096, 2187, 16384, 6561, 65536, 19683, 262144, 59049, 1048576, 177147, 4194304, 531441, 16777216, 1594323, 67108864, 4782969, 268435456, 14348907, 1073741824, 43046721, 4294967296, 129140163, 17179869184

Offset: 0

Paul Curtz, Sep 04 2024

Index entries for linear recurrences with constant coefficients, signature (0,7,0,-12).

Cf. A000244 and A000302 interleaved.

Cf. A000035, A059841, A072345, A098293.

seq[len_] := Module[{m = Ceiling[len/2] - 1}, Riffle @@ Map[#^Range[0, m] &, {3, 4}]]; seq[36] (* Amiram Eldar, Sep 05 2024 *)

def A375966(n): return 1<<(n^1) if n&1 else 3**(n>>1) # Chai Wah Wu, Sep 24 2024

a(n) = 7*a(n-2) - 12*a(n-4) for n >= 4.
From Stefano Spezia, Sep 06 2024: (Start)
G.f.: (1 + x - 4*x^2 - 3*x^3)/((1 - 2*x)*(1 + 2*x)*(1 - 3*x^2)).
a(n) = (4*3^(n/2)*A059841(n) - (-2)^n + 2^n)/4.
E.g.f.: cosh(sqrt(3)*x) + cosh(x)*sinh(x). (End)

A375476 a(3n)=A001045(n+1), a(3n+1)=A084214(n), a(3*n+2)=A000079(n) for n >= 0.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 4, 4, 5, 6, 8, 11, 14, 16, 21, 26, 32, 43, 54, 64, 85, 106, 128, 171, 214, 256, 341, 426, 512, 683, 854, 1024, 1365, 1706, 2048, 2731, 3414, 4096, 5461, 6826, 8192, 10923, 13654, 16384, 21845, 27306, 32768, 43691, 54614, 65536

Offset: 0

Paul Curtz, Aug 18 2024

Index entries for linear recurrences with constant coefficients, signature (0,0,1,0,0,2).

Cf. A000079, A001045, A084214.

G.f. 1 + x - x^2*(1+x+x^2+x^3+2*x^4+3*x^5) / ( (1+x)*(2*x^3-1)*(x^2-x+1) ).

A374927 a(n) = A135318(n)^2.

Original entry on oeis.org

1, 1, 1, 4, 9, 16, 25, 64, 121, 256, 441, 1024, 1849, 4096, 7225, 16384, 29241, 65536, 116281, 262144, 466489, 1048576, 1863225, 4194304, 7458361, 16777216, 29822521, 67108864, 119311929, 268435456, 477204025, 1073741824, 1908903481, 4294967296, 7635439161

Offset: 0

Paul Curtz, Jul 24 2024

A374098 terms swapped by pairs.

Index entries for linear recurrences with constant coefficients, signature (0,3,0,6,0,-8).

Cf. A000302, A112387, A135318, A139818, A374098.

a(2*n) = A139818(n+1).
a(2*n+1) = 4^n = A000302(n).
a(n) = 3*a(n-2) +6*a(n-4) -8*a(n-6).

A374098 a(n) = A112387(n)^2.

Original entry on oeis.org

1, 1, 4, 1, 16, 9, 64, 25, 256, 121, 1024, 441, 4096, 1849, 16384, 7225, 65536, 29241, 262144, 116281, 1048576, 466489, 4194304, 1863225, 16777216, 7458361, 67108864, 29822521, 268435456, 119311929, 1073741824, 477204025, 4294967296, 1908903481, 17179869184

Offset: 0

Paul Curtz, Jun 28 2024

Index entries for linear recurrences with constant coefficients, signature (0,3,0,6,0,-8).

Cf. A000302, A003683, A084175, A112387, A139818.

LinearRecurrence[{0, 3, 0, 6, 0, -8}, {1, 1, 4, 1, 16, 9}, 35] (* Amiram Eldar, Jul 01 2024 *)

a(2*n) = A000302(n); a(2*n+1) = A139818(n+1).
(a(2*n) + a(2*n-1))^2 = A084175(n+1)^2 + 16*A003683(n)^2, for n >= 1. - Thomas Scheuerle, Jun 28 2024
G.f. ( 1+x+x^2-2*x^3-2*x^4 ) / ( (x-1)*(2*x+1)*(2*x-1)*(1+x)*(2*x^2+1) ). - R. J. Mathar, Aug 02 2024

cp's OEIS Frontend

User: Paul Curtz

A383953 a(0) = 4, a(n) = 2*a(n-1) + (-1)^n.

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A385178 Triangle T(n,k) read by rows in which the n-th diagonal lists the n-th differences of A001047, 0 <= k <= n.

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A380384 a(0)=0, a(n) = 2*(a(n-1) + ceiling(n/2)) - 1 for n>0.

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A379530 a(n) = (A135318(3*n) + A135318(3*n+1) + A135318(3*n+2))/3.

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A378120 a(n) = (A000217(n) + A005132(n))/2.

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A377748 a(n) = (A000217(n) - A005132(n))/2.

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A375966 Powers of 3 alternating with powers of 4.

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A375476 a(3*n)=A001045(n+1), a(3*n+1)=A084214(n), a(3*n+2)=A000079(n) for n >= 0.

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A374927 a(n) = A135318(n)^2.

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A379530 a(n) = (A135318(3n) + A135318(3n+1) + A135318(3*n+2))/3.

A375476 a(3n)=A001045(n+1), a(3n+1)=A084214(n), a(3*n+2)=A000079(n) for n >= 0.