A172002 -id:A172002 - cp's OEIS frontend

A168388 First number in the n-th row of A172002.

1, 3, 5, 13, 21, 39, 57, 89, 121, 171, 221, 293, 365, 463, 561, 689, 817, 979, 1141, 1341, 1541, 1783, 2025, 2313, 2601, 2939, 3277, 3669, 4061, 4511, 4961, 5473, 5985, 6563, 7141, 7789, 8437, 9159, 9881, 10681, 11481, 12363, 13245, 14213, 15181, 16239, 17297

Offset: 1

Paul Curtz, Nov 24 2009

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2, 1, -4, 1, 2, -1).

Cf. A168234.

[(12+n+3*(-1)^n*n+2*n^3)/12: n in [1..60]]; // Vincenzo Librandi, Jul 20 2016

LinearRecurrence[{2,1,-4,1,2,-1},{1,3,5,13,21,39},50] (* Harvey P. Dale, Nov 29 2014 *)
Table[(n + 1) (3 + 2 n^2 + 4 n - 3 (-1)^n)/12 + 1, {n, 0, 46}] (* Michael De Vlieger, Jul 19 2016, after Vincenzo Librandi at A168380 *)

a(2*n+1) = A166464(n)        a(2*n) = A166911(n).
a(n+1) - a(n) = A093907(n-1), n>1.
a(n) = 2*a(n-1) +a(n-2) -4*a(n-3) +a(n-4) +2*a(n-5) -a(n-6).
G.f.: x*(1 - x^2 + 2*x)*(1 - x + x^2 + x^3)/( (1+x)^2 * (x-1)^4).
a(n+1) = A168380(n)+1.
From G. C. Greubel, Jul 19 2016: (Start)
a(n) = (12 + n + 3*(-1)^n*n + 2*n^3)/12.
E.g.f.: (1/12)*( -3*x - 12*exp(x) + (12 + 3*x + 6*x^2 + 2*x^3)*exp(2*x) )*exp(-x). (End)

Edited and extended by R. J. Mathar, Mar 25 2010

A138509 Janet (or left-step) periodic table from right to left. 120 terms. 8 rows, 32 columns without spaces.

Original entry on oeis.org

2, 1, 4, 3, 12, 11, 10, 9, 8, 7, 6, 5, 20, 19, 18, 17, 16, 15, 14, 13, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 56, 55, 54, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 88, 87, 86, 85, 84, 83, 82, 81, 80, 79, 78, 77, 76, 75, 74, 73, 72, 71, 70, 69, 68, 67, 66, 65, 64, 63, 62, 61, 60, 59, 58, 57, 120, 119, 118, 117, 116, 115, 114, 113, 112, 111, 110, 109, 108, 107, 106, 105, 104, 103, 102, 101, 100, 99, 98, 97, 96, 95, 94, 93, 92, 91, 90, 89

Offset: 1

Paul Curtz, May 10 2008

Cf. A137583, A168380 (first number in each row), A171710, A172002.

Table[Reverse@ Range[2 Ceiling[n/2]^2] + (# + 1) (3 + 2 #^2 + 4 # - 3 (-1)^#)/12 &[n - 1], {n, 8}] // Flatten (* Michael De Vlieger, Jul 20 2016 *)

There are respectively A137583 = 2, 2, 8, 8, 18, 18, 32, 32 terms from right to left.

A166911 a(n) = (9 + 14n + 12n^2 + 4*n^3)/3.

Original entry on oeis.org

3, 13, 39, 89, 171, 293, 463, 689, 979, 1341, 1783, 2313, 2939, 3669, 4511, 5473, 6563, 7789, 9159, 10681, 12363, 14213, 16239, 18449, 20851, 23453, 26263, 29289, 32539, 36021, 39743, 43713, 47939, 52429, 57191, 62233, 67563, 73189, 79119, 85361, 91923

Offset: 0

Paul Curtz, Oct 23 2009

The inverse binomial transform yields the quasi-finite sequence 3,10,16,8,0,.. (0 continued).
These are the bottom-left numbers in the blocks (each with 2 rows) shown in A172002, the
atomic number of the leftmost element in the 2nd, 4th, 6th etc. row of the Janet table.

