A001477 -id:A001477 - cp's OEIS frontend

A130845 a(4n) = a(4n+1) = a(4n+2) = A001477(n), a(4n+3) = A005408(n).

0, 0, 0, 1, 1, 1, 1, 3, 2, 2, 2, 5, 3, 3, 3, 7, 4, 4, 4, 9, 5, 5, 5, 11, 6, 6, 6, 13, 7, 7, 7, 15, 8, 8, 8, 17, 9, 9, 9, 19, 10, 10, 10, 21, 11, 11, 11, 23, 12, 12, 12, 25, 13, 13, 13, 27, 14, 14, 14, 29, 15, 15, 15, 31, 16, 16, 16, 33, 17, 17, 17

Offset: 0

Paul Curtz, Jul 20 2007

nonn

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

CoefficientList[Series[x^3(1+x+x^2+x^3+x^4)/((1-x)^2(1+x)^2(1+x^2)^2),{x,0,80}],x] (* or *) LinearRecurrence[{0,0,0,2,0,0,0,-1},{0,0,0,1,1,1,1,3},80] (* Harvey P. Dale, Mar 04 2012 *)

O.g.f.: x^3*(1+x+x^2+x^3+x^4)/((1-x)^2*(1+x)^2*(1+x^2)^2). - R. J. Mathar, Aug 22 2008
a(0)=0, a(1)=0, a(2)=0, a(3)=1, a(4)=1, a(5)=1, a(6)=1, a(7)=3, a(n)=2*a(n-4)- a(n-8). - Harvey P. Dale, Mar 04 2012
a(n) = cos(n*Pi/2)/4-(n-1)*(2*sin(n*Pi/2)+(-1)^n-5)/16. - Wesley Ivan Hurt, May 05 2021

Edited by N. J. A. Sloane, Sep 28 2007

A182440 Table, read by antidiagonals, in which the n-th row comprises A214206(n) in position 0 followed by a second order recursive series G in which each product G(i)*G(i+1) lies in the same row of A001477 (interpreted as a square array).

Original entry on oeis.org

0, 14, 4, 0, 14, 7, 16, 1, 14, 8, 126, 40, 2, 14, 10, 770, 287, 60, 3, 14, 11, 4524, 1730, 420, 72, 4, 14, 12, 26404, 10141, 2522, 497, 88, 5, 14, 13, 153930, 59164, 14774, 2978, 602, 100, 6, 14, 14, 897206

Offset: 0

Kenneth J Ramsey, Apr 28 2012

This is a table related to A001477 interpreted as a square array of the onnegative integers (A001477). Each row k contains A003056(14*k) in column 0 and a corresponding 2nd order recursive sequence G(k) beginning at position a(k,1) such that G(i) = a(k,i+1). If the product 14*k appears in row "r" of the square array A001477, then the product of adjacent terms G(i)*G(i+1) if greater than (r^2 + 3*r - 2)/2, is always in row "r" of square array A001477.
A property of this table is that a(k+1,i)-a(k,i) directly depends on the value of a(k+1,0)-a(k,0) in the same manner regardless of the value of k. For instance, a(k+1,i+1)-a(k,i+1 = A210695(i) if a(k + 1,0) - a(k,0) = 1; while a(k+1,i+1)-a(k,i+1 = A001108(i) if a(k+1,0) - a(k,0) = 0.
A related property is that a(k+1,1+n) - a(k,1+n) is divisible by A143608(n) for all k.

			For i = 1,2,3,4 ..., a(1,i)*a(1,i+1) = 14*1,1*40,40*287,287*1730, ...; and, each product is 4 more than a triangular number and thus lies in row 4 of square array A001477.

