A144864 -id:A144864 - cp's OEIS frontend

A371100 Array A read by upward antidiagonals in which the entry A(n,k) in row n and column k is defined by A(n, k) = 4^n(6k - 3 - 2*(-1)^n) + (4^n - 1)/3, n,k >= 1.

21, 21, 45, 341, 117, 69, 341, 725, 213, 93, 5461, 1877, 1109, 309, 117, 5461, 11605, 3413, 1493, 405, 141, 87381, 30037, 17749, 4949, 1877, 501, 165, 87381, 185685, 54613, 23893, 6485, 2261, 597, 189, 1398101, 480597, 283989, 79189, 30037, 8021, 2645, 693, 213, 1398101, 2970965, 873813, 382293, 103765, 36181, 9557, 3029, 789, 237

Offset: 1

Antti Karttunen and Ali Sada, Apr 18 2024

			The top left corner of the array:
n\k|      1       2       3        4        5        6        7        8
---+--------------------------------------------------------------------------
1  |     21,     45,     69,      93,     117,     141,     165,     189, ...
2  |     21,    117,    213,     309,     405,     501,     597,     693, ...
3  |    341,    725,   1109,    1493,    1877,    2261,    2645,    3029, ...
4  |    341,   1877,   3413,    4949,    6485,    8021,    9557,   11093, ...
5  |   5461,  11605,  17749,   23893,   30037,   36181,   42325,   48469, ...
6  |   5461,  30037,  54613,   79189,  103765,  128341,  152917,  177493, ...
7  |  87381, 185685, 283989,  382293,  480597,  578901,  677205,  775509, ...
8  |  87381, 480597, 873813, 1267029, 1660245, 2053461, 2446677, 2839893, ...
...

Paolo Xausa, Table of n, a(n) for n = 1..11325 (first 150 antidiagonals, flattened).

Cf. A007283, A079319, A257852, A371094, A371101.
Cf. A372351 (same terms, in different order), A372290 (sorted into ascending order, without duplicates), A372293 (odd numbers that do not occur here).
Leftmost column is A144864 duplicated, without its initial 1.
Row 1: A102603.

A371100[n_, k_] := 4^n*(6*k - 3 - 2*(-1)^n) + (4^n - 1)/3;
Table[A371100[n - k + 1, k], {n, 10}, {k, n}] (* Paolo Xausa, Apr 21 2024 *)

up_to = 55;
A371100sq(n,k) = 4^n*(6*k - 3 - 2*(-1)^n) + (4^n - 1)/3;
A371100list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A371100sq((a-(col-1)),col))); (v); };
v371100 = A371100list(up_to);
A371100(n) = v371100[n];

A(n, k) = A007283(n)*A257852(n,k) + A079319(n).
A(n, k) = A371094(A257852(n,k)).
A(n+2, k) = 5 + 16*A(n,k).

A195156 a(n) = (16^n-1)/3.

Original entry on oeis.org

0, 5, 85, 1365, 21845, 349525, 5592405, 89478485, 1431655765, 22906492245, 366503875925, 5864062014805, 93824992236885, 1501199875790165, 24019198012642645, 384307168202282325, 6148914691236517205, 98382635059784275285, 1574122160956548404565

Offset: 0

Omar E. Pol, Sep 10 2011

Numbers of A002450 that are multiples of 5. Also sequence found by reading the line from 0, in the direction 0, 5,..., in the square spiral whose edges are the Jacobsthal numbers A001045 and whose vertices are the numbers A000975. This is a semi-diagonal in the spiral.
In binary, these numbers are 101...01 (see A031982). - Alonso del Arte, May 20 2017
0 together with Jacobsthal numbers ending with the decimal digit 5. - Jianing Song, Aug 30 2022

Vincenzo Librandi, Table of n, a(n) for n = 0..800
Index entries for linear recurrences with constant coefficients, signature (17,-16).

Bisection of A002450.
Cf. A000975, A001025, A013777, A031982, A131865.
First quadrisection of Jacobsthal numbers A001045; the other quadrisections are A139792 (second), A144864 (third), and A141060 (fourth).