Charles Janet, La structure du noyau de l'atome .., Nov 1927, page 15.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

[(9+14*n+12*n^2+4*n^3)/3: n in [0..40] ]; // Vincenzo Librandi, Aug 06 2011

LinearRecurrence[{4,-6,4,-1}, {3, 13, 39, 89}, 100] (* G. C. Greubel, May 28 2016 *)

a(n)=n*(4*n^2+12*n+14)/3+3 \\ Charles R Greathouse IV, Dec 21 2011

First differences: a(n)-a(n-1) = 2+4*n+4*n^2 = 1+(1+2n)^2 = 1 + A016754(n+1) = A069894(n+1).
Second differences: a(n) - 2*a(n-1) + a(n-2) = 8*n = A008590(n+2).
Third differences: a(n) - 3*a(n-1) + 3*a(n-2) - a(n-3) = 8.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
G.f.: (3 + x + 5*x^2 - x^3)/(1-x)^4.
a(n) = A166464(n) + 2*(n+1)^2 = A166464(n) + A001105(n+1).
E.g.f.: (1/3)*(9 + 30*x + 24*x^2 + 4*x^3)*exp(x). - G. C. Greubel, May 28 2016

Edited and extended by R. J. Mathar, Mar 02 2010

A138100 The atomic numbers read along the odd-indexed rows of the Janet table of the elements.

Original entry on oeis.org

1, 2, 5, 6, 7, 8, 9, 10, 11, 12, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88

Offset: 1

Paul Curtz, May 03 2008

The union with A138101 gives the first 120 terms of A000027.

			Starts with 1 and 2 of the first row (H and He). Next come 5 to 12 of the 3rd row (B to Mg).

Albert Tarantola, PSE of Elements (Janet form).

Cf. A000027, A138101 (even-indexed rows), A138509 (Left-step Janet periodic table), A137583, A172002.

Table[Range[2 Ceiling[n/2]^2] + (# + 1) (3 + 2 #^2 + 4 # - 3 (-1)^#)/12 &[n - 1], {n, 1, 7, 2}] // Flatten (* Michael De Vlieger, Jul 21 2016 *)

Edited by R. J. Mathar, Oct 07 2009

A138101 The atomic numbers read along the even-indexed rows of the Janet table of the elements.

Original entry on oeis.org

3, 4, 13, 14, 15, 16, 17, 18, 19, 20, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120

Offset: 1

Paul Curtz, May 03 2008

The union with A138100 gives the first 120 terms of A000027.

			Starts with 3 and 4 of the 2nd row (Li and Be). Next come 13 to 20 of the 4th row (Al to Ca).

Albert Tarantola, PSE of Elements (Janet form).
Anonymous, Janet periodic table, Web Elements Chemistry Nexus

Cf. A000027, A138100 (odd-indexed rows), A138509 (Left-step Janet periodic table), A137583, A172002.

Table[Range[2 Ceiling[n/2]^2] + (# + 1) (3 + 2 #^2 + 4 # - 3 (-1)^#)/12 &[n - 1], {n, 2, 8, 2}] // Flatten (* Michael De Vlieger, Jul 21 2016 *)

Edited by R. J. Mathar, Oct 07 2009

A172482 a(n) = (1+n)(9 + 11n + 4*n^2)/3.

Original entry on oeis.org

3, 16, 47, 104, 195, 328, 511, 752, 1059, 1440, 1903, 2456, 3107, 3864, 4735, 5728, 6851, 8112, 9519, 11080, 12803, 14696, 16767, 19024, 21475, 24128, 26991, 30072, 33379, 36920, 40703, 44736, 49027, 53584, 58415, 63528, 68931, 74632, 80639, 86960, 93603

Offset: 0

Paul Curtz, Feb 04 2010

One of the bisections of the left central column in the Janet table A172002.
Row 1 of the convolution array A213844. - Clark Kimberling, Jul 05 2012
With offset 2, this is 4*n^3/3 - 3*n^2 + 8*n/3 - 1, the number of divisions of a 2 X n board into 3 pieces where the rightmost squares separate. See Jacob Brown article. - Michel Marcus, Jun 29 2021

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Jacob Brown, Counting Divisions of a 2×n Rectangular Grid, arXiv:2106.14755 [math.CO], 2021.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A100178, A131941.