Cf. A001108, A210695, A143608, A182431, A182439, A182441

highTri = Compile[{{S1,_Integer}},Module[{xS0=0,xS1=S1},
While[xS1-xS0*(xS0+1)/2>xS0,xS0++];
xS0]];
overTri = Compile[{{S2,_Integer}},Module[{xS0=0,xS2=S2},
While[xS2-xS0*(xS0+1)/2>xS0,xS0++];
xS2 - (xS0*(1+xS0)/2)]];
K1 = 0;
m = 14;table=Reap[While[K1<16,J1=highTri[m*K1];X = 2*(m+K1+(J1*2+1));K2 = (6 K1 - m + X);K3 = 6 K2 - K1 + X;
K4 =  6 K3 - K2 + X; K5 = 6 K4 -K3 + X; K6 = 6*K5 - K4 + X;K7 = 6*K6-K5+X; K8 = 6*K7-K6+X; Sow[J1,c];Sow[m,d];
Sow[K1,e];Sow[K2,f];Sow[K3,g];Sow[K4,h];
  Sow[K5,i]; Sow[K6,j];Sow[K7,k];Sow[K8,l];
K1++]][[2]];
a=1;
list5 = Reap[While[a<11,b=a;
While[b>0,Sow[table[[b,a+1-b]]];b--];a++]][[2,1]];
list5

a(k,0) equals the positive argument of the largest triangular number equal to or less than 14*k (= A214206(k) which = A003056(14*k)).
a(k,1) equals 14; a(k,2) = k.
For i > 2, a(k,i) = 6*a(k,i-1) -a (k,i-2) + G_k where G_k is a constant equal to 28 + 2*k + 2 + 4*A214206(k).

A213268 Denominators of the Inverse semi-binomial transform of A001477(n) read downwards antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 4, 4, 1, 2, 2, 8, 2, 1, 1, 4, 1, 16, 16, 1, 2, 1, 8, 8, 32, 16, 1, 1, 4, 4, 16, 8, 64, 64, 1, 2, 2, 8, 4, 32, 32, 128, 16, 1, 1, 4, 2, 16, 16, 64, 8, 256, 256, 1, 2, 1, 8, 8, 32, 4, 128, 128, 512, 256, 1, 1, 4, 4, 16, 2, 64, 64, 256, 128, 1024, 1024

Offset: 0

Paul Curtz, Jun 08 2012

Starting from any sequence a(k) in the first row, define the array T(n,k) of the inverse semi-binomial transform  by T(0,k) = a(k), T(n,k) = T(n-1,k+1) -T(n-1,k)/2, n>=1.
Here, where the first row is the nonnegative integers, the array is
0        1      2      3    4      5      6      7     8   =A001477(n)
1       3/2     2    5/2    3    7/2      4    9/2     5   =A026741(n+2)/A000034(n)
1       5/4   3/2    7/4    2    9/4    5/2   11/4     3   =A060819(n+4)/A176895(n)
3/4     7/8     1    9/8  5/4   11/8    3/2   13/8   7/4   =A106609(n+6)/A205383(n+6)
1/2    9/16   5/8  11/16  3/4  13/16    7/8  15/16     1   =A106617(n+8)/TBD
5/16  11/32   3/8  13/32 7/16  15/32    1/2  17/32  9/16
3/16  13/64  7/32  15/64  1/4  17/64   9/32  19/64  5/16
7/64 15/128   1/8 17/128 9/64 19/128   5/32 21/128 11/64
1/16 17/256 9/128 19/256 5/64 21/256 11/128 23/256  3/32.
The first column contains 0, followed by fractions A000265/A084623, that is Oresme numbers n/2^n multiplied by 2 (see A209308).

			The array of denominators starts:
  1   1   1   1   1   1   1   1   1   1   1 ...
  1   2   1   2   1   2   1   2   1   2   1 ...
  1   4   2   4   1   4   2   4   1   4   2 ...
  4   8   1   8   4   8   2   8   4   8   1 ...
  2  16   8  16   4  16   8  16   1  16   8 ...
16  32   8  32  16  32   2  32  16  32   8 ...
16  64  32  64   4  64  32  64  16  64  32 ...
64 128   8 128  64 128  32 128  64 128  16 ...
16 256 128 256  64 256 128 256  32 256 128 ...
256 512 128 512 256 512  64 512 256 512 128 ...
All entries are powers of 2.

