A195828 -id:A195828 - cp's OEIS frontend

A195838 Triangle read by rows which arises from A195828, in the same way as A175003 arises from A195310. Column k starts at row A001082(k+1).

1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 4, 1, -1, 4, 1, -1, 4, 2, -1, 5, 3, -1, 7, 4, -1, 10, 4, -2, 12, 4, -3, 13, 5, -4, 14, 7, -4, -1, 16, 10, -4, -1, 21, 12, -5, -1, 27, 13, -7, -1, 32, 14, -10, -1, 35, 16, -12, -2, 1, 38, 21, -13, -3, 1, 44, 32, -14, -4, 1

Offset: 1

Omar E. Pol, Sep 24 2011

The sum of terms of row n is equal to the leftmost term of row n+1. This sequence is related to the generalized octagonal numbers A001082, A195828 and A195848 in the same way as A175003 is related to the generalized pentagonal numbers A001318, A195310 and A000041. See comments in A195825.

			Written as a triangle:
.  1;
.  1;
.  1;
.  1;
.  1,   1;
.  2,   1;
.  3,   1;
.  4,   1,  -1;
.  4,   1,  -1;
.  4,   2,  -1;
.  5,   3,  -1;
.  7,   4,  -1;
. 10,   4,  -2;
. 12,   4,  -3;
. 13,   5,  -4;
. 14,   7,  -4,  -1;
. 16,  10,  -4,  -1;
. 21,  12,  -5,  -1;
. 27,  13,  -7,  -1;
. 32,  14, -10,  -1;
. 35,  16, -12,  -2,   1;
. 38,  21, -13,  -3,   1;

Row sums give A195848.

Cf. A001082, A175003, A195828, A195836, A195837.

A195825 Square array T(n,k) read by antidiagonals, n>=0, k>=1, which arises from a generalization of Euler's Pentagonal Number Theorem.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 5, 2, 1, 1, 1, 7, 3, 1, 1, 1, 1, 11, 4, 2, 1, 1, 1, 1, 15, 5, 3, 1, 1, 1, 1, 1, 22, 7, 4, 2, 1, 1, 1, 1, 1, 30, 10, 4, 3, 1, 1, 1, 1, 1, 1, 42, 13, 5, 4, 2, 1, 1, 1, 1, 1, 1, 56, 16, 7, 4, 3, 1, 1, 1, 1, 1, 1, 1, 77, 21, 10, 4

Offset: 0

Omar E. Pol, Sep 24 2011

In the infinite square array the column k is related to the generalized m-gonal numbers, where m = k+4. For example: the first column is related to the generalized pentagonal numbers A001318. The second column is related to the generalized hexagonal numbers A000217 (note that A000217 is also the entry for the triangular numbers). And so on ... (see the program in which A195152 is a table of generalized m-gonal numbers).
In the following table Euler's Pentagonal Number Theorem is represented by the entries A001318, A195310, A175003 and A000041 (see below the first row of the table):
========================================================
.                                          Column k of
.                                          this square
.       Generalized   Triangle  Triangle   array A195825
k    m    m-gonal       "A"       "B"      [row sums of
.         numbers                          triangle "B"
.                                          with a(0)=1]
========================================================
1    5    A001318     A195310   A175003      A000041
2    6    A000217     A195826   A195836      A006950
3    7    A085787     A195827   A195837      A036820
4    8    A001082     A195828   A195838      A195848
5    9    A118277     A195829   A195839      A195849
6   10    A074377     A195830   A195840      A195850
7   11    A195160     A195831   A195841      A195851
8   12    A195162     A195832   A195842      A195852
9   13    A195313     A195833   A195843      A196933
10  14    A195818     A210944   A210954      A210964
...
It appears that column 2 of the square array is A006950.
It appears that column 3 of the square array is A036820.
Conjecture: if k is odd then column k contains (k+1)/2 plateaus whose levels are the first (k+1)/2 terms of A210843 and whose lengths are k+1, k-1, k-3, k-5, ... 2. Otherwise, if k is even then column k contains k/2 plateaus whose levels are the first k/2 terms of A210843 and whose lengths are k+1, k-1, k-3, k-5, ... 3. The sequence A210843 gives the levels of the plateaus of column k, when k -> infinity. For the visualization of the plateaus see the graph of a column, for example see the graph of A210964. - Omar E. Pol, Jun 21 2012