[(1+n)*(9+11*n+4*n^2)/3: n in [0..40]]; // Vincenzo Librandi, Aug 04 2011

CoefficientList[Series[(x + 3) (1 + x)/(x - 1)^4, {x, 0, 40}], x] (* Michael De Vlieger, Nov 02 2018 *)

a(n)=(1+n)*(9+11*n+4*n^2)/3 \\ Charles R Greathouse IV, Aug 04 2011

a(n) = A131941(2n+2), where A100178(n) = A131941(2n-1).
a(n) = 4*a(n) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) mod 10 = 3, 6, 7, 4, 5, 8, 1, 2, 9, 0 (and repeat periodically).
G.f.: (x+3)*(1+x)/(x-1)^4.
E.g.f.: exp(x)*(9 + 39*x + 27*x^2 + 4*x^3)/3. - Stefano Spezia, Mar 02 2025

Edited by R. J. Mathar, Feb 24 2010

A168142 Count downwards from 2, then from 8, then from 18, then from ... 2*k^2, k>=1.

Original entry on oeis.org

2, 1, 8, 7, 6, 5, 4, 3, 2, 1, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31

Offset: 1

Paul Curtz, Nov 19 2009

nonn

Janet's extended enumeration of the periodic table of the elements.

The table is read from the right to the left.

Charles Janet, La structure du Noyau de l'atome,consideree dans la Classification periodique des elements chimiques, Nov. 1927, N. 2, Beauvais, 67 pages, 3 leafleats, see page 15.
Charles Janet, Considerations sur la structure du noyau de l'atome, Dec 1929, N 5, Beauvais, 2+45 pp.,4 leaflets, see leaflets 2 and 3.

Cf. A138100, A138101, A093907, A137583, A172002.

Table[Reverse@ Range[2 n^2], {n, 5}] // Flatten (* Michael De Vlieger, Jul 22 2016 *)

Edited by R. J. Mathar, Feb 15 2010

A168574 a(n) = (4n + 3)(1 + 2*n^2)/3.

Original entry on oeis.org

1, 7, 33, 95, 209, 391, 657, 1023, 1505, 2119, 2881, 3807, 4913, 6215, 7729, 9471, 11457, 13703, 16225, 19039, 22161, 25607, 29393, 33535, 38049, 42951, 48257, 53983, 60145, 66759, 73841, 81407, 89473, 98055, 107169, 116831, 127057, 137863, 149265, 161279

Offset: 0

Paul Curtz, Nov 30 2009

Binomial transform of quasi-finite sequence 1, 6, 20, 16, 0, 0, ... (0 continued).

a(n+1) is the sum of the first and last number at the bottom (2nd row) of each block in A172002, 3+4, 13+20, 39+56, ...

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A168547, A168234.

[(4*n+3)*(1+2*n^2)/3 : n in [0..40]]; // Vincenzo Librandi, Aug 06 2011

Table[ (4*n+3)*(1+2*n^2)/3 , {n,0,25}] (* G. C. Greubel, Jul 26 2016 *)
LinearRecurrence[{4,-6,4,-1},{1,7,33,95},40] (* Harvey P. Dale, May 16 2019 *)

a(n)=(4*n+3)*(1+2*n^2)/3 \\ Charles R Greathouse IV, Jul 26 2016

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 16.
a(n) = A168582(2*n+1) .
a(n+1) = A166911(n) + A002492(n+1).
G.f.: (1 + 3*x + 11*x^2 + x^3)/(1 - x)^4.
E.g.f.: (1/3)*(3 + 18*x + 30*x^2 + 8*x^3)*exp(x). - G. C. Greubel, Jul 26 2016

Edited and extended by R. J. Mathar, Mar 25 2010

A168582 a(n) = (4n^3 - 6n^2 + 8n + 9 + 3(-1)^n)/12.