A213268frac := proc(n,k)
        if n = 0 then
                return k ;
        else
                return procname(n-1,k+1)-procname(n-1,k)/2 ;
        end if;
end proc:
A213268 := proc(n,k)
        denom(A213268frac(n,k)) ;
end proc: # R. J. Mathar, Jun 30 2012

T[0, k_] := k; T[n_, k_] := T[n, k] = T[n-1, k+1] - T[n-1, k]/2; Table[T[n-k, k] // Denominator, {n, 0, 11}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Sep 12 2014 *)

A224791 Triangle T(n,k) read by rows: left edge is 0, 1, 2, ... (cf. A001477); otherwise each entry is sum of entry to left and entries immediately above it to left and right, with 1 for the missing right term at right edge.

Original entry on oeis.org

0, 1, 2, 2, 5, 8, 3, 10, 23, 32, 4, 17, 50, 105, 138, 5, 26, 93, 248, 491, 630, 6, 37, 156, 497, 1236, 2357, 2988, 7, 50, 243, 896, 2629, 6222, 11567, 14556, 8, 65, 358, 1497, 5022, 13873, 31662, 57785, 72342, 9, 82, 505, 2360, 8879, 27774, 73309, 162756

Offset: 0

Reinhard Zumkeller, Apr 18 2013

			Triangle begins:
  0;
  1,  2;
  2,  5,  8;
  3, 10, 23,  32;
  4, 17, 50, 105, 138;

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
Index entries for triangles and arrays related to Pascal's triangle

Cf. A051601, A059283.

a224791 n k = a224791_tabl !! n !! k
a224791_row n = a224791_tabl !! n
a224791_tabl = iterate
   (\row -> scanl1 (+) $ zipWith (+) ([1] ++ row) (row ++ [1])) [0]

T:= proc(n, k) option remember;
      if k=0 then n
    elif k=n then T(n,n-1) + T(n-1,n-1) + 1
    else T(n,k-1) + T(n-1,k-1) + T(n-1, k)
      fi
    end:
seq(seq(T(n, k), k=0..n), n=0..12); # G. C. Greubel, Nov 12 2019

T[n_, k_]:= T[n, k]= If[k==0, n, If[k==n , T[n, n-1] + T[n-1, n-1] + 1, T[n, k-1] + T[n-1, k-1] + T[n-1, k]]]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 12 2019 *)

T(n,k) = if(k==0, n, if(k==n, T(n,n-1) + T(n-1,n-1) + 1, T(n,k-1) + T(n-1,k-1) + T(n-1, k) )); \\ G. C. Greubel, Nov 12 2019

@CachedFunction
def T(n, k):
    if (k==0): return n
    elif (k==n): return T(n,n-1) + T(n-1,n-1) + 1
    else: return T(n,k-1) + T(n-1,k-1) + T(n-1, k)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 12 2019

T(n,0) = n, T(n+1,k) = T(n+1,k-1) + T(n,k-1) + T(n,k) (0 < k <= n) and T(n+1,n+1) = T(n+1,n) + T(n,n) + 1.

A131508 2*A000027 (natural numbers) sandwiched by tripled A001477 (nonnegative numbers).

Original entry on oeis.org

0, 0, 0, 2, 1, 1, 1, 4, 2, 2, 2, 6, 3, 3, 3, 8, 4, 4, 4, 10, 5, 5, 5, 12, 6, 6, 6, 14, 7, 7, 7, 16, 8, 8, 8, 18, 9, 9, 9, 20, 10, 10, 10, 22, 11, 11, 11, 24, 12, 12, 12, 26, 13, 13, 13, 28, 14, 14, 14, 30, 15, 15, 15, 32, 16, 16, 16, 34, 17, 17, 17, 36, 18, 18, 18, 38, 19, 19, 19, 40, 20

Offset: 0

Paul Curtz, Aug 13 2007

Muniru A Asiru, Table of n, a(n) for n = 0..5000

a:=[0,0,0,2,1,1,1,4];; for n in [9..85] do a[n]:=2*a[n-4]-a[n-8]; od; a; # Muniru A Asiru, Oct 24 2018

seq(coeff(series(x^3*(x^3+x^2+x+2)/(x^8-2*x^4+1),x,n+1), x, n), n = 0 .. 85); # Muniru A Asiru, Oct 24 2018

