A079824 -id:A079824 - cp's OEIS frontend

A185787 Sum of first k numbers in column k of the natural number array A000027; by antidiagonals.

1, 7, 25, 62, 125, 221, 357, 540, 777, 1075, 1441, 1882, 2405, 3017, 3725, 4536, 5457, 6495, 7657, 8950, 10381, 11957, 13685, 15572, 17625, 19851, 22257, 24850, 27637, 30625, 33821, 37232, 40865, 44727, 48825, 53166, 57757, 62605, 67717, 73100, 78761, 84707, 90945, 97482, 104325, 111481, 118957, 126760, 134897, 143375

Offset: 1

Clark Kimberling, Feb 03 2011

This is one of many interesting sequences and arrays that stem from the natural number array A000027, of which a northwest corner is as follows:
  1....2.....4.....7...11...16...22...29...
  3....5.....8....12...17...23...30...38...
  6....9....13....18...24...31...39...48...
  10...14...19....25...32...40...49...59...
  15...20...26....33...41...50...60...71...
  21...27...34....42...51...61...72...84...
  28...35...43....52...62...73...85...98...
Blocking out all terms below the main diagonal leaves columns whose sums comprise A185787.  Deleting the main diagonal and then summing give A185787.  Analogous treatments to the left of the main diagonal give A100182 and A101165.  Further sequences obtained directly from this array are easily obtained using the following formula for the array: T(n,k)=n+(n+k-2)(n+k-1)/2.
Examples:
row 1:  A000124
row 2:  A022856
row 3:  A016028
row 4:  A145018
row 5:  A077169
col 1:  A000217
col 2:  A000096
col 3:  A034856
col 4:  A055998
col 5:  A046691
col 6:  A052905
col 7:  A055999
diag. (1,5,...) ...... A001844
diag. (2,8,...) ...... A001105
diag. (4,12,...)...... A046092
diag. (7,17,...)...... A056220
diag. (11,23,...) .... A132209
diag. (16,30,...) .... A054000
diag. (22,38,...) .... A090288
diag. (3,9,...) ...... A058331
diag. (6,14,...) ..... A051890
diag. (10,20,...) .... A005893
diag. (15,27,...) .... A097080
diag. (21,35,...) .... A093328
antidiagonal sums:  (1,5,15,34,...)=A006003=partial sums of A002817.
Let S(n,k) denote the n-th partial sum of column k.  Then
S(n,k)=n*(n^2+3k*n+3*k^2-6*k+5)/6.
S(n,1)=n(n+1)(n+2)/6
S(n,2)=n(n+1)(n+5)/6
S(n,3)=n(n+2)(n+7)/6
S(n,4)=n(n^2+12n+29)/6
S(n,5)=n(n+5)(n+10)/6
S(n,6)=n(n+7)(n+11)/6
S(n,7)=n(n+10)(n+11)/6
Weight array of T: A144112
Accumulation array of T: A185506
Second rectangular sum array of T: A185507
Third rectangular sum array of T: A185508
Fourth rectangular sum array of T: A185509

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000027, A185788, A100182, A101165, A079824.

[n*(7*n^2-6*n+5)/6: n in [1..50]]; // Vincenzo Librandi, Jul 04 2012

f[n_,k_]:=n+(n+k-2)(n+k-1)/2;
s[k_]:=Sum[f[n,k],{n,1,k}];
Factor[s[k]]
Table[s[k],{k,1,70}]  (* A185787 *)
CoefficientList[Series[(3*x^2+3*x+1)/(1-x)^4,{x,0,50}],x] (* Vincenzo Librandi, Jul 04 2012 *)

a(n)=n*(7*n^2-6*n+5)/6.

G.f.: x*(3*x^2+3*x+1)/(1-x)^4. - Vincenzo Librandi, Jul 04 2012

Edited by Clark Kimberling, Feb 25 2023

A056536 Mapping from half-antidiagonal reading of the triangle (as used in A028297) to the column-by-column reading of the triangular tables.

Original entry on oeis.org

1, 2, 4, 3, 7, 5, 11, 8, 6, 16, 12, 9, 22, 17, 13, 10, 29, 23, 18, 14, 37, 30, 24, 19, 15, 46, 38, 31, 25, 20, 56, 47, 39, 32, 26, 21, 67, 57, 48, 40, 33, 27, 79, 68, 58, 49, 41, 34, 28, 92, 80, 69, 59, 50, 42, 35, 106, 93, 81, 70, 60, 51, 43, 36, 121, 107, 94, 82, 71, 61, 52

Offset: 1

Antti Karttunen, Jun 20 2000

nonn

Moves squares (A000290) to triangular numbers (A000217). See 1st formula.