Original entry on oeis.org

1, 1, 3, 7, 17, 33, 59, 95, 145, 209, 291, 391, 513, 657, 827, 1023, 1249, 1505, 1795, 2119, 2481, 2881, 3323, 3807, 4337, 4913, 5539, 6215, 6945, 7729, 8571, 9471, 10433, 11457, 12547, 13703, 14929, 16225, 17595, 19039, 20561

Offset: 0

Paul Curtz, Nov 30 2009

Starting with a(2), the sum of the first and last term in row n-1 of the Janet table A172002.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Cf. A137928 (first differences).

[2*n/3 +3/4 -n^2/2 +n^3/3 +(-1)^n/4: n in [0..40]]; // Vincenzo Librandi, Aug 06 2011

Table[(4*n^3 - 6*n^2 + 8*n + 9 + 3*(-1)^n)/12, {n,0,50}] (* G. C. Greubel, Jul 26 2016 *)

a(n)=(4*n^3-6*n^2+8*n+9+3*(-1)^n)/12 \\ Charles R Greathouse IV, Jul 26 2016

a(n+2) = A168388(n) + A168380(n), n >= 0.
a(2n) = A168547(n);
a(2n+1) = A168574(n).
G.f.: (1 - 2*x + x^4 + 2*x^2 + 2*x^3)/((1+x)*(x-1)^4). - R. J. Mathar, Jun 27 2011
E.g.f.: (1/12)*((4*x^3 + 6*x^2 + 6*x + 9)*exp(x) + 3*exp(-x)). - G. C. Greubel, Jul 26 2016

A171710 Union of A168234 and A171219, sorted.

Original entry on oeis.org

3, 3, 5, 5, 13, 13, 13, 13, 13, 13, 13, 13, 21, 21, 21, 21, 21, 21, 21, 21, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89

Offset: 1

Paul Curtz, Dec 16 2009

Consider a table T(n,k) similar to A168142 = {2,1}, {8,7,6,...,2,1}, {18,17,...,2,1},... that repeats each row. Thus T(n,k) = {2,1}, {2,1}, {8,7,6,...,2,1}, {8,7,6,...,2,1}, {18,17,...,2,1}, etc. The rows of T(n,k) decrement from 2*ceiling(n/2)^2 to 1. Then we can construct the table of atomic numbers in the Janet periodic table A138509(n) = T(n,k) + a(n), with k=2*ceiling(n/2)^2 - 1 down to 1 by step -1.

Michael De Vlieger, Table of n, a(n) for n = 1..10680 (first 39 rows)

Cf. A138509 (Janet periodic table, rows n > 1 end in the repeated numbers in this sequence), A168234 (odd rows), A168380 (repeated numbers k - 1), A171219 (even rows), A172002 (smallest values of rows n > 1 are the repeated numbers in this sequence).

Table[(n + 1) (3 + 2 n^2 + 4 n - 3 (-1)^n)/12 + 1, {n, 7}, {k, 2 Ceiling[n/2]^2}] // Flatten (* Michael De Vlieger, Jul 20 2016 *)

May be regarded as an irregular triangle read by rows, defined by T(n,k) = A168380(n) + 1, with 1 <= k <= ceiling(n/2)^2. - Michael De Vlieger, Jul 20 2016

cp's OEIS Frontend

A168388 First number in the n-th row of A172002.

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A138509 Janet (or left-step) periodic table from right to left. 120 terms. 8 rows, 32 columns without spaces.

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A166911 a(n) = (9 + 14*n + 12*n^2 + 4*n^3)/3.

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A138100 The atomic numbers read along the odd-indexed rows of the Janet table of the elements.

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A138101 The atomic numbers read along the even-indexed rows of the Janet table of the elements.

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A172482 a(n) = (1+n)*(9 + 11*n + 4*n^2)/3.

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A168142 Count downwards from 2, then from 8, then from 18, then from ... 2*k^2, k>=1.

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A168574 a(n) = (4*n + 3)*(1 + 2*n^2)/3.

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A166911 a(n) = (9 + 14n + 12n^2 + 4*n^3)/3.

A172482 a(n) = (1+n)(9 + 11n + 4*n^2)/3.

A168574 a(n) = (4n + 3)(1 + 2*n^2)/3.

A168582 a(n) = (4n^3 - 6n^2 + 8n + 9 + 3(-1)^n)/12.