From Chai Wah Wu, Oct 24 2018: (Start)
a(n) = 2*a(n-4) - a(n-8) for n > 7.
G.f.: x^3*(x^3 + x^2 + x + 2)/(x^8 - 2*x^4 + 1). (End)

A342385 Triangle T(n,k) read by rows (n >= 0, 0 <= k <= n): first column is A001477 and column k > 0 is k*A028310.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 3, 2, 2, 3, 4, 3, 4, 3, 4, 5, 4, 6, 6, 4, 5, 6, 5, 8, 9, 8, 5, 6, 7, 6, 10, 12, 12, 10, 6, 7, 8, 7, 12, 15, 16, 15, 12, 7, 8, 9, 8, 14, 18, 20, 20, 18, 14, 8, 9, 10, 9, 16, 21, 24, 25, 24, 21, 16, 9, 10, 11, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 11

Offset: 0

Paul Curtz, Mar 10 2021

Row sums: A177787(n) with A177787(0) = 0.
First subtriangle: A003991(n+1).
Second subtriangle: A173395(n+1).
A000290(n) is a subsequence via (2*n+1)-th rows.

			Triangle begins:
  n\k|  0   1   2   3   4   5   6   7   8   9 ...
  ---+-------------------------------------------
   0 |  0;
   1 |  1,  1;
   2 |  2,  1,  2;
   3 |  3,  2,  2,  3;
   4 |  4,  3,  4,  3,  4;
   5 |  5,  4,  6,  6,  4,  5;
   6 |  6,  5,  8,  9,  8,  5,  6;
   7 |  7,  6, 10, 12, 12, 10,  6,  7;
   8 |  8,  7, 12, 15, 16, 15, 12,  7,  8;
   9 |  9,  8, 14, 18, 20, 20, 18, 14,  8,  9;
  ...

Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of the triangle, flattened)

Cf. A000290, A001477, A003991, A028310, A173395, A177787.

With[{rowmax=10},Table[If[0Paolo Xausa, Nov 15 2023 *)

From Paolo Xausa, Nov 15 2023: (Start)
T(n,0) = T(n,n) = n.
T(n,k) = k*(n-k), for 0 < k < n. (End)

Name edited by Paolo Xausa, Nov 15 2023

A358711 Autobiographical numbers: let the k-th digit count the k-th nonnegative integer (A001477(k)) (possibly overlapping) occurrences in the term.

Original entry on oeis.org

1210, 2020, 21200, 3211000, 42101000, 521001000, 6210001000, 53110100002, 62200010001, 541011000021, 6401101000310, 74011001003100, 840110001031000, 9321000001201000, 94201000012110000

Offset: 1

Marc Morgenegg, Nov 28 2022

The k-th digit must count the k-th nonnegative integer (A001477(k)) appearances in the term.
This sequence is in base b=10. The number of appearances of any integer is always less than b in a term. E.g., the integer '0' can appear at most 9 times in a term.
There are no further terms. This was verified with a computer search of all (permutations of) partitions of d = 1..90 using up to 9 of any digit 0..9 and all (permutations of) "completions" of the remaining d-10 digits consistent with these digit counts. It was verified in each of the two cases for counting appearances: without overlaps (1111 has 2 11's) and with overlaps allowed (1111 has 3 11's). - Michael S. Branicky, Dec 02 2022

			6401101000310 is a term: we have six 0's, four 1's, zero 2's, one 3, one 4, one 6, three 10's and one 11 as integers in the term, visualized as follows:
  Digits(k): 0 1 2 3 4 5 6 7 8 9 10 11 12 (also the Integers(k))
  term:      6 4 0 1 1 0 1 0 0 0  3  1  0
Note that overlapping integers are counted as well: e.g., 110 is one 11, one 10. 111 is two 11's.

Michael S. Branicky, Python program

Cf. A046043, A138480, A001477.

a(8) inserted and a(10)-a(15) by Michael S. Branicky, Nov 28 2022

A131308 Alternate A001477 and tripled 2*A000027.