[(16^n-1)/3:n in [0..20]]; // Vincenzo Librandi, Sep 20 2011

A195156:=n->(16^n-1)/3; seq(A195156(k), k=0..50); # Wesley Ivan Hurt, Oct 24 2013

Table[(16^n - 1)/3, {n, 0, 63}] (* Wesley Ivan Hurt, Oct 24 2013 *)

for(n=0,50, print1((16^n - 1)/3, ", ")) \\ G. C. Greubel, Oct 11 2017

From Bruno Berselli, Sep 19 2011: (Start)
G.f.: 5*x/((1-x)*(1-16*x)).
a(n) = A002450(2n) = (16^n-1)/3.
a(n) = 5*A131865(n-1) = a(n-1) + 5*A001025(n-1) = 16*a(n-1) + 5 for n > 0. (End)
From Jianing Song, Aug 30 2022: (Start)
a(n) = A001045(4*n).
a(n+1) - a(n) = 10*A013777(n-1) = 80*A001025(n-1) for n >= 1. (End)
E.g.f.: exp(x)*(exp(15*x) - 1)/3. - Stefano Spezia, Dec 17 2022

New sequence name suggested by Charles R Greathouse IV using Berselli's formula. - Sep 19 2011

A141060 Fourth quadrisection of Jacobsthal numbers A001045: a(n)=16a(n-1)-5.

Original entry on oeis.org

3, 43, 683, 10923, 174763, 2796203, 44739243, 715827883, 11453246123, 183251937963, 2932031007403, 46912496118443, 750599937895083, 12009599006321323, 192153584101141163, 3074457345618258603, 49191317529892137643

Offset: 0

Paul Curtz, Jul 30 2008

Jacobsthal numbers ending with the decimal digit 3. - Jianing Song, Aug 30 2022

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (17,-16).

The other quadrisections of A001045 are A195156 (first), A139792 (second), and A144864 (third).

[(1/3)*(1+8*16^n): n in [0..25]]; // Vincenzo Librandi, May 25 2011

LinearRecurrence[{17,-16},{3,43},30] (* Harvey P. Dale, Mar 16 2015 *)

a(n)=8*16^n\3+1 \\ Charles R Greathouse IV, May 25 2011

def A141060(n): return ((1<<(n<<2)+3)|1)//3 # Chai Wah Wu, Apr 18 2025

a(n) = A139792(n) + A013776(n).
a(n+1) - a(n) = 10*A013709(n) = 40*A001025(n).
G.f.: (3-8*x)/((1-x)*(1-16*x)). [Colin Barker, Apr 05 2012]
a(0)=3, a(1)=43, a(n)=17*a(n-1)-16*a(n-2). - Harvey P. Dale, Mar 16 2015
From Jianing Song, Aug 30 2022: (Start)
a(n) = A001045(4*n+3).
a(n) = 10*A141032(n) + 3 = 20*A098704(n+1) + 1 = 40*A131865(n-1) + 1 for n >= 1. (End)

A139792 First quadrisection of A139763 (1, 2, 3, 4, 11, ...).

Original entry on oeis.org

1, 11, 171, 2731, 43691, 699051, 11184811, 178956971, 2863311531, 45812984491, 733007751851, 11728124029611, 187649984473771, 3002399751580331, 48038396025285291, 768614336404564651, 12297829382473034411, 196765270119568550571, 3148244321913096809131

Offset: 0

Paul Curtz, May 21 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (17,-16)

Second quadrisection of Jacobsthal numbers A001045; the other quadrisections are A195156 (first), A144864 (third), and A141060 (fourth).

[(1+2*16^n)/3: n in [0..20]]; // Vincenzo Librandi, Aug 09 2011

Table[(1 + 2^(4*n+1))/3, {n,0,20}] (* G. C. Greubel, Nov 03 2018 *)

vector(20, n, n--; (1 + 2^(4*n+1))/3) \\ G. C. Greubel, Nov 03 2018

a(n) = 16*a(n-1) - 5.
a(n) = 10*A131865(n) + 1.
G.f.: ( 1-6*x ) / ( (16*x-1)*(x-1) ). - R. J. Mathar, Feb 06 2011
E.g.f.: (exp(x) + 2*exp(16*x))/3. - G. C. Greubel, Nov 03 2018
From Jianing Song, Aug 30 2022: (Start)
a(n) = A001045(4*n+1).
a(n+1) - a(n) = 10*A001025(n). (End)

A372292 Numbers that occur more than once in the odd bisection of A371094.