			Array begins:
    1,  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
    1,  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
    2,  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
    3,  2,  1,  1,  1,  1,  1,  1,  1,  1, ...
    5,  3,  2,  1,  1,  1,  1,  1,  1,  1, ...
    7,  4,  3,  2,  1,  1,  1,  1,  1,  1, ...
   11,  5,  4,  3,  2,  1,  1,  1,  1,  1, ...
   15,  7,  4,  4,  3,  2,  1,  1,  1,  1, ...
   22, 10,  5,  4,  4,  3,  2,  1,  1,  1, ...
   30, 13,  7,  4,  4,  4,  3,  2,  1,  1, ...
   42, 16, 10,  5,  4,  4,  4,  3,  2,  1, ...
   56, 21, 12,  7,  4,  4,  4,  4,  3,  2, ...
   77, 28, 14, 10,  5,  4,  4,  4,  4,  3, ...
  101, 35, 16, 12,  7,  4,  4,  4,  4,  4, ...
  135, 43, 21, 13, 10,  5,  4,  4,  4,  4, ...
  176, 55, 27, 14, 12,  7,  4,  4,  4,  4, ...
  ...
Column 1 is A000041 which starts: [1, 1], 2, 3, 5, 7, 11, ... The column contains only one plateau: [1, 1] which has level 1 and length 2.
Column 3 is A036820 which starts: [1, 1, 1, 1], 2, 3, [4, 4], 5, 7, 10, ... The column contains two plateaus: [1, 1, 1, 1], [4, 4], which have levels 1, 4 and lengths 4, 2.
Column 6 is A195850 which starts: [1, 1, 1, 1, 1, 1, 1], 2, 3, [4, 4, 4, 4, 4], 5, 7, 10, 12, [13, 13, 13], 14, 16, 21, ... The column contains three plateaus: [1, 1, 1, 1, 1, 1, 1], [4, 4, 4, 4, 4], [13, 13, 13], which have levels 1, 4, 13 and lengths 7, 5, 3.

Leonhard Euler, De mirabilibus proprietatibus numerorum pentagonalium
Leonhard Euler, On the remarkable properties of the pentagonal numbers, arXiv:math/0505373 [math.HO], 2005.
Eric Weisstein's World of Mathematics, Pentagonal Number Theorem
Wikipedia, Pentagonal number theorem

Columns (1-10): A000041, A006950, A036820, A195848, A195849, A195850, A195851, A195852, A196933, A210964.
For another version see A211970.
Cf. A057077, A195152, A210843.

Column k is asymptotic to exp(Pi*sqrt(2*n/(k+2))) / (8*sin(Pi/(k+2))*n). - Vaclav Kotesovec, Aug 14 2017

A195848 Expansion of 1 / f(-x^1, -x^5) in powers of x where f() is Ramanujan's two-variable theta function.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 4, 4, 4, 5, 7, 10, 12, 13, 14, 16, 21, 27, 32, 35, 38, 44, 54, 67, 78, 86, 94, 107, 128, 153, 176, 194, 213, 241, 282, 331, 376, 415, 456, 512, 590, 680, 767, 845, 928, 1037, 1180, 1345, 1506, 1657, 1818, 2020, 2278, 2570, 2862, 3142, 3442

Offset: 0

Omar E. Pol, Sep 24 2011

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Also column 4 of A195825, therefore this sequence contains two plateaus: [1, 1, 1, 1, 1], [4, 4, 4]. For more information see A210843 and other sequences of this family. - Omar E. Pol, Jun 26 2012
The number of partitions of n into parts congruent to 0, 1 or 5 ( mod 6 ). - Peter Bala, Dec 09 2020