This sequence may be regarded as a triangular array read by rows: 1; 2; 4, 3; 7, 5; 11, 8, 6; 16, 12, 9; 22, 17, 13, 10; .... with row sums: A079824 = [1, 2, 7, 12, 25, 37, 62, 84, ...]. - Philippe Deléham, Feb 16 2004

			As a triangular array read by rows:
    1;
    2;
    4,  3;
    7,  5;
   11,  8, 6;
   16, 12, 9;
   22, 17, 13, 10;
   29, 23, 18, 14;
   37, 30, 24, 19, 15;
   46, 38, 31, 25, 20;
   56, 47, 39, 32, 26, 21;
   67, 57, 48, 40, 33, 27;
   79, 68, 58, 49, 41, 34, 28;
   92, 80, 69, 59, 50, 42, 35;
  106, 93, 81, 70, 60, 51, 43, 36;
  ...

Index entries for sequences that are permutations of the natural numbers

Cf. A000217, A000290, A056537 (inverse), A079824, A091018.

triang_perm := proc(upto_d) local a,i,j; a := []; for i from 1 to upto_d do for j from 1 to floor((i+1)/2) do a := [op(a),binomial((i-j)+1,2)+j]; od; od; RETURN(a); end;

a(A000290(i)) = A000217(i) for all i >= 1.

a(n) = A091018(n-1) + 1.

A185788 Sum of the first k-1 numbers in the k-th column of the natural number array A000027, by antidiagonals.

Original entry on oeis.org

0, 2, 12, 37, 84, 160, 272, 427, 632, 894, 1220, 1617, 2092, 2652, 3304, 4055, 4912, 5882, 6972, 8189, 9540, 11032, 12672, 14467, 16424, 18550, 20852, 23337, 26012, 28884, 31960, 35247, 38752, 42482, 46444, 50645, 55092, 59792, 64752, 69979, 75480, 81262, 87332, 93697, 100364, 107340, 114632, 122247, 130192, 138474

Offset: 1

Clark Kimberling, Feb 03 2011

See A185787.

			Start from
  1.....2....4.....7...11...16...22...29...
  3.....5....8....12...17...23...30...38...
  6.....9...13....18...24...31...39...48...
  10...14...19....25...32...40...49...59...
  15...20...26....33...41...50...60...71...
  21...27...34....42...51...61...72...84...
  28...35...43....52...62...73...85...98...
Block out all terms starting at and below the main diagonal then sum up the remaining terms.
  .....2.....4.....7...11...16...22...29...
  ...........8....12...17...23...30...38...
  ................18...24...31...39...48...
  .....................32...40...49...59...
  ..........................50...60...71...
  ...............................72...84...
  ....................................98...

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1)

Cf. A000027, A185787, A079824.

f[n_,k_]:=n+(n+k-2)(n+k-1)/2;
s[k_]:=Sum[f[n,k],{n,1,k-1}];
Factor[s[k]]
Table[s[k],{k,1,70}]
Table[(n - 1)*(7*n^2 - 11*n + 6)/6, {n, 1, 50}] (* G. C. Greubel, Jul 12 2017 *)

for(n=1,50, print1((n-1)*(7*n^2 - 11*n + 6)/6, ", ")) \\ G. C. Greubel, Jul 12 2017

a(n) = (n-1)*(7*n^2 - 11*n + 6)/6. - Corrected by Manfred Arens, Mar 11 2016

G.f.: x^2*(2+4*x+x^2) / (x-1)^4 . - R. J. Mathar, Aug 23 2012

A079823 Consider the triangle shown below; sequence contains the concatenation of numbers read at a 45-degree angle upwards with horizontal beginning with the first term of a row.

Original entry on oeis.org

1, 2, 43, 75, 1186, 16129, 22171310, 29231814, 3730241915, 4638312520, 564739322621, 675748403327, 79685849413428, 92806959504235, 10693817060514336, 121107948271615244, 137122108958372625345, 1541381231099684736354, 1721551391241109785746455

Offset: 1

Amarnath Murthy, Feb 11 2003

   1
   2  3
   4  5  6
   7  8  9 10
  11 12 13 14 15
  16 17 18 19 20 21
  ...
a(n) also is the concatenation of the terms of the n-th row of A056536. - Michel Marcus, Dec 14 2023

G. C. Greubel, Table of n, a(n) for n = 1..395

Cf. A056536, A079808, A079824.

read("transforms"):
A079823aux := proc(n,k)
    A000124(n)+k ;
end proc:
A079823 := proc(n)
    local L,k,n0 ;
    n0 := n-1 ;
    L := [] ;
    for k from 0 do
        if k > n0-k then
            break;
        end if;
        L := [op(L),A079823aux(n0-k,k)] ;
    end do:
    digcatL(L) ;
end proc: # R. J. Mathar, Aug 23 2012
# second Maple program:
T:= (i, j)-> i*(i-1)/2+j:
a:= n-> parse(cat(seq(T(n-j,j+1), j=0..(n-1)/2))):
seq(a(n), n=1..23);  # Alois P. Heinz, Aug 03 2022

Table[FromDigits[Join@@IntegerDigits[Table[Binomial[n-k+1,2] + k, {k, Ceiling[n/2]}]]], {n,30}] (* G. C. Greubel, Dec 13 2023 *)

More terms from Jason D. W. Taff (jtaff(AT)jburroughs.org), Oct 31 2003

Corrected by Philippe Deléham, Feb 16 2004

A249490 a(n) = Sum_{m=0..floor((n-1)/2)} prime((n-m)(n-m-1)/2+m+1).