Original entry on oeis.org

21, 117, 213, 309, 341, 405, 501, 597, 693, 789, 885, 981, 1077, 1173, 1269, 1365, 1461, 1557, 1653, 1749, 1845, 1877, 1941, 2037, 2133, 2229, 2325, 2421, 2517, 2613, 2709, 2805, 2901, 2997, 3093, 3189, 3285, 3381, 3413, 3477, 3573, 3669, 3765, 3861, 3957, 4053, 4149, 4245, 4341, 4437, 4533, 4629, 4725, 4821, 4917

Offset: 1

Antti Karttunen, Apr 26 2024

nonn

Numbers that occur more than once in array A371100.

			21 is present because A371094(1) = A371094(3) = 21.
87381 is present because A371094(85) = A371094(213) = A371094(7281) = A371094(14563) = 87381.
185685 is present because A371094(469) = A371094(15473) = A371094(30947) = 185685.

Antti Karttunen, Table of n, a(n) for n = 1..15532

Setwise difference A372290 \ A372291.

Cf. A144864 (subsequence after its initial 1), A371094, A371100.

A371094(n) = { my(m=1+3*n, e=valuation(m,2)); ((m*(2^e)) + (((4^e)-1)/3)); };
isA372292(n) = if(!(n%2),0,my(c=0); forstep(k=1,n,2,if(A371094(k)==n,c++)); (c>1));

search_up_to = 1398101;
A371094(n) = { my(m=1+3*n, e=valuation(m,2)); ((m*(2^e)) + (((4^e)-1)/3)); };
A372292list(up_to_n) = { my(v=vector((1+up_to_n)/2), x, lista=List([])); forstep(k=1,up_to_n,2,x=A371094(k); if(x <= up_to_n, v[(x+1)/2]++)); for(i=1,(1+up_to_n)/2,if(v[i]>1, listput(lista,i+i-1))); Vec(lista); };
v372292 = A372292list(search_up_to);
A372292(n) = v372292[n];

A144863 Start with 1, then at each step prepend 10 and append 01.

Original entry on oeis.org

1, 10101, 101010101, 1010101010101, 10101010101010101, 101010101010101010101, 1010101010101010101010101, 10101010101010101010101010101, 101010101010101010101010101010101

Offset: 1

Artur Jasinski, Sep 23 2008, Sep 25 2008

Bisection of A094028. - Omar E. Pol, Nov 12 2008
a(n) is also A144864(n) written in base 2. - Omar E. Pol, Nov 13 2008
Quadrisection of A147759. - Omar E. Pol, Nov 16 2008

Index entries for linear recurrences with constant coefficients, signature (10001,-10000).

Cf. A056830, A094028, A135576, A144864, A147759.

a = {}; k = {1}; Do[x = FromDigits[k, 2]; AppendTo[a, FromDigits[RealDigits[x, 2]]]; AppendTo[k, 0]; AppendTo[k, 1]; PrependTo[k, 0]; PrependTo[k, 1], {n, 1, 100}];
Table[FromDigits[RealDigits[1/12 (-4 + 16^n), 2]], {n, 1, 10}]
a = {}; k = 1; Do[AppendTo[a, k]; k = 10000 k + 101, {n, 1, 10}]; a
Table[1/99 (-1 + 100^(-1 + 2 n)), {n, 1, 20}]
LinearRecurrence[{10001,-10000},{1,10101},20] (* Harvey P. Dale, Aug 22 2014 *)

a(n) = (-1+100^(-1+2*n))/99.
If a(n) is interpreted as binary number, (-4+16^n)/12 gives the decimal representation of a(n).
a(n) = 10000*a(n-1)+101, n>1.
G.f.: x*(1+100*x) / ( (10000*x-1)*(x-1) ).

cp's OEIS Frontend

A371100 Array A read by upward antidiagonals in which the entry A(n,k) in row n and column k is defined by A(n, k) = 4^n*(6*k - 3 - 2*(-1)^n) + (4^n - 1)/3, n,k >= 1.

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A195156 a(n) = (16^n-1)/3.

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A141060 Fourth quadrisection of Jacobsthal numbers A001045: a(n)=16a(n-1)-5.

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A139792 First quadrisection of A139763 (1, 2, 3, 4, 11, ...).

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A372292 Numbers that occur more than once in the odd bisection of A371094.

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A144863 Start with 1, then at each step prepend 10 and append 01.

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A371100 Array A read by upward antidiagonals in which the entry A(n,k) in row n and column k is defined by A(n, k) = 4^n(6k - 3 - 2*(-1)^n) + (4^n - 1)/3, n,k >= 1.