			G.f. = 1 + x + x^2 + x^3 + x^4 + 2*x^5 + 3*x^6 + 4*x^7 + 4*x^8 + 4*x^9 + 5*x^10 + ...
G.f. = 1/q + q^2 + q^5 + q^8 + q^11 + 2*q^14 + 3*q^17 + 4*q^20 + 4*q^23 + 4*q^26 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
K. Bringmann, J. Lovejoy, and K. Mahlburg, A partition identity and the universal mock theta function g_2(x;q), Mathematical Research Letters, 23 (2016), 67-80.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Column 1 of triangle A195838. Also 1 together with the row sums of triangle A195838. Column 4 of array A195825.
Cf. A000041, A001082, A006950, A036820, A057077, A195825, A195828, A195849, A195850, A195851, A195852, A196933, A210843, A210964, A211971.
Cf. A089802.

A001082 := proc(n)
        if type(n,'even') then
                n*(3*n-4)/4 ;
        else
                (n-1)*(3*n+1)/4 ;
        end if;
end proc:
A195838 := proc(n,k)
        option remember;
        local ks,a,j ;
        if A001082(k+1) > n then
                0 ;
        elif n <= 5 then
                return 1;
        elif k = 1 then
                a := 0 ;
                for j from 1 do
                        if A001082(j+1) <= n-1 then
                                a := a+procname(n-1,j) ;
                        else
                                break;
                        end if;
                end do;
                return a;
        else
                ks := A001082(k+1) ;
                (-1)^floor((k-1)/2)*procname(n-ks+1,1) ;
        end if;
end proc:
A195848 := proc(n)
        A195838(n+1,1) ;
end proc:
seq(A195848(n),n=0..60) ; # R. J. Mathar, Oct 07 2011

a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] QPochhammer[ x^3] / (QPochhammer[ x] QPochhammer[ x^6]^2), {x, 0, n}]; (* Michael Somos, Oct 18 2014 *)
a[ n_] := SeriesCoefficient[ 2 q^(3/8) / (QPochhammer[ q, q^2] EllipticTheta[ 2, 0, q^(3/2)]), {q, 0, n}]; (* Michael Somos, Oct 18 2014 *)
nmax = 60; CoefficientList[Series[Product[(1+x^k) / ((1+x^(3*k)) * (1-x^(6*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 08 2015 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A) / (eta(x + A) * eta(x^6 + A)^2), n))}; /* Michael Somos, Jun 07 2012 */
From Omar E. Pol, Jun 10 2012: (Start)
(GW-BASIC)' A program with two A-numbers:
10 Dim A001082(100), A057077(100), a(100): a(0)=1
20 For n = 1 to 58: For j = 1 to n
30 If A001082(j) <= n then a(n) = a(n) + A057077(j-1)*a(n - A001082(j))
40 Next j: Print a(n-1);: Next n (End)

Expansion of 1 / (psi(x^3) * chi(-x)) in powers of x where psi(), chi() are Ramanujan theta functions. - Michael Somos, Jun 07 2012
Expansion of q^(1/3) * eta(q^2) * eta(q^3) / (eta(q) * eta(q^6)^2) in powers of q. - Michael Somos, Jun 07 2012
Euler transform of period 6 sequence [ 1, 0, 0, 0, 1, 1, ...]. - Michael Somos, Oct 18 2014
Convolution inverse of A089802. - Michael Somos, Oct 18 2014
a(n) ~ exp(Pi*sqrt(n/3))/(4*n). - Vaclav Kotesovec, Nov 08 2015
a(n) = (1/n)*Sum_{k=1..n} A284362(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017
From Peter Bala, Dec 09 2020: (Start)
O.g.f.: 1/( Product_{n >= 1} (1 - x^(6*n-5))*(1 - x^(6*n-1))*(1 - x^(6*n)) ).
a(n) = a(n-1) + a(n-5) - a(n-8) - a(n-16) + + - - ... (with the convention a(n) = 0 for negative n), where 1, 5, 8, 16, ... is the sequence of generalized octagonal numbers A001082. (End)