Original entry on oeis.org

0, 2, 2, 2, 1, 4, 4, 4, 2, 6, 6, 6, 3, 8, 8, 8, 4, 10, 10, 10, 5, 12, 12, 12, 6, 14, 14, 14, 7, 16, 16, 16, 8, 18, 18, 18, 9, 20, 20, 20, 10, 22, 22, 22, 11, 24, 24, 24, 12, 26, 26, 26, 13, 28, 28, 28, 14, 30, 30, 30, 15, 32, 32, 32, 16, 34, 34, 34, 17, 36, 36, 36, 18, 38, 38, 38, 19

Offset: 0

Paul Curtz, Sep 29 2007

nonn

A168496 The positions of non-single or nonisolated numbers in A001477.

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 39, 40, 41, 42, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 67, 69, 70, 71, 72, 74, 75, 76, 77, 78, 79, 81, 82, 83, 85, 86, 87

Offset: 1

Juri-Stepan Gerasimov, Nov 27 2009

Cf. A001477, A167707.

a(n) = 1+A167707(n).

Formula corrected by R. J. Mathar, Jun 04 2010

A360489 Convolution of A000219 and A001477.

Original entry on oeis.org

0, 1, 3, 8, 19, 43, 91, 187, 369, 711, 1335, 2459, 4442, 7904, 13851, 23965, 40958, 69248, 115872, 192097, 315652, 514485, 832112, 1336214, 2131099, 3377178, 5319290, 8330147, 12973662, 20100411, 30986772, 47542096, 72609729, 110410791, 167186826, 252138816, 378781852

Offset: 0

Vaclav Kotesovec, Feb 09 2023

nonn

In general, for 0 < p < 1, delta > 1, beta > -1, the convolution of (delta^(n^p) * n^alfa) and n^beta is asymptotic to delta^(n^p) * n^(alfa + (1-p)*(beta+1)) * Gamma(beta+1) / (p^(beta+1) * log(delta)^(beta+1)).

For p = 1 is the convolution of (delta^(n^p) * n^alfa) and n^beta asymptotic to delta^n * n^alfa * polylog(-beta, 1/delta).

Cf. A000219, A001477, A014153, A095944, A161870.

b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-j)*numtheory[sigma][2](j), j=1..n)/n)
    end:
a:= n-> add(b(n-j)*j, j=0..n):
seq(a(n), n=0..42);  # Alois P. Heinz, Feb 09 2023

nmax = 50; CoefficientList[Series[x/(1-x)^2 * Product[1/(1 - x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]

a(n) = Sum_{k=0..n} A000219(k) * (n-k).
G.f.: x/(1-x)^2 * Product_{k>=1} 1/(1 - x^k)^k.
a(n) ~ exp(1/12 + 3*zeta(3)^(1/3) * n^(2/3) / 2^(2/3)) / (A * sqrt(3*Pi) * 2^(35/36) * zeta(3)^(17/36) * n^(1/36)), where A is the Glaisher-Kinkelin constant A074962.

cp's OEIS Frontend

A130845 a(4n) = a(4n+1) = a(4n+2) = A001477(n), a(4n+3) = A005408(n).

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A182440 Table, read by antidiagonals, in which the n-th row comprises A214206(n) in position 0 followed by a second order recursive series G in which each product G(i)*G(i+1) lies in the same row of A001477 (interpreted as a square array).

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A213268 Denominators of the Inverse semi-binomial transform of A001477(n) read downwards antidiagonals.

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A224791 Triangle T(n,k) read by rows: left edge is 0, 1, 2, ... (cf. A001477); otherwise each entry is sum of entry to left and entries immediately above it to left and right, with 1 for the missing right term at right edge.

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A131508 2*A000027 (natural numbers) sandwiched by tripled A001477 (nonnegative numbers).

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A342385 Triangle T(n,k) read by rows (n >= 0, 0 <= k <= n): first column is A001477 and column k > 0 is k*A028310.

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A358711 Autobiographical numbers: let the k-th digit count the k-th nonnegative integer (A001477(k)) (possibly overlapping) occurrences in the term.

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A131308 Alternate A001477 and tripled 2*A000027.

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A168496 The positions of non-single or nonisolated numbers in A001477.

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