Original entry on oeis.org

2, 3, 12, 28, 63, 113, 208, 296, 473, 657, 946, 1236, 1661, 2071, 2688, 3228, 4059, 4811, 5898, 6858, 8209, 9433, 11116, 12572, 14637, 16395, 18872, 21046, 23935, 26405, 29836, 32742, 36695, 40007, 44480, 48254, 53405, 57681, 63488, 68284, 74791, 80149, 87374

Offset: 1

George Stagg, Oct 30 2014

Arrange the prime numbers into a triangle, with 2 at the top, 3 and 5 in the second row, 7, 11 and 13 in the third row, and so on:
2
3 5
7 11 13
17 19 23 29
31 37 41 43 47
...
The n-th term in the sequence is then the sum of the numbers in the upward diagonal beginning on the n-th row of this triangle.

			a(1) = 2; a(2) = 3; a(3) = 7+5 = 12; a(4) = 17+11 = 28.

George Stagg, Table of n, a(n) for n = 1..1000

Cf. A007468, A079824.

A249490:=n->add(ithprime((n-m)*(n-m-1)/2+m+1), m=0..floor((n-1)/2)): seq(A249490(n), n=1..50); # Wesley Ivan Hurt, Nov 07 2014

Table[Sum[Prime[(n - m) (n - m - 1)/2 + m + 1], {m, 0, Floor[(n - 1)/2]}], {n, 50}] (* Wesley Ivan Hurt, Nov 07 2014 *)

a(n) = sum(m=0,(n-1)\2, prime((n-m)*(n-m-1)/2+m+1)); \\ Michel Marcus, Nov 04 2014

a(n) = Sum_{m=0..floor((n-1)/2)} prime((n-m)(n-m-1)/2+m+1).

A356288 Sum of numbers in n-th upward diagonal of triangle the sum of {1; 2,3; 4,5,6; 7,8,9,10; ...} and {1; 2,3; 3,4,5; 4,5,6,7; ...}.

Original entry on oeis.org

2, 4, 13, 20, 40, 55, 90, 116, 170, 210, 287, 344, 448, 525, 660, 760, 930, 1056, 1265, 1420, 1672, 1859, 2158, 2380, 2730, 2990, 3395, 3696, 4160, 4505, 5032, 5424, 6018, 6460, 7125, 7620, 8360, 8911, 9730, 10340, 11242, 11914, 12903, 13640, 14720, 15525, 16700

Offset: 1

Torlach Rush, Aug 02 2022

			   2 = A079824(1) + A093005(1) =  1 + 1.
   4 = A079824(2) + A093005(2) =  2 + 2.
  13 = A079824(3) + A093005(3) =  7 + 6.
  20 = A079824(4) + A093005(4) = 12 + 8.

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

Cf. A079824, A093005.

def a(n): return (n * ((n + n % 2) // 2)) + (15 + 25*n + 15*(n**2) + 14*(n**3) - 3*(((-1)**n))*(5 + n*(3 + n))) // 96

a(n) = (n * ceiling(n/2)) + ((15 + 25*n + 15*n^2 + 14*n^3 - 3*(((-1)^n))*(5 + n*(3 + n))) / 96).
a(n) = A079824(n) + A093005(n).
G.f.: x*(2 + 2*x + 3*x^2 + x^3 - x^4)/((1 - x)^4*(1 + x)^3). - Stefano Spezia, Aug 19 2022

cp's OEIS Frontend

A185787 Sum of first k numbers in column k of the natural number array A000027; by antidiagonals.

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A056536 Mapping from half-antidiagonal reading of the triangle (as used in A028297) to the column-by-column reading of the triangular tables.

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A185788 Sum of the first k-1 numbers in the k-th column of the natural number array A000027, by antidiagonals.

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A079823 Consider the triangle shown below; sequence contains the concatenation of numbers read at a 45-degree angle upwards with horizontal beginning with the first term of a row.

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A249490 a(n) = Sum_{m=0..floor((n-1)/2)} prime((n-m)(n-m-1)/2+m+1).

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A356288 Sum of numbers in n-th upward diagonal of triangle the sum of {1; 2,3; 4,5,6; 7,8,9,10; ...} and {1; 2,3; 3,4,5; 4,5,6,7; ...}.

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