New sequence name from Michael Somos, Oct 18 2014

A211970 Square array read by antidiagonal: T(n,k), n >= 0, k >= 0, which arises from a generalization of Euler's Pentagonal Number Theorem.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 4, 2, 1, 1, 6, 3, 1, 1, 1, 10, 5, 2, 1, 1, 1, 16, 7, 3, 1, 1, 1, 1, 24, 11, 4, 2, 1, 1, 1, 1, 36, 15, 5, 3, 1, 1, 1, 1, 1, 54, 22, 7, 4, 2, 1, 1, 1, 1, 1, 78, 30, 10, 4, 3, 1, 1, 1, 1, 1, 1, 112, 42, 13, 5, 4, 2, 1, 1, 1, 1, 1, 1

Offset: 0

Omar E. Pol, Jun 10 2012

In the infinite square array if k is positive then column k is related to the generalized m-gonal numbers, where m = k+4. For example: column 1 is related to the generalized pentagonal numbers A001318. Column 2 is related to the generalized hexagonal numbers A000217 (note that A000217 is also the entry for the triangular numbers). And so on...
In the following table Euler's Pentagonal Number Theorem is represented by the entries A001318, A195310, A175003 and A000041. It seems unusual that the partition numbers are located in a middle column (see below row 1 of the table):
========================================================
.                                          Column k of
.                                          this square
.       Generalized   Triangle  Triangle   array A211970
k    m    m-gonal       "A"       "B"      [row sums of
.         numbers                          triangle "B"
.        (if k>=1)                         with a(0)=1,
.                                          if k >= 0]
========================================================
0    4    A008794        -         -         A211971
1    5    A001318     A195310   A175003      A000041
2    6    A000217     A195826   A195836      A006950
3    7    A085787     A195827   A195837      A036820
4    8    A001082     A195828   A195838      A195848
5    9    A118277     A195829   A195839      A195849
6   10    A074377     A195830   A195840      A195850
7   11    A195160     A195831   A195841      A195851
8   12    A195162     A195832   A195842      A195852
9   13    A195313     A195833   A195843      A196933
10  14    A195818     A210944   A210954      A210964
...
It appears that column 2 of the square array is A006950.
It appears that column 3 of the square array is A036820.
The partial sums of column 0 give A015128. - Omar E. Pol, Feb 09 2014

			Array begins:
1,     1,   1,   1,   1,   1,  1,  1,  1,  1,  1, ...
1,     1,   1,   1,   1,   1,  1,  1,  1,  1,  1, ...
2,     2,   1,   1,   1,   1,  1,  1,  1,  1,  1, ...
4,     3,   2,   1,   1,   1,  1,  1,  1,  1,  1, ...
6,     5,   3,   2,   1,   1,  1,  1,  1,  1,  1, ...
10,    7,   4,   3,   2,   1,  1,  1,  1,  1,  1, ...
16,   11,   5,   4,   3,   2,  1,  1,  1,  1,  1, ...
24,   15,   7,   4,   4,   3,  2,  1,  1,  1,  1, ...
36,   22,  10,   5,   4,   4,  3,  2,  1,  1,  1, ...
54,   30,  13,   7,   4,   4,  4,  3,  2,  1,  1, ...
78,   42,  16,  10,   5,   4,  4,  4,  3,  2,  1, ...
112,  56,  21,  12,   7,   4,  4,  4,  4,  3,  2, ...
160,  77,  28,  14,  10,   5,  4,  4,  4,  4,  3, ...
224, 101,  35,  16,  12,   7,  4,  4,  4,  4,  4, ...
312, 135,  43,  21,  13,  10,  5,  4,  4,  4,  4, ...
432, 176,  55,  27,  14,  12,  7,  4,  4,  4,  4, ...
...

L. Euler, De mirabilibus proprietatibus numerorum pentagonalium
L. Euler, On the remarkable properties of the pentagonal numbers, arXiv:math/0505373 [math.HO], 2005.
Eric Weisstein's World of Mathematics, Pentagonal Number Theorem

Columns (0-10): A211971, A000041, A006950, A036820, A195848, A195849, A195850, A195851, A195852, A196933, A210964.
For another version see A195825.
Cf. A057077, A195152.

T(n,k) = A211971(n), if k = 0.

T(n,k) = A195825(n,k), if k >= 1.

A195826 Triangle read by rows with T(n,k) = n - A000217(k), n>=1, k>=1, if (n - A000217(k))>=0.

Original entry on oeis.org

0, 1, 2, 0, 3, 1, 4, 2, 5, 3, 0, 6, 4, 1, 7, 5, 2, 8, 6, 3, 9, 7, 4, 0, 10, 8, 5, 1, 11, 9, 6, 2, 12, 10, 7, 3, 13, 11, 8, 4, 14, 12, 9, 5, 0, 15, 13, 10, 6, 1, 16, 14, 11, 7, 2, 17, 15, 12, 8, 3, 18, 16, 13, 9, 4, 19, 17, 14, 10, 5, 20, 18, 15, 11, 6, 0

Offset: 1

Omar E. Pol, Sep 24 2011

Also triangle read by rows in which column k lists the nonnegative integers A001477 starting at the row A000217(k).

This sequence is related to the generalized hexagonal numbers (A000217), A195836 and A006950 in the same way as A195310 is related to the generalized pentagonal numbers A001318, A175003 and A000041. See comments in A195825.

			Written as a triangle:
.  0;
.  1;
.  2,  0;
.  3,  1;
.  4,  2;
.  5,  3,  0;
.  6,  4,  1;
.  7,  5,  2;
.  8,  6,  3;
.  9,  7,  4,  0;
. 10,  8,  5,  1;
. 11,  9,  6,  2;
. 12, 10,  7,  3;
. 13, 11,  8,  4;
. 14, 12,  9,  5,  0;
. 15, 13,  10, 6,  1;
. 16, 14,  11, 7,  2;
. 17, 15,  12, 8,  3;

Cf. A000217, A006950, A195310, A195825, A195827, A195828, A195829, A195830, A195831, A195832, A195833, A195836.

A195829 Triangle read by rows with T(n,k) = n - A118277(k), n>=1, k>=1, if (n - A118277(k))>=0.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 0, 6, 1, 7, 2, 8, 3, 0, 9, 4, 1, 10, 5, 2, 11, 6, 3, 12, 7, 4, 13, 8, 5, 14, 9, 6, 15, 10, 7, 16, 11, 8, 17, 12, 9, 18, 13, 10, 0, 19, 14, 11, 1, 20, 15, 12, 2, 21, 16, 13, 3, 22, 17, 14, 4, 23, 18, 15, 5, 0, 24, 19, 16, 6, 1, 25, 20, 17, 7, 2

Offset: 1

Omar E. Pol, Sep 24 2011

Also triangle read by rows in which column k lists the nonnegative integers A001477 starting at the row A118277(k).

This sequence is related to the generalized enneagonal numbers A118277, A195839 and A195849 in the same way as A195310 is related to the generalized pentagonal numbers A001318, A175003 and A000041. See comments in A195825.

			Written as a triangle:
.  0;
.  1;
.  2;
.  3;
.  4;
.  5,  0;
.  6,  1;
.  7,  2;
.  8,  3,  0;
.  9,  4,  1;
. 10,  5,  2;
. 11,  6,  3;
. 12,  7,  4;
. 13,  8,  5;
. 14,  9,  6;
. 15, 10,  7;
. 16, 11,  8;
. 17, 12,  9;
. 18, 13, 10,  0;
. 19, 14, 11,  1;
. 20, 15, 12,  2;

Cf. A118277, A195310, A195825, A195826, A195827, A195828, A195830, A195831, A195832, A195833, A195839, A195849.

A195830 Triangle read by rows with T(n,k) = n - A074377(k), n>=1, k>=1, if (n - A074377(k))>=0.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 0, 7, 1, 8, 2, 9, 3, 0, 10, 4, 1, 11, 5, 2, 12, 6, 3, 13, 7, 4, 14, 8, 5, 15, 9, 6, 16, 10, 7, 17, 11, 8, 18, 12, 9, 19, 13, 10, 20, 14, 11, 21, 15, 12, 0, 22, 16, 13, 1, 23, 17, 14, 2, 24, 18, 15, 3, 25, 19, 16, 4, 26, 20, 17, 5, 0

Offset: 1

Omar E. Pol, Sep 24 2011

Also triangle read by rows in which column k lists the nonnegative integers A001477 starting at the row A074377(k).

This sequence is related to the generalized decagonal numbers A074377, A195840 and A195850 in the same way as A195310 is related to the generalized pentagonal numbers A001318, A175003 and A000041. See comments in A195825.

			Written as a triangle:
.  0;
.  1;
.  2;
.  3;
.  4;
.  5;
.  6,  0;
.  7,  1;
.  8,  2;
.  9,  3,  0;
. 10,  4,  1;
. 11,  5,  2;
. 12,  6,  3;
. 13,  7,  4;
. 14,  8,  5;
. 15,  9,  6;
. 16, 10,  7;
. 17, 11,  8;
. 18, 12,  9;
. 19, 13, 10;
. 20, 14, 11;
. 21, 15, 12,  0;
. 22, 16, 13,  1;
. 23, 17, 14,  2;

Cf. A074377, A195310, A195825, A195826, A195827, A195828, A195829, A195831, A195832, A195833, A195840, A195850.

A195831 Triangle read by rows with T(n,k) = n - A195160(k), n>=1, k>=1, if (n - A195160(k))>=0.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 0, 8, 1, 9, 2, 10, 3, 0, 11, 4, 1, 12, 5, 2, 13, 6, 3, 14, 7, 4, 15, 8, 5, 16, 9, 6, 17, 10, 7, 18, 11, 8, 19, 12, 9, 20, 13, 10, 21, 14, 11, 22, 15, 12, 23, 16, 13, 24, 17, 14, 0, 25, 18, 15, 1, 26, 19, 16, 2, 27, 20, 17, 3, 28, 21, 18, 4

Offset: 1

Omar E. Pol, Sep 24 2011

Also triangle read by rows in which column k lists the nonnegative integers A001477 starting at the row A195160(k).

This sequence is related to the generalized hendecagonal numbers A195160, A195841 and A195851 in the same way as A195310 is related to the generalized pentagonal numbers A001318, A175003 and A000041. See comments in A195825.

			.  0;
.  1;
.  2;
.  3;
.  4;
.  5;
.  6;
.  7,  0;
.  8,  1;
.  9,  2;
. 10,  3,  0;
. 11,  4,  1;
. 12,  5,  2;
. 13,  6,  3;
. 14,  7,  4;
. 15,  8,  5;
. 16,  9,  6;
. 17, 10,  7;
. 18, 11,  8;
. 19, 12,  9;
. 20, 13, 10;

Cf. A195160, A195310, A195825, A195826, A195827, A195828, A195829, A195830, A195832, A195833, A195841, A195851.

A195832 Triangle read by rows with T(n,k) = n - A195162(k), n>=1, k>=1, if (n - A195162(k))>=0.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 9, 1, 10, 2, 11, 3, 0, 12, 4, 1, 13, 5, 2, 14, 6, 3, 15, 7, 4, 16, 8, 5, 17, 9, 6, 18, 10, 7, 19, 11, 8, 20, 12, 9, 21, 13, 10, 22, 14, 11, 23, 15, 12, 24, 16, 13, 25, 17, 14, 26, 18, 15, 27, 19, 16, 0, 28, 20, 17, 1, 29, 21, 18, 2

Offset: 1

Omar E. Pol, Sep 24 2011

Also triangle read by rows in which column k lists the nonnegative integers A001477 starting at the row A195162(k).

This sequence is related to the generalized dodecagonal numbers A195162, A195842 and A195852 in the same way as A195310 is related to the generalized pentagonal numbers A001318, A175003 and A000041. See comments in A195825.

			Written as a triangle:
.  0;
.  1;
.  2;
.  3;
.  4;
.  5;
.  6;
.  7;
.  8,  0;
.  9,  1;
. 10,  2;
. 11,  3,  0;
. 12,  4,  1;
. 13,  5,  2;
. 14,  6,  3;
. 15,  7,  4;
. 16,  8,  5;
. 17,  9,  6;
. 18, 10,  7;

Cf. A195162, A195310, A195825, A195826, A195827, A195828, A195829, A195830, A195831, A195833, A195842, A195852.

A195833 Triangle read by rows with T(n,k) = n - A195313(k), n>=1, k>=1, if (n - A195313(k))>=0.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 10, 1, 11, 2, 12, 3, 0, 13, 4, 1, 14, 5, 2, 15, 6, 3, 16, 7, 4, 17, 8, 5, 18, 9, 6, 19, 10, 7, 20, 11, 8, 21, 12, 9, 22, 13, 10, 23, 14, 11, 24, 15, 12, 25, 16, 13, 26, 17, 14, 27, 18, 15, 28, 19, 16, 29, 20, 17, 30, 21, 18, 0

Offset: 1

Omar E. Pol, Sep 24 2011

Also triangle read by rows in which column k lists the nonnegative integers A001477 starting at the row A195313(k).

This sequence is related to the generalized tridecagonal numbers A195313, A195843 and A196933 in the same way as A195310 is related to the generalized pentagonal numbers A001318, A175003 and A000041. See comments in A195825.

			Written as a triangle:
.  0;
.  1;
.  2;
.  3;
.  4;
.  5;
.  6;
.  7;
.  8;
.  9, 0;
. 10, 1;
. 11, 2;
. 12, 3, 0;
. 13, 4, 1;
. 14, 5, 2;
. 15, 6, 3;
. 16, 7, 4;

Cf. A195310, A195313, A195825, A195826, A195827, A195828, A195829, A195830, A195831, A195832, A195843, A196933.

cp's OEIS Frontend

A195838 Triangle read by rows which arises from A195828, in the same way as A175003 arises from A195310. Column k starts at row A001082(k+1).

Views

Author

Keywords

Comments

Examples

Crossrefs

A195825 Square array T(n,k) read by antidiagonals, n>=0, k>=1, which arises from a generalization of Euler's Pentagonal Number Theorem.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A195848 Expansion of 1 / f(-x^1, -x^5) in powers of x where f() is Ramanujan's two-variable theta function.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A211970 Square array read by antidiagonal: T(n,k), n >= 0, k >= 0, which arises from a generalization of Euler's Pentagonal Number Theorem.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A195826 Triangle read by rows with T(n,k) = n - A000217(k), n>=1, k>=1, if (n - A000217(k))>=0.

Views

Author

Keywords

Comments

Examples

Crossrefs

A195829 Triangle read by rows with T(n,k) = n - A118277(k), n>=1, k>=1, if (n - A118277(k))>=0.

Views

Author

Keywords

Comments

Examples

Crossrefs

A195830 Triangle read by rows with T(n,k) = n - A074377(k), n>=1, k>=1, if (n - A074377(k))>=0.

Views

Author

Keywords

Comments

Examples

Crossrefs

A195831 Triangle read by rows with T(n,k) = n - A195160(k), n>=1, k>=1, if (n - A195160(k))>=0.

Views

Author

Keywords

Comments

Examples

Crossrefs

A195832 Triangle read by rows with T(n,k) = n - A195162(k), n>=1, k>=1, if (n - A195162(k))>=0.

Views

Author

Keywords

Comments

Examples

Crossrefs

A195833 Triangle read by rows with T(n,k) = n - A195313(k), n>=1, k>=1, if (n - A195313(k))>=0.

Views

Author

Keywords